###### Abstract

Weak-value amplification (WVA) is a metrological protocol that effectively
amplifies ultra-small physical effects, making it highly applicable in the
fields of quantum sensing and metrology. However, the amplification effect is
achieved through post-selection, which leads to a significant decrease in
signal intensity. Consequently, there is a heated debate regarding the trade-
off between the amplification effect and the success probability of post-
selection, questioning whether WVA surpasses conventional measurement (CM) in
terms of measurement precision. Extensive research indicates that the specific
theoretical assumptions and experimental conditions play crucial roles in
determining the respective advantages of WVA and CM. WVA provides new
perspectives for recognizing the important role of post-selection in precision
metrology. It demonstrates significant advantages in two aspects: (i) WVA
based on the phase space interaction provides feasible strategies to
practically achieve the Heisenberg-scaling precision using only classical
resources. (ii) WVA exhibits robustness against certain types of technical
noise and imperfections of detectors. Moreover, WVA allows for various
modifications to extend the applicable scope and enhance the metrological
performance in corresponding situations. Despite substantial progress in
recent years, the inherent connection between the advantages of WVA and its
unique features remains incompletely understood. In this paper, we
systematically review the recent advances in the WVA scheme, with a particular
focus on the ultimate precision of WVA under diverse conditions. Our objective
is to provide a comprehensive perspective on the benefits of WVA in precision
measurement and facilitate the realization of its full potential.

weak measurement, weak value, quantum metrology, Fisher information

###### Contents

  1. I 1\. Introduction
  2. II 2\. Basic principles of Weak-value Amplification
    1. II.1 2.1 Standard weak-value amplification
    2. II.2 2.2 Applications of weak-value amplification with photonic systems
    3. II.3 2.3 Generalization of weak-value amplification
  3. III 3\. Precision analysis on the weak-value amplification
    1. III.1 3.1 Quantum metrology theory
    2. III.2 3.2 Metrological Limits of Weak Value Amplification
  4. IV 4\. Weak-value amplification with Heisenberg-limited precision
    1. IV.1 4.1 Heisenberg-limited weak-value amplification in linear optical system
    2. IV.2 4.2 weak-value amplification with phase-space interaction
  5. V 5\. Advantages of weak-value amplification in the presence of practical imperfections
    1. V.1 5.1 Weak-value amplification against time-correlated noise
    2. V.2 5.2 Imaginary weak-value amplification against technical noise
    3. V.3 5.3 Weak-value amplification with imperfect detectors
  6. VI 6\. Various modifications of weak-value amplification schemes
    1. VI.1 6.1 Inverse weak-value amplification
    2. VI.2 6.2 Almost-balanced weak-value amplification
    3. VI.3 6.3 Biased weak-value amplification
    4. VI.4 6.4 Power-recycling weak value amplification
  7. VII 7\. Conclusion and Outlook

##  I 1\. Introduction

The precise measurement of ultra-small physical parameters has been
instrumental in advancing fundamental research Abbott  _et al._ (2016); Baker
_et al._ (2006); Bulatowicz  _et al._ (2013) and driving the revolution of
advanced technology Budker and Romalis (2007); Kornack  _et al._ (2005);
Budker and Romalis (2007). However, experimental imperfections unavoidably
diminish the practical sensitivity and precision of measurement protocols,
making it challenging to effectively extract weak signals. To enhance the
sensitivity, researchers have pursued two complementary approaches: designing
and manufacturing sophisticated equipment to isolate noise sources, and
developing noise-robust measurement protocols. Among the most notable noise-
robust measurement protocols, weak-value amplification (WVA) has demonstrated
remarkable success in detecting ultra-small physical effects and is
increasingly indispensable in the fields of quantum sensing and metrology.

The concept of weak value originates from the influential work of Aharonov,
Albert and Vaidman (AAV) in 1988 Aharonov  _et al._ (1988), wherein they
extended the framework of Von Neumann measurement (VNM) Mackey (2013). In the
VNM framework, the quantum measurement is described as a coupling process
between a quantum system (QS) and a meter state (MS) under an interaction
Hamiltonian. The Hamiltonian that describes this interaction typically takes
the form of $\hat{H}_{\text{int}}=g\hat{S}\otimes\hat{M}$, where $g$ is the
coupling strength, and $\hat{S}$ ($\hat{M}$) is the observable of the QS (MS).
After the interaction, the shift of the MS is then proportional to the
eigenvalue of the observable $\hat{S}$ multiplied by the coupling strength
$g$. Strong measurement is typically employed to accurately identify
measurement results, requiring the coupling strength being much larger than
the uncertainty of the MS. In this case, the QS inevitably collapses into the
corresponding eigenstate. The AAV extension encompasses two main aspects: (i)
weak measurement is performed by setting the coupling strength to be much
smaller than the uncertainty of the MS; (ii) The QS is post-selected after the
weak measurement with only the surviving MS being observed. In this scenario,
both the initial (pre-selected) and final (post-selected) states of the QS
jointly determine the average shifts of the post-selected MS, which can be
quantified by the weak value of the observable.

The pre- and post-selection is fundamentally connected to the two-state vector
formalism (TSVF), which provides a time-symmetric description of the QS. In
TSVF, the pre-selected state evolves forward from the past to the future while
the post-selected state evolves backward from the future to the past Aharonov
_et al._ (1964); Aharonov and Vaidman (1990, 1998); Muga  _et al._ (2007);
Reznik and Aharonov (1995). To reveal the properties of such time-symmetric
systems, weak measurement is performed to extract partial information about
the QS with minimal disturbance. Thus, weak values maintain the time-symmetric
formulation and can be obtained by measuring the post-selected meter states.
The measurement of weak values opens up new avenues for exploring fundamental
quantum phenomena, including Hardy’s paradox Aharonov  _et al._ (2002);
Lundeen and Steinberg (2009), the three-box paradox Resch  _et al._ (2004),
the Leggett-Garg inequality Williams and Jordan (2008); Dressel  _et al._
(2011), quantum contextuality Pusey (2014); Hofmann (2020), the observation of
the Bohm trajectory Kocsis  _et al._ (2011); Mahler  _et al._ (2016); Xiao
_et al._ (2017, 2019) and more Wiseman (2002); Solli  _et al._ (2004); Huang
_et al._ (2019a); Yu  _et al._ (2020); Cho  _et al._ (2019); Chen  _et al._
(2019a).

The anomalous weak values that lie outside the eigenvalue spectrum or possess
complex values demonstrate promising potential for the development of novel
quantum technologies. A comprehensive explanation of weak values from a
pragmatic standpoint can be found in Ref. Dressel  _et al._ (2014). As
generalized measurement outcomes, weak values can be used for direct
measurement of wavefunction Lundeen  _et al._ (2011), which also inspires
approaches for direct characterization of generalized quantum systems,
including various quantum states, quantum processes and quantum measurements
Lundeen and Bamber (2012); Salvail  _et al._ (2013); Malik  _et al._ (2014);
Denkmayr  _et al._ (2017); Calderaro  _et al._ (2018); Kim  _et al._ (2018);
Pan  _et al._ (2019); Xu  _et al._ (2021a); Chen  _et al._ (2021a); Xu  _et
al._ (2021b, 2024). When the pre- and post-selected states are nearly
orthogonal, the large weak value serves as an amplification factor for the
coupling strength, giving rise to the WVA scheme. On one hand, this
amplification effect helps overcome experimental imperfections and technical
noise, contributing to the remarkable achievements of WVA in measuring ultra-
small physical parameters. Owing to its advantages, weak value amplification
is extensively employed for highly sensitive sensing applications Li  _et al._
(2021); Qu  _et al._ (2020); Luo  _et al._ (2020a); Li  _et al._ (2018); Cao
_et al._ (2024). On the other hand, the amplified outcomes are counterbalanced
by a reduction in signal intensity due to post-selection, leading to a
potential confusion where WVA may be perceived as suboptimal compared to the
CM. The resolution of this contradiction necessitates a detailed evaluation of
the ultimate precision achieved by both WVA and CM under various conditions.

This review employs universal metrics, such as the Fisher information (FI) and
the signal-to-noise ratio (SNR) in quantum metrology to evaluate both the
performance of WVA and CM. The analysis reveals the distribution of
metrological information before and after the post-selection. The optimal WVA
allows to achieve almost the same precision as the CM by detecting only a
small fraction of successfully post-selected signals. Moreover, the analytical
results provide WVA with possible approaches for surpassing the classical
limit with only classical resources. The comparison between WVA and CM
enhances the understanding of the metrological advantages of WVA under certain
conditions. The analytical procedure described here is generally applicable to
emerging modified WVA schemes and other metrological protocols that involve
post-selection. We hope that this review provides a clear perspective on how
the intrinsic properties of WVA contribute to its metrological advantages and
inspires the development of noise-robust measurement schemes.

The structure of this paper is as follows: section 2 provides an introduction
to the theoretical framework of the standard WVA and its generalization. It
also presents a comprehensive review of the typical applications of WVA based
on different physical implementations. In section 3, we provide a brief
overview of the basic framework of quantum metrology, which includes the
universal metrics and key precision limits. These metrics are subsequently
employed to evaluate the ultimate precision of WVA. Section 4 summarizes
various WVA schemes that can achieve Heisenberg-scaling precision, with or
without the utilization of quantum resources. Section 5 examines the ultimate
precision of WVA in the presence of experimental imperfections, illustrating
the advantages of WVA over CM under specific conditions. In section 6, we
introduce several modified WVA schemes and highlight their distinct advantages
compared to the standard WVA. Finally, the last section concludes our
discussion on the advancements made in WVA and proposes potential avenues for
future research.

##  II 2\. Basic principles of Weak-value Amplification

###  II.1 2.1 Standard weak-value amplification

The typical procedure of WVA is depicted in Fig. 1 (a). A two-level QS is
first pre-selected by
$|\psi_{i}\rangle=\cos{(\theta_{i}/2)}|0\rangle+\sin(\theta_{i}/2)e^{i\phi_{i}}|1\rangle$.
The initial MS is prepared as a Gaussian superposition state given by

$\displaystyle|\Phi\rangle$ $\displaystyle=$ $\displaystyle\int
dq\frac{1}{(2\pi\sigma^{2})^{1/4}}\exp[-\frac{q^{2}}{4\sigma^{2}}]|q\rangle{}$
(1) $\displaystyle=$ $\displaystyle\int
dp\Big{(}\frac{2\sigma^{2}}{\pi}\Big{)}^{\frac{1}{4}}\exp(-\sigma^{2}p^{2})|p\rangle,$

where $q$ and $p$ are conjugate variables (e.g., momentum and position,
respectively). Correspondingly, $|q\rangle$ ($|p\rangle$) denotes the
eigenstate of the observable $\hat{Q}$ ($\hat{P}$) of the MS. The QS interacts
with the MS through the Hamiltonian
$\hat{H}=g\delta(t-t_{0})\hat{A}\otimes\hat{P}$, where $\hat{A}$ represents an
observable of the QS and $g$ is the coupling strength that needs to be
estimated. This interaction leads to the entangled joint state

$|\Psi_{jt}\rangle=\exp(-i\int\hat{H}dt)|\psi_{i}\rangle\otimes|\Phi\rangle.$
(2)

Next, the joint state is post-selected by
$|\psi_{f}\rangle=\cos{(\theta_{f}/2)}|0\rangle+\sin(\theta_{f}/2)e^{i\phi_{f}}|1\rangle$
of the QS. The surviving MS after post-selection is given by
$|\Phi_{f}\rangle=\langle\psi_{f}|\Psi_{jt}\rangle/\sqrt{p_{f}}$ with the
success probability $p_{f}=|\langle\psi_{f}|\Psi_{jt}\rangle|^{2}$. In the
weak-measurement regime ($g\ll\sigma$), the final MS can be approximated to

$\displaystyle|\Phi_{f}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p_{f}}}\langle\psi_{f}|\exp(-ig\hat{A}\hat{P})|\psi_{i}\rangle|\Phi\rangle{}$
(3) $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{p_{f}}}\langle\psi_{f}|(1-ig\hat{A}\hat{P})|\psi_{i}\rangle|\Phi\rangle{}$
$\displaystyle=$
$\displaystyle\frac{\langle\psi_{f}|\psi_{i}\rangle}{\sqrt{p_{f}}}(1-ig\langle\hat{A}\rangle_{w}\hat{P})|\Phi\rangle,$

where the weak value is defined as

$\langle\hat{A}\rangle_{w}=\frac{\langle\psi_{f}|\hat{A}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}.$
(4)

If the additional AAV approximate condition

$\text{max}_{n}g\frac{|\langle\psi_{f}|\hat{A}^{n}|\psi_{i}\rangle|^{1/n}}{|\langle\psi_{f}|\psi_{i}\rangle|}\ll\sigma\quad\text{for}\quad
n=1,2,...,$ (5)

is satisfied, then the final MS can be further simplified as

$|\Phi_{f}\rangle\approx\exp(-ig\langle\hat{A}\rangle_{w}\hat{P})|\Phi\rangle.$
(6)

When the pre- and post-selected states approach being orthogonal, indicated by
$|\langle\psi_{f}|\psi_{i}\rangle|\rightarrow 0$, the modulus of the weak
value $|\langle\hat{A}\rangle_{w}|$ becomes significantly larger than the
maximum eigenvalue of the observable $\hat{A}$. Moreover, weak value can also
take complex numbers. In both of these cases, the weak value is referred to as
anomalous. Both the real and imaginary parts of the large weak value can be
obtained by the average value of the corresponding observables ($\hat{Q}$ or
$\hat{P}$) of the MS Jozsa (2007):

$\displaystyle\langle\hat{Q}\rangle_{f}$ $\displaystyle=$ $\displaystyle
g\text{Re}\langle\hat{A}\rangle_{w},{}$
$\displaystyle\langle\hat{P}\rangle_{f}$ $\displaystyle=$
$\displaystyle\frac{g}{2\sigma^{2}}\text{Im}\langle\hat{A}\rangle_{w},$ (7)

where $\langle\cdot\rangle_{f}$ denotes the mean value of the observable in
$|\Phi_{f}\rangle$. This amplification form represents the initial and widely
adopted form of WVA. Therefore, we refer to it as the standard WVA. Since the
observables $\hat{Q}$ and $\hat{P}$ are incompatible, the measurement of
either observable may not extract the complete information of a complex weak
value. Researchers typically adopt a purely real or imaginary weak value in
WVA. For instance, by setting the parameters of the pre- and post-selected
states as $\theta_{i}=\pi/2-\epsilon$, $\theta_{f}=-\pi/2$ and
$\phi_{i}=\phi_{f}=0$ ($\theta_{i}=\pi/2$, $\theta_{f}=-\pi/2$ and
$\phi_{i}-\phi_{f}=\phi$) with a small $\epsilon$ ($\phi$), a large real
(imaginary) weak value $2/\epsilon$ ($-2i/\phi$) can be obtained. However,
this amplification comes at the cost of a corresponding decrease in the
success probability of post-selection, approximately given by
$p_{f}\approx\epsilon^{2}$ and $p_{f}\approx\epsilon^{2}$ for real and
imaginary weak value, respectively.

Figure 1: (a) The basic procedure of the standard weak-value amplification
scheme. Here, the observable of the QS is set to be
$\hat{A}=\hat{\sigma}_{z}=|0\rangle\langle 0|-|1\rangle\langle 1|$. (b)-(e)
The Wigner function and the marginal distribution of the initial meter state
and the post-selected meter states with different weak values. Here, we set
the parameter of the initial MS $\sigma=\sqrt{2}/2$ and the corresponding
parameters of the QS $\langle\hat{A}\rangle_{w}=200$: $\theta_{i}=\pi/2$,
$\theta_{f}=-\pi/2+\epsilon$, $\phi_{f}-\phi_{i}=0$;
$\langle\hat{A}\rangle_{w}=-200i$: $\theta_{i}=\pi/2$,
$\theta_{f}=-\pi/2$,$\phi_{f}-\phi_{i}=\phi$;
$\langle\hat{A}\rangle_{w}=100-100i$: $\theta_{i}=\pi/2$,
$\theta_{f}=-\pi/2+\epsilon$,$\phi_{f}-\phi_{i}=\phi$ with
$\epsilon=\phi=0.01$ and $g=\sigma/400$. The yellow lines refer to the optimal
observables $\hat{S}_{\theta}$ in different situations.

In some cases where a complex weak value is necessary, researchers have also
optimized the observables of the MS to achieve the maximal amplification of
the parameter $g$. Given that a complex weak value induces shifts of the MS in
both $q$ and $p$, recent works Knee and Briggs (2018); Pan  _et al._ (2020)
have utilized the Wigner function Case (2008) to describe the final MS. This
representation can be expressed as:

$W_{\text{WVA}}(q,p)=p_{f}W_{m}(q-g\text{Re}\langle\hat{A}\rangle_{w},p-\frac{g}{2\sigma^{2}}\text{Im}\langle\hat{A}\rangle_{w})$
(8)

with
$W_{m}(q,p)=\exp\big{[}-q^{2}/(2\sigma^{2})\big{]}\exp(-2\sigma^{2}p^{2})/\pi$
denoting the Wigner function of the initial MS $|\Phi\rangle$.

For a general observable of the MS
$\hat{S}_{\theta}=\hat{Q}\cos\theta+\hat{P}\sin\theta$, the integration of its
conjugate observable $\hat{T}_{\theta}=-\hat{Q}\sin\theta+\hat{P}\cos\theta$
over the distribution $W_{\text{WVA}}(q,p)$ leads to the Gaussian marginal
probability distribution:

$P_{\hat{S}_{\theta}}(s_{\theta})=\frac{1}{\sqrt{2\pi\sigma_{\theta}^{2}}}\exp\Big{[}-\frac{[s_{\theta}-\langle\hat{S}_{\theta}\rangle_{f}]^{2}}{2\sigma_{\theta}^{2}}\Big{]},$
(9)

with the mean value $\langle\hat{S}_{\theta}\rangle_{f}$ and variance
$\sigma^{2}_{\theta}=\sigma^{2}(2\sigma^{2}\cos^{2}\theta+\frac{1}{2\sigma^{2}}\sin^{2}\theta)$.
By expressing the weak value in polar form as
$\langle\hat{A}\rangle_{w}=|\langle\hat{A}\rangle_{w}|\cos\phi_{wv}+i|\langle\hat{A}\rangle_{w}|\sin\phi_{wv}$,
the average shift of the final MS can be simplified as

$\langle\hat{S}_{\theta}\rangle_{f}=g|\langle\hat{A}\rangle_{w}|\cos\phi_{wv}\cos\theta+\frac{g}{2\sigma^{2}}|\langle\hat{A}\rangle_{w}|\sin\phi_{wv}\sin\theta.$
(10)

Thus, the maximum shift of the MS
$g|\langle\hat{A}\rangle_{w}|\sqrt{\cos^{2}\phi_{wv}+\sin^{2}\phi_{wv}/(4\sigma^{4})}$
can be attained by satisfying the condition
$2\sigma^{2}\tan\theta=\tan\phi_{wv}$. In Fig. 1 (b)-(e), the Wigner function
of the initial MS $|\Phi\rangle$ and the final MS $|\Phi_{f}\rangle$ with
various weak values are visually presented to demonstrate the optimal
observables $\hat{S}_{\theta}$.

###  II.2 2.2 Applications of weak-value amplification with photonic systems

The applications of WVA primarily focus on observing ultra-small physical
effects and sensing weak signals. The photonic system holds a crucial position
in quantum metrology and sensing due to its good coherence, as well as its
ease of manipulation and detection. Consequently, the practical applications
of WVA are typically implemented by the interaction between two different
degrees of freedom (DOF) of photons. The discrete variable DOF, such as
polarization and path are usually referred to as the QS, while the continuous
variable DOF, including the spatial, the spectrum and the temporal modes are
denoted as the MS. The ultra-small coupling strength between the QS and the MS
can be amplified using either the real or imaginary weak value. Fig. 2
illustrates the most representative applications of WVA with photonic systems.

In 1991, N. W. M. Ritchie et al. first implemented the measurement of a real
weak value in an optical system Ritchie  _et al._ (1991) using the
experimental setup depicted in Fig. 2 (a). The polarization and transverse
spatial mode of photons are used as the QS and the MS, respectively. The
transversal displacement of the light beam induced by the birefringence is
amplified up to 20 times larger than its actual displacement, demonstrating
the potential of weak value to amplify the small physical effects. Such
interaction has been widely used to investigate the birefringence effects Rhee
_et al._ (2013); Qiu  _et al._ (2016) as well as the Goos-Hänchen shift Das
and Pradhan (2021); Zhou  _et al._ (2021).

In 2008, Hosten and Kwiat utilized WVA scheme to amplify the spin Hall effect
of light by four orders of magnitude, achieving a sensitivity of 1 angstrom
without isolating the air disturbance and mechanical vibration Hosten and
Kwiat (2008). The experimental setup is illustrated in Fig. 2 (b). The spin,
represented by the left- or right-circularly polarization, and the transverse
spatial mode of photons serves as the QS and the MS, respectively. Through
observing the momentum of the post-selected photons, the transversal
displacement of light caused by the SHEL is amplified by the imaginary weak
value. This experiment marked the first practical application of WVA in
amplifying small physical effects. Subsequently, WVA has become a popular
approach to study the spin Hall effect of light, as evidenced by numerous
studies Chen  _et al._ (2015); Zhou  _et al._ (2012); Qin  _et al._ (2020);
Pfeifer and Fischer (2011); Qin  _et al._ (2019); Wang  _et al._ (2013);
Gorodetski  _et al._ (2012); Neugebauer  _et al._ (2018).

Figure 2: Applications of weak-value amplification. (a) First realization of
the measurement of the weak value Ritchie  _et al._ (1991). (b) The
observation of the spin Hall effect of light Hosten and Kwiat (2008). (c)
Ultrasensitive measurement of the beam deflection Dixon  _et al._ (2009). (d)
Phase estimation using a white light source Brunner and Simon (2010). (e)
Measurement of the longitude velocities Viza  _et al._ (2013). (f) achieving
incompatible quantum limits on multiparameter estimation using Hermite-
Gaussian states Xia  _et al._ (2023). (g) Amplification of angular rotations
Magaña-Loaiza  _et al._ (2014). (h) Amplification of the non-linear effect of
a single photon Hallaji  _et al._ (2017).

In 2009, Dixon et al. applied the WVA protocol to sensing the ultra-small tilt
of a mirror, achieving a sensitivity of 400-frad sensitivity Dixon  _et al._
(2009); Howell  _et al._ (2010) with the experimental setup depicted in Fig. 2
(c). The which-path information, clockwise or anticlockwise, inside the Sagnac
interferometer and the transverse spatial distribution of photons are
considered as the QS and the MS, respectively. A small tilt of the mirror
driven by the Piezo couples the QS and the MS. The post-selection is
implemented by monitoring the dark port of the interferometer. The imaginary
weak value maps the ultra-small tilt angle of the mirror to the shift of the
transverse beam position. This work extended the scope of the applications of
WVA from studying physical effects to the sensing of practical parameters. In
2010, Starling et. al. measured the beam deflection induced by a change in
optical frequency, achieving a sensitivity of $129\pm
7\text{kHz}/\sqrt{\text{Hz}}$ with 2 mW continuous wave Starling  _et al._
(2010a). In 2011, a quasi-autocollimator was designed by Turner et al., which
attained the sensitivity better than 10 $\text{prad}/\sqrt{\text{Hz}}$ between
10 and 200 Hz Turner  _et al._ (2011). Hogan et al. incorporated an optical
lever into the WVA protocol, substantially reducing the noise floor down to
1.3 $\text{prad}/\sqrt{\text{Hz}}$ at 2.4 kHz Hogan  _et al._ (2011). In 2012,
Egan et al. reported a temperature sensor based on WVA by implanting a prism
filled with thermo-optic fluid into the interferometer, achieving the
thermostat sensitivity to 0.2 mK at room temperature Egan and Stone (2012).
The above-mentioned WVA schemes mainly focus on the phase modulation of the MS
during the unitary interaction. The optical loss that modulates the amplitude
of the MS also carries useful information, corresponding to the nonunitary
process. Liu et al. theoretically extend WVA to the nonunitary regime Liu  _et
al._ (2019). In 2022, Li et al. experimentally achieved the efficiency of the
nonunitary WVA nearly ninefold that of unitary WVA by observing the quadratic
relation between the meter shift and the norm of the weak value Li  _et al._
(2022).

In 2010, Brunner et. al. proposed a WVA protocol for measuring the longitude
phase shift of photons. This approach, combining the imaginary weak value and
frequency-domain measurement, was shown to outperform the CM by several orders
of magnitude Brunner and Simon (2010). In 2013, Xu et al. achieved high-
precision phase estimation using a white light source Xu  _et al._ (2013), as
illustrated in Fig. 2 (d). The polarization and frequency spectrum of photons
are denoted as the QS and the MS, respectively. The wide spectrum distribution
of the white light source significantly contributes to the amplification
effect of the longitude phase. The experimental results also demonstrate the
robustness of phase estimation based on imaginary WVA against chromatic
dispersion. This configuration of the QS and the MS is commonly employed in
measuring relative time delay caused by temperature shifts Salazar-Serrano
_et al._ (2015), rotatory dispersion effects Zhou  _et al._ (2020), glucose
concentration Li  _et al._ (2016) and other effects Fang  _et al._ (2016);
Salazar-Serrano  _et al._ (2014a, b); Li  _et al._ (2021); Zanetto  _et al._
(2021); Huang  _et al._ (2021a); Wang  _et al._ (2021); Zhong  _et al._
(2021); Liu  _et al._ (2021).

In 2013, Viza et al. demonstrated an imaginary WVA scheme capable of measuring
the Doppler frequency shift produced by a moving mirror in the time domain
with the experimental setup shown in Fig. 2 (e) Viza  _et al._ (2013). The
which-path information in a Michelson Interferometer and the non-Fourier
limited Gaussian temporal pulse of photons generated by the acousto-optic
modulator are referred to as the QS and the MS, respectively. For the MS, the
temporal length of the pulse can be much larger than the coherence length of
the laser. The reported sensitivity of the velocity measurement is down to 400
$\text{fs}/\text{s}$. Recent studies have improved the sensitivity and SNR of
the velocity measurement using the Vernier-effect principal Huang  _et al._
(2021b). The time-domain measurement of the imaginary WVA can be also applied
to measure ultra-small parameters such as group delay Mirhosseini  _et al._
(2016) and angular velocity Fang  _et al._ (2021), and demonstrate interesting
physical effects like fast-light effect Mirhosseini  _et al._ (2016). The
applications related to the Fig. 2 (d) and (e) indicate that the incoherent MS
can contribute to the amplification of the coupling strength when combined
with the imaginary weak value.

Apart from the conventional Gaussian-shaped meter state, the high-order
Hermite-Gaussian and Laguerre-Gaussian modes provide WVA with significant
potential for measuring complex signals. In Fig. 2 (f), Xia et al. proposed a
multi-parameter WVA scheme using $n$-order Hermite-Gaussian states as the MS
Xia  _et al._ (2020, 2023). This approach improves the precision by a factor
of $2n+1$ compared to the Gaussian MS. Moreover, it enables simultaneous
estimation of spatial displacement and angular tilt of light, achieving the
precisions up to 1.45 nm and 4.08 nrad, respectively. In Fig. 2 (g),
Maga$\tilde{\text{n}}$a-Loaiza et al. demonstrated WVA in the azimuthal DOF
Magaña-Loaiza  _et al._ (2014), estimating the angular rotation by utilizing
the polarization DOF and the spatial mode with orbital angular momentum of
photons as the QS and the MS, respectively. The rotation of a Dove prism
inside the Sagnac interferometer that implements the spin-orbit coupling is
amplified by about two orders of magnitude.

In most applications of WVA as listed above, the coupling between the QS and
the MS is typically achieved through the interaction between different DOFs of
single photons. This scenario of WVA can be explained using classical
electromagnetic theory, which has raised a debate regarding the quantum nature
of weak values. To address this argument, researchers have attempted to
measure weak values without resort to classical interactions. For example, the
observation of anomalous weak values has been demonstrated in massive-particle
systems Sponar  _et al._ (2015). In 2005, Pryde et al. successfully measured
weak values of photons based on nonclassical interaction, e.g., Hong-Ou-Mandle
interference between photons. In 2013, Feizpour et. al. proposed the
theoretical WVA protocol to amplify the optical non-linearities at the single-
photon level Feizpour  _et al._ (2011). In 2017, Hallaji et al. experimentally
achieved an eightfold amplification of the non-linear phase shift caused by a
single photon Hallaji  _et al._ (2017) with the experimental setup depicted in
Fig. 2 (h). The amplification of the nonlinear interaction strength has also
inspired practical Heisenberg-scaling metrology without the need for quantum
resources Zhang  _et al._ (2015); Chen  _et al._ (2018a, b). In order to
clarify the role of statistical averaging in generating anomalous weak values,
Rebufello et al. designed a single-click experiment to obtain the anomalous
weak value with the amplified outcomes largely surpassing the uncertainty of
the MS Rebufello  _et al._ (2021). This finding enriches the meaning of weak
values beyond a statistical concept of conditional expectation value, thus
paving the way for the robust WVA applications.

###  II.3 2.3 Generalization of weak-value amplification

In the standard WVA, whether the shifts of post-selected MS are proportional
to weak values relies on the validity of two crucial approximate conditions,
namely, the weak-measurement approximation ($g\ll\sigma$) and the AAV
approximation condition presented in Eq. (5). The former provides the
possibility of generating anomalous weak values that surpass the eigenvalue
spectrum of the observable. As a contrast, the latter determines that the
amplified shifts in the final MS cannot be arbitrarily large for a fixed
coupling strength. The generalization of the standard WVA relies on rigorously
deriving the shifts of the post-selected MS without approximations. In 2011,
Koike et. al. conducted a study on the practical limits of WVA by considering
all orders of the coupling strength $g$. They derived the optimal overlap of
the pre- and post-selected states to achieve the maximum shifts of the post-
selected MS on the Sagnac interferometer Koike and Tanaka (2011). Zhu et al.
derived the rigorous expressions for the shifts of the post-selected MS in WVA
without restricting the coupling strength. The maximum shifts of MS in WVA are
also obtained for a qubit system Zhu  _et al._ (2011). Wu and pang et al.
focused on the WVA protocol with asymptotically or exactly orthogonal pre- and
post-selected states Pang  _et al._ (2012); Wu and Li (2011). A general
rigorous framework for the weak measurement beyond the AAV approximation
condition was established. Depending on the values of the coupling strength
$g$, the standard derivation of the MS $\sigma$ and the weak value
$\langle\hat{A}\rangle_{w}$, the actual shifts of the post-selected MS can be
classified into three regions: (i) strong-measurement region ($g\gg\sigma$);
(ii) WVA region ($g<g|\langle\hat{A}\rangle_{w}|\ll\sigma$); (iii) inverse WVA
region ($g\ll\sigma\ll g|\langle\hat{A}\rangle_{w}|$). In the following, we
adopt the derivations of Wu et al. Wu and Li (2011) to clarify the
generalization of the WVA scheme.

With a general pre-selected state $\rho_{s}$ and a measurement operator
$\hat{\Pi}_{f}$ implementing the post-selection, Wu et al. defined the high-
order weak value of the observable $\hat{A}$ as

$\langle\hat{A}\rangle^{m,l}_{w}=\frac{\text{Tr}(\hat{\Pi}_{f}\hat{A}^{m}\rho_{s}\hat{A}^{l})}{\text{Tr}(\hat{\Pi}_{f}\rho_{s})},$
(11)

By taking up to the second-order approximation about the coupling strength
$g$, the shifts of the final MS can be expressed with the high-order weak
values as follows:

$\displaystyle\langle\hat{Q}\rangle_{f}$ $\displaystyle\approx$
$\displaystyle\frac{g\text{Re}\langle\hat{A}\rangle_{w}+g\text{Im}\langle\hat{A}\rangle_{w}\langle\\{\hat{Q},\hat{P}\\}\rangle}{1+g^{2}\text{Var}(\hat{P})(\langle\hat{A}\rangle_{w}^{1,1}-\text{Re}\langle\hat{A}^{2}\rangle_{w})}=\frac{4g\text{Re}\langle\hat{A}\rangle_{w}\sigma^{2}}{4\sigma^{2}+g^{2}(|\langle\hat{A}\rangle_{w}|^{2}-1)}{}$
$\displaystyle\langle\hat{P}\rangle_{f}$ $\displaystyle\approx$
$\displaystyle\frac{2g\text{Im}\langle\hat{A}\rangle_{w}\text{Var}(\hat{P})}{1+g^{2}\text{Var}(\hat{P})(\langle\hat{A}\rangle_{w}^{1,1}-\text{Re}\langle\hat{A}^{2}\rangle_{w})}=\frac{2g\text{Im}\langle\hat{A}\rangle_{w}}{4\sigma^{2}+g^{2}(|\langle\hat{A}\rangle_{w}|^{2}-1)}.$
(12)

Here, the simplified results are obtained by considering the observable
$\hat{A}=\hat{\sigma}_{z}=|0\rangle\langle 0|-|1\rangle\langle 1|$ and the
Gaussian MS in Eq. (1). When the coupling strength $g$ and weak value
$\langle\hat{A}\rangle_{w}$ satisfy both the weak-measurement and AAV
approximation conditions, Eq. (12) degrades to Eq. (7), corresponding to the
standard WVA. However, increasing the modular of weak value continuously can
lead to a violation of the AAV approximation condition. This results in an
initial increase followed by a rapid decrease in the shift of the post-
selected MS. During this process, the measurement scheme transitions from the
WVA region to the inverse WVA region. The transition points correspond to the
maximum values of $\langle\hat{Q}\rangle_{f}$ and $\langle\hat{P}\rangle_{f}$
in the real and imaginary WVA schemes, respectively. According to Eq. (12),
these maximum values are derived as $\max\langle\hat{Q}\rangle_{f}=\sigma$ and
$\max\langle\hat{P}\rangle_{f}=1/(2\sigma)$, which aligns well with the
rigorous derivations by Zhu et. al. Zhu  _et al._ (2011). When
$|\langle\hat{A}\rangle_{w}|$ increases such that
$g|\langle\hat{A}\rangle_{w}|\gg\sigma$, Eq. (12) can be approximated as
$\langle\hat{Q}\rangle_{f}\approx
4\sigma^{2}/(g\text{Re}\langle\hat{A}\rangle_{w})$ and
$\langle\hat{P}\rangle_{f}\approx 2/(g\text{Im}\langle\hat{A}\rangle_{w})$ for
the real and imaginary weak values, respectively. In such cases, the shifts
$\langle\hat{Q}\rangle_{f}$ and $\langle\hat{P}\rangle_{f}$ are inversely
proportional to the weak value, and the measurement protocol operates in the
inverse WVA region. In this region, the weak value and the associated
parameters (e.g., $\theta_{i}$, $\theta_{f}$) in the QS can be effectively
amplified by a factor of $1/g$, which will be discussed further in section 6.

The weak-to-strong measurement transition of the pre- and post-selected QS was
experimentally examined in a trap-ion system in 2020 Pan  _et al._ (2020). We
adopt their experimental parameters to demonstrate the actual shifts of the
final MS within all three regions and two critical points between them. In
their work, the QS utilized the internal electronic state of a single trapped
${}^{40}\text{Ca}^{+}$ ion, while the axial vibrational motion of the ion
serves as the MS. The pre-selected state of the QS, denoted as
$|\psi_{i}\rangle=|1\rangle$, interacts with the Gaussian MS $|\Phi\rangle$
under the Hamiltonian $\hat{H}=\gamma_{0}\hat{\sigma}_{x}\otimes\hat{P}$,
where $\hat{\sigma}_{x}=|1\rangle\langle 0|+|0\rangle\langle 1|$ and $\hat{P}$
represent the observables of the QS and the MS, respectively. The interaction
time $t$ can be adjusted to control the coupling strength $g=\gamma_{0}t$. The
joint state is then post-selected by
$|\psi_{f}\rangle=\cos\theta|0\rangle-\sin\theta|1\rangle$, resulting in a
real weak value $\langle\hat{\sigma}_{x}\rangle_{w}=-\cot\theta$.
Consequently, the shift of the post-selected MS in $q$ space is given by

$\langle\hat{Q}\rangle_{f}=-\frac{\gamma_{0}t\sin
2\theta}{1-\cos(2\theta)e^{-\Gamma^{2}/2}},$ (13)

where $\Gamma=\gamma_{0}t/\sigma$ characterizes the relative coupling
strength. The average shift $\langle\hat{Q}\rangle_{f}$ during the strong-to-
weak measurement transition are illustrated in Fig. 3 (a) with different
coupling strength $\Gamma$ and the parameter $\theta$ in the post-selection.

Figure 3: (a) The normalized shifts of the final meter state
$\langle\hat{Q}\rangle_{f}/(\gamma_{0}t)$ are plotted with different post-
selection angle $\theta$ during the weak-to-strong measurement transition in a
trapped-ion system. In (b) and (c), the probability density of Gaussian
initial meter states (the dotted black lines), the post-selected meter states
in standard (the dashed red lines) and orthogonal (the solid green lines) WVA
are compared in $q$ and $p$ representations, respectively. In (b) and (c),
both the weak value $\langle\hat{A}\rangle_{w}$ and the orthogonal weak value
$\langle\hat{A}\rangle_{ow}^{1,0}$ are set to $0.2+0.1i$.

When the coupling strength is strong ($\Gamma\gg 1$), the shift of the final
MS can be approximated as
$\langle\hat{Q}\rangle_{f}|_{\Gamma\rightarrow\inf}=-\gamma_{0}t\sin
2\theta=\gamma_{0}t\langle\psi_{f}|\hat{\sigma}_{x}|\psi_{f}\rangle$, which is
directly proportional to the expectation value of the observable
$\hat{\sigma}_{x}$ in the post-selected state $|\psi_{f}\rangle$. In this
strong measurement region, the ratio of $\langle\hat{Q}\rangle_{f}$ to
$\gamma_{0}t$ remains within the eigenvalue spectrum of the observable. As the
coupling strength weakens to satisfy the AAV condition
($\Gamma|\langle\hat{\sigma}_{x}\rangle_{w}|\ll 1$), the ratio
$\langle\hat{Q}\rangle_{f}/(\gamma_{0}t)_{|\Gamma\rightarrow
0}\approx-\cot\theta\approx\langle\hat{\sigma}_{x}\rangle_{w}$ approaches the
weak value. Furthermore, as the parameter $\theta$ decreases, the AAV
approximate condition is violated due to a large weak value. The shift
$\langle\hat{Q}\rangle_{f}$ gradually deviates from the standard WVA. This
trend suggests an extreme value of $\langle\hat{Q}\rangle_{f}$, which can be
achieved by $\theta=\text{arccos}(e^{-\Gamma^{2}/2})/2$. After reaching this
extreme value, the shift of the MS enters the inverse WVA region, where
$\langle\hat{Q}\rangle_{f}\approx-4\sigma^{2}\cot\theta/g$.

Besides the previously mentioned regions, there exists a special case where
the pre- and post-selected states are strictly orthogonal, i.e.,
$\langle\psi_{f}|\psi_{i}\rangle=0$. In this scenario, Wu et.al. introduced
the concept of orthogonal weak value to describe the shifts of the post-
selected MS Wu and Li (2011). With a general pre-selected state $\rho_{i}$ and
post-selected operator $\hat{\Pi}_{f}$ satisfying
$\text{Tr}(\rho_{i}\hat{\Pi}_{f})=0$, the orthogonal weak value of the
observable $\hat{A}$ is defined as

$\langle\hat{A}\rangle_{ow}^{m,l}=\frac{\text{Tr}(\hat{\Pi}_{f}\hat{A}^{m+1}\rho_{i}\hat{A}^{l+1})}{(m+1)(l+1)\text{Tr}(\hat{\Pi}_{f}\hat{A}\rho_{i}\hat{A})}.$
(14)

When considering a Gaussian initial MS $|\Phi\rangle$, the shifts of the post-
selected MS in the orthogonal weak measurement scenario can be expressed
similarly to the standard WVA, given by
$\langle\hat{Q}\rangle_{f}=g\text{Re}\langle\hat{A}\rangle_{ow}^{1,0}$ and
$\langle\hat{P}\rangle_{f}=3g\text{Im}\langle\hat{A}\rangle_{ow}^{1,0}/(2\sigma^{2})$.
The probability distribution of the post-selected MS in both the standard WVA
and the orthogonal weak measurement can be compared. As shown in Fig. 3 (b)
and (c), the average shifts $\langle\hat{Q}\rangle_{f}$ and
$\langle\hat{P}\rangle_{f}$ in orthogonal weak measurement result from the
displacement and the relative intensity variation of bimodal-shape
distribution, respectively.

Apart from the average shift of the post-selected MS, Parks et.al. derived the
variance of an arbitrary observable $\hat{M}$ under the post-selected MS
$|\Phi_{f}\rangle$ with a complex weak value. They found that the variance of
the observable $\hat{P}$ and $\hat{Q}$ are only related to the imaginary part
of weak value and third central moment of $\hat{P}$ relative to the initial MS
Parks and Gray (2011). The exact formulas that include all-order effects of
the coupling strength with a typical observables $\hat{A}$ (satisfying
$\hat{A}^{2}=\hat{I}$) are provided to evaluate the shifts and the probability
distributions of post-selected MS Nakamura  _et al._ (2012). Additionally,
Lorenzo et. al. derived the characteristic function for the moments of the
post-selected MS to fully describe the statistics of WVA protocol Di Lorenzo
(2012). For a general derivation of the Von Neumann measurement in the
framework of pre- and post-selection, we suggest to refer to the review paper
Ref. Kofman  _et al._ (2012) for further reading.

Though the Gaussian superposition state $|\Phi\rangle$ is most prevalent in
both the theoretical investigations and experimental applications, the WVA
scheme can also be implemented with more general types of MS. In 2004,
Johansen has provided the derivation of weak value with the arbitrary MS
including the mixed states Johansen (2004). Moreover, the wavefunction of the
MS can be optimized to acquire the maximal amplification factor in WVA for a
given weak value, which removes the maximal shifts restricted by the Gaussian
MS Susa  _et al._ (2012). Specifically, the theoretical schemes were developed
to realize WVA with the MS such as a qubit Wu and Mølmer (2009), orbital-
angular-momentum states Puentes  _et al._ (2012), Hermite-Gaussian states,
Laguerre-Gaussian states Turek  _et al._ (2015), thermal states, mixed states
Cho  _et al._ (2010) and etc Luo  _et al._ (2020b). The concept of modular
values are proposed to describe the coupling between the qubit MS and the QS
with the arbitrary strength Kedem and Vaidman (2010). In order to adapt the
requirements of WVA in more general cases, e.g., in the presence of the
decoherence, Shikano et. al. introduced a $\hat{W}$ operator and the
associated quantum operations on it to formally describe the weak value
Shikano and Hosoya (2009). Recent work has also shown that the weak value can
be operationally formulated as the response of the pre- and post-selected QS
without resort to the MS Ogawa  _et al._ (2020).

##  III 3\. Precision analysis on the weak-value amplification

As WVA scheme plays an increasingly crucial role in practical precision
metrology, it is imperative to develop generic approaches to evaluate its
precision. This research not only improves our understanding of the potential
advantages of WVA but also provides guidance for further enhancing its
performance. In this section, we first review the quantum metrology theory for
single-parameter estimation and introduce the Fisher information (FI) metric
as a means to determine the ultimate precision of general measurement
protocols. Moreover, we compare the widely used signal-to-noise ratio (SNR)
metric in the field of sensing with FI, clarifying their consistency and
differences in evaluating measurement protocols. Subsequently, we apply the FI
metric to analyze the ultimate precision of WVA and discuss the attainability
of its optimal precision.

###  III.1 3.1 Quantum metrology theory

In quantum metrology, the general procedure to estimate an unknown parameter
$g$ is presented in Fig. 4 Giovannetti  _et al._ (2011); Degen  _et al._
(2017); Polino  _et al._ (2020). Initially, a quantum state $\rho_{0}$ is
prepared, which evolves under the parameter-encoding process $\hat{U}_{g}$ to
obtain the final state $\rho_{g}=\hat{U}_{g}\rho_{0}\hat{U}_{g}^{\dagger}$.
Subsequently, a general quantum measurement, known as a positive-operator-
valued measure (POVM) $\\{\hat{\Pi}_{x}\\}$ is performed on the final state to
transfer the information about the parameter $g$ from $\rho_{g}$ to the
probability distribution $P(x|g)$. The conditional probability for each
measurement result $x$ is determined by Born’s rule:
$P(x|g)=\text{Tr}(\rho_{g}\hat{\Pi}_{x})$. After repeating the measurement
process $\nu$ times, a sequence of measurement results
$\vec{x}=(x_{1},...,x_{\nu})$ is obtained. Finally, the parameter is estimated
using an appropriate estimator $g_{\text{est}}=G(\vec{x})$.

The precision of parameter estimation is typically assessed by the variance of
the estimated parameter, denoted as $\text{Var}(\cdot)$. The ultimate
precision of unbiased estimation of the parameter $g$ from the sequence
$\vec{x}$ is determined by the Cramér-Rao bound (CRB):

$\text{Var}(g)\geq\frac{1}{\nu F_{g}}.$ (15)

Here, the Fisher information (FI), defined as:

$F_{g}=\int\frac{1}{P(x|g)}\Big{[}\frac{\partial P(x|g)}{\partial
g}\Big{]}^{2}dx,$ (16)

quantifies the sensitivity of the probability distribution $P(x|g)$ to the
variations in the parameter $g$ Fisher (1922). A larger FI indicates a
potential for achieving better estimation precision. The CRB can be
asymptotically attained with a sufficiently large number of measurements
($\nu$) and an efficient estimator, such as the maximum likelihood estimation
(MLE) Lehmann and Casella (2006), Bayesian estimation Pezze and Smerzi (2014);
Trees and Bell (2007); Le Cam (2012), etc. For a given final state $\rho_{g}$,
the probability distribution $P_{l}(x|g)$ varies as the POVM
$\\{\hat{\Pi}^{(l)}_{x}\\}$ changes, resulting in different FI $F_{g}^{(l)}$.
By maximizing the FI over all possible POVM, we can obtain the quantum Fisher
information (QFI): $Q_{g}=\max_{\\{\hat{\Pi}_{x}^{(l)}\\}}F_{g}$ Lee  _et al._
(2002). The QFI quantifies the sensitivity of the parameter-encoded quantum
state $\rho_{g}$ to the parameter $g$. When both the quantum measurement is
optimal and the estimator is efficient, the precision of estimating the
parameter $g$ can achieve the quantum CRB $\Delta^{2}g\geq 1/(\nu Q_{g})$.

Figure 4: The basic procedure of quantum metrology for single-parameter
estimation. Some related abbreviations: averaging the measurement results
(AMR), maximum likelihood estimation (MLE), Cramér-Rao bound (CRB), standard
quantum limit (SQL), Heisenberg limit (HL).

To rigorously determine the QFI $Q_{g}$ of a general state $\rho_{g}$, a super
operator called the symmetric logarithmic derivative (SLD)
$\mathcal{R}_{\rho_{g}}^{-1}(\partial_{g}\rho_{g})$ is introduced with the
definition
$\partial_{g}\rho_{g}=[\rho_{g}\mathcal{R}_{\rho_{g}}^{-1}(\partial_{g}\rho_{g})+\mathcal{R}_{\rho_{g}}^{-1}(\partial_{g}\rho_{g})\rho_{g}]/2$.
By expressing the SLD $\mathcal{R}_{\rho_{g}}^{-1}(O)$ in the diagonal
representation of $\rho_{g}=\sum_{j}\lambda_{j}|j\rangle\langle j|$ as

$\mathcal{R}_{\rho_{g}}^{-1}(O)=\sum_{j,k:\lambda_{j}+\lambda_{k}\neq
0}\frac{2O_{jk}|j\rangle\langle k|}{\lambda_{j}+\lambda_{k}},$ (17)

we can obtain the QFI as Helstrom (1969); Holevo (2011); Lee  _et al._ (2002)

$Q_{g}=\text{Tr}[\mathcal{R}_{\rho_{g}}^{-1}(\partial_{g}\rho_{g})\rho_{g}\mathcal{R}_{\rho_{g}}^{-1}(\partial_{g}\rho_{g})].$
(18)

The definition of the QFI provides it with two fundamental properties:
convexity and additivity: (i) For a mixed state
$\rho_{g}=\sum_{j}c_{j}\rho_{g}^{(j)}$, we have
$Q_{g}\leq\sum_{j}c_{j}Q_{g}^{(j)}$. (ii) For a direct product state
$\rho_{g}=\otimes_{j}\rho_{g}^{(j)}$, we have $Q_{g}=\sum_{j}Q_{g}^{(j)}$,
where $Q_{g}^{(j)}$ represents the QFI of $\rho_{g}^{(j)}$. These properties
allows for the convenient derivation of upper bounds for the QFI in compound
quantum systems. When $\rho_{g}=|\psi_{g}\rangle\langle\psi_{g}|$ is a pure
state, the expression for the QFI simplifies to:

$Q_{g}=4\Big{[}\frac{d\langle\psi_{g}|}{dg}\frac{d|\psi_{g}\rangle}{dg}-\Big{|}\frac{d\langle\psi_{g}|}{dg}|\psi_{g}\rangle\Big{|}^{2}\Big{]}.$
(19)

If we consider the parameter-encoding process as a unitary evolution
$\hat{U}=\exp(-ig\hat{\mathcal{H}})$ with the generator $\hat{\mathcal{H}}$,
the QFI depends solely on the pure initial state
$\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$:

$Q_{g}=4[\langle\psi_{0}|\hat{\mathcal{H}}^{2}|\psi_{0}\rangle-\langle\psi_{0}|\hat{\mathcal{H}}|\psi_{0}\rangle^{2}]=4\text{Var}_{|\psi_{0}\rangle}(\hat{\mathcal{H}}),$
(20)

Here, the quantum CRB coincides with the generalized uncertainty relation
$\text{Var}(g)\text{Var}_{|\psi_{0}\rangle}(\hat{\mathcal{H}})\geq 1/4$. Both
the QFI and the uncertainty relation indicate that the optimal precision of
$g$ is achieved with the maximum variance
$\text{Var}_{|\psi_{0}\rangle}(\hat{\mathcal{H}})$ Braunstein  _et al._
(1996).

The linear parameter-encoding process $\hat{U}_{g}$ introduces fundamental
bounds on measurement precision when optimizing the initial state
$|\psi_{0}\rangle$. Fig. 5 (a) illustrates $N$ classically correlated states
$|\psi_{0}^{(j)}\rangle$ ($j=1,2,...,N$) composing the total product state
$|\Psi_{0}\rangle_{\text{tot}}^{(c)}=\otimes_{j=1}^{N}|\psi_{0}^{(j)}\rangle$.
By considering the maximal ($h_{\text{max}}$) and minimal ($h_{\text{min}}$)
eigenvalues of the generator $\hat{\mathcal{H}}$, the optimal state
$|\psi_{0}^{(j)}\rangle=(|h_{\text{min}}\rangle+|h_{\text{max}}\rangle)/\sqrt{2}$
yields a maximum QFI of $Q_{g}^{(j)}=(h_{\text{max}}-h_{\text{min}})^{2}$.
When $N$ copies of the optimal states are used, the total QFI is maximized as
follows:

$Q_{\text{tot}}^{(c)}=4\text{Var}_{|\Psi_{0}\rangle_{\text{tot}}^{(c)}}(\hat{\mathcal{H}}^{\otimes
N})=N(h_{\text{max}}-h_{\text{min}})^{2}.$ (21)

Consequently, the precision of $g$ ultimately scales as $\text{Var}(g)\propto
1/N$ Giovannetti  _et al._ (2006); Pezzé and Smerzi (2009), similar to the
central limit theorem, which is referred to as the standard quantum limit
(SQL). Then, we consider a quantum-correlated system denoted by the total
state $|\Psi_{0}\rangle_{\text{tot}}^{(q)}$. By globally maximizing the total
QFI, we obtain:

$Q_{\text{tot}}^{(q)}=\text{max}\text{Var}_{|\Psi_{0}\rangle_{\text{tot}}^{(q)}}(\hat{\mathcal{H}})=N^{2}(h_{\text{max}}-h_{\text{min}})^{2}.$
(22)

The corresponding optimal state is the maximally entangled state
$|\Psi_{0}\rangle_{\text{tot}}^{(q)}=(|h_{\text{max}}\rangle^{N}+|h_{\text{min}}\rangle^{N})/\sqrt{2}$,
shown in Fig. 5 (b). In this case, the precision of $g$ scales as
$\text{Var}(g)\propto 1/N^{2}$ Berry and Wiseman (2000); Górecki  _et al._
(2020); Bužek  _et al._ (1999). Compared to the SQL, the maximally entangled
state enhances the precision limit by a factor of $N$, known as the Heisenberg
Limit (HL).

Figure 5: (a) $N$ classically correlated states are employed to measure the
parameter $g$ with the precision attaining the standard quantum limit (SQL).
(b) A maximally entangled state is employed to measure the parameter $g$ with
the precision achieving Heisenberg limit. (c) The phase space distribution of
the coherent state. (d) A Mach-Zender interferometer that measures an unknown
phase $\phi$ with single-pass and multi-pass scenarios Higgins  _et al._
(2007).

In Fig. 5 (d), we present the measurement of the relative phase $\phi$ between
two paths (labelled as 1 and 2) in a Mach-Zehnder interferometer. The
generator for the parameter $\phi$ is the photon number operator associated
with the relative path, denoted as $\hat{\mathcal{H}}=\hat{n}_{1}$. As
discussed earlier, the SQL and HL can be achieved by using $N$ independent
single photons in the state
$|\psi_{N}\rangle=2^{-N/2}(|1\rangle_{1}|0\rangle_{2}+|0\rangle_{1}|1\rangle_{2})^{\otimes
N}$ and the ‘N00N’ state
$|\psi_{N00N}\rangle=(|N\rangle_{1}|0\rangle_{2}+|0\rangle_{1}|N\rangle_{2})/\sqrt{2}$,
respectively. Another commonly used classical photonic state is the coherent
state $|\alpha\rangle$, whose phase-space distribution is shown in Fig. 5 (c).
When estimating the relative phase $\phi$ using the coherent state, the
precision is ultimately limited by the variance of the photon number operator
$\hat{n}$ in $|\alpha\rangle$: $\text{Var}(\phi)\propto
1/\text{Var}(\hat{n})_{|\alpha\rangle}=1/\bar{n}$, where
$\bar{n}=|\langle\alpha|\alpha\rangle|^{2}$ is the mean photon number. This
precision limit is referred to as the shot noise limit (SNL) Braunstein
(1992). To reduce the photon number fluctuations, the phase-space distribution
can be squeezed along a specific direction. The squeezing operator
$S_{2}(r)=\exp(r^{*}\hat{a}_{1}\hat{a}_{2}-r\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger})$
acting on the two-mode vacuum state leads to the two-mode squeezed vacuum
states $|r\rangle_{2}=S_{2}(r)|0\rangle_{1}|0\rangle_{2}$. The average and
variance of the photon number operator in $|r\rangle_{2}$ are given by
$\bar{n}=2\sinh^{2}|r|$ and
$\text{Var}_{|r\rangle_{2}}(\hat{n}_{1})=2(\bar{n}^{2}+\bar{n})$,
respectively. Consequently, the precision of $\phi$ can be improved to
$\text{Var}(\phi)=1/[8(\bar{n}^{2}+\bar{n})]$, attaining Heisenberg-scaling
precision with large $\bar{n}$ Yurke  _et al._ (1986). However, the
preparation of quantum-correlated states (e.g., N00N states and squeezed
states) with large photon numbers remains a significant experimental
challenge, hindering the practical implementation of quantum metrology
enhanced by quantum resources. To circumvent these technical problems, Higgins
et al. proposed an entanglement-free Heisenberg-limited phase-estimation
protocol by replacing the single-pass phase shift with the multiple-pass
scenario, as shown in Fig. 5 (d).

The signal-to-noise ratio (SNR) is also a commonly used metric for evaluating
the performance of metrological protocols, which depends significantly on the
specific measurement schemes Barnett  _et al._ (2003). For instance, we
consider the conventional scheme (CM) to measure the transversal displacement
of a Gaussian beam $|\Phi\rangle$ due to the unitary evolution
$\hat{U}_{g}=\exp(-ig\hat{P})$. The final state $|\Phi_{g}\rangle$ is measured
with the POVM $\\{\hat{\Pi}_{x}\\}$, which yields the measurement result $x$
subjecting to the probability distribution
$P(x|g)=\langle\Phi_{g}|\hat{\Pi}_{x}|\Phi_{g}\rangle$. The terms ”signal” and
”noise” correspond to the average shift and the standard deviation of $x$,
respectively. Therefore, the SNR of total $\nu$ measurements is given by

$\text{SNR}=\frac{\sqrt{\nu}|\langle x\rangle-x_{0}|}{\sqrt{\langle
x^{2}\rangle-\langle x\rangle^{2}}},$ (23)

where $\langle x\rangle=\int P(x|g)xdx$, and
$x_{0}=\langle\Phi|\hat{\Pi}_{x}|\Phi\rangle$ is the average of $x$ for the
initial state. The SNR metric naturally defines both the quantum measurement
$\\{\hat{\Pi}_{x}\\}$ and the estimator of averaging the measurement results
(AMR) Pang and Brun (2015a). If the AMR estimator efficiently saturates the
CRB, the SNR metric is equivalent to the FI in quantifying the metrological
information of the parameter in a probability distribution $P(x|g)$. In this
scenario, the MLE is equivalent to the AMR estimator with a Gaussian MS
$|\Phi\rangle$. Specifically, the SNR can be related to the FI $F_{g}$ about
$g$ through $\text{SNR}=\sqrt{\nu}g/\sigma=g\sqrt{\nu F_{g}}$. Furthermore, if
$\hat{\Pi}_{x}$ is also the optimal quantum measurement to attain the QFI, the
SNR metric captures the whole QFI of the final state $|\Phi_{g}\rangle$.

###  III.2 3.2 Metrological Limits of Weak Value Amplification

The schematic diagram for analysing the ultimate precision of WVA is presented
in Fig. 6 (a). The parameter $g$ is encoded into the joint state
$|\Psi_{jt}\rangle$ through the interaction between the QS and the MS. By
substituting the Eq. (2) ($|\Psi_{jt}\rangle$) into Eq. (19), we obtain the
QFI of the joint state

$\displaystyle Q_{jt}$ $\displaystyle=$ $\displaystyle
4\Big{[}\langle\hat{A}^{2}\rangle_{s}\langle\hat{P}^{2}\rangle_{m}-\langle\hat{A}\rangle_{s}^{2}\langle\hat{P}\rangle_{m}^{2}\Big{]}{}$
(24) $\displaystyle=$ $\displaystyle
4\Big{[}\langle\hat{A}^{2}\rangle_{s}\text{Var}_{m}(\hat{P})+\text{Var}_{s}(\hat{A})\langle\hat{P}\rangle_{m}^{2}\Big{]}{}$
$\displaystyle=$ $\displaystyle
4\Big{[}\text{Var}_{s}(\hat{A})\langle\hat{P}^{2}\rangle_{m}+\langle\hat{A}\rangle_{s}^{2}\text{Var}_{m}(\hat{P})\Big{]},$

where
$\text{Var}(\hat{O})=\langle\hat{O}^{2}\rangle-\langle\hat{O}\rangle^{2}$ is
the variance of the operator $\hat{O}$ and $\langle\cdot\rangle$ denotes the
average value in the initial state of the QS (subscript $s$) or the MS
(subscript $m$). If the initial state of the QS (MS) is a balanced state,
i.e., $\langle\hat{A}\rangle_{s}=0$ ($\langle\hat{P}\rangle_{m}=0$), the QFI
of the joint state can be simplified as
$Q_{jt}=\text{Var}_{s}(\hat{A})\langle\hat{P}^{2}\rangle_{m}$
($Q_{jt}=\langle\hat{A}^{2}\rangle_{s}\text{Var}_{m}(\hat{P})$). Additionally,
both $|\psi_{i}\rangle$ and $|\Phi\rangle$ are balanced states, resulting in
$Q_{jt}=4\text{Var}_{s}(\hat{A})\text{Var}_{m}(\hat{P})$. The subsequent post-
selection of the QS can be regarded as part of the measurements on the joint
state, which cannot enhance QFI Zhang  _et al._ (2015). Thus, the QFI of the
joint state $Q_{jt}$ sets the upper bound for the measurement precision in the
WVA scheme. Furthermore, the expression of $Q_{jt}$ also provides the possible
approaches to improve the ultimate precision of WVA.

Figure 6: (a) The distribution of the FI during the post-selection process in
WVA. (b) The ratio of the QFI of WVA $Q_{\text{WVA}}$ to $Q_{jt}$ and the
ratio of the classical FI $F_{p}$ to $Q_{jt}$ are compared while varying the
parameter $\theta_{i}$. The post-selected state keeps the optimal choice
$|\psi_{f}^{\text{opt}}\rangle$ and $g/(2\sigma)=0.1$. (c) and (d) The maximum
FI of the real and the imaginary WVA, $p_{f}F_{f}^{(q)}$ and
$p_{f}F_{f}^{(p)}$, over each probability of post-selection are compared with
different $g$ and $\sigma=0.5$.

To account for the metrological contributions of all components during the
post-selection process, we describe the post-selection by performing a POVM
$\\{|\psi_{f}\rangle\langle\psi_{f}|\otimes\hat{I}_{m},|\psi_{r}\rangle\langle\psi_{r}|\otimes\hat{I}_{m}\\}$
($\langle\psi_{r}|\psi_{f}\rangle=0$) measurement on the joint state. This
post-selection generates the success and failure modes of the final meter
states $|\Phi_{f}\rangle$ and $|\Phi_{r}\rangle$, corresponding to the post-
selection probabilities $p_{f}$ and $p_{r}=1-p_{f}$, respectively. By
substituting the expression of $|\Phi_{f}\rangle$ into Eq. (19), we obtain the
QFI of the post-selected MS $|\Phi_{f}\rangle$ with respect to $g$:

$Q_{f}=4\Big{[}\langle\hat{J}^{\dagger}\hat{J}\rangle_{m}-\Big{|}\langle\hat{J}^{\dagger}\hat{K}\rangle_{m}\Big{|}^{2}\Big{]}.$
(25)

Here, we denote
$\hat{J}=\langle\psi_{f}|\hat{A}\hat{P}\exp(-ig\hat{A}\hat{P})|\psi_{i}\rangle/\sqrt{p_{f}}$
and
$\hat{K}=\langle\psi_{f}|\exp(-ig\hat{A}\hat{P})|\psi_{i}\rangle/\sqrt{p_{f}}$
Alves  _et al._ (2015). The QFI $Q_{r}$ of $|\Phi_{r}\rangle$ can be derived
similarly by replacing $|\Phi_{f}\rangle$ and $p_{f}$ with $|\Phi_{r}\rangle$
and $p_{r}$, respectively. Furthermore, the probability distribution
$\\{p_{f},p_{r}\\}$ in the post-selection process also contains information
about $g$ quantified by the classical FI

$F_{p}=\frac{1}{p_{f}(1-p_{f})}\Big{(}\frac{dp_{f}}{dg}\Big{)}^{2}.$ (26)

All three contributions constitute the QFI of the joint state
$Q_{jt}=p_{f}Q_{f}+p_{r}Q_{r}+F_{p}$ Zhang  _et al._ (2015). Since the WVA
scheme typically focuses on the amplified outcomes in the post-selected MS
$|\Phi_{f}\rangle$, we regard $Q_{\text{WVA}}=p_{f}Q_{f}$ as the QFI of WVA.

In order to realize the optimal precision of WVA, it is necessary to maximize
the total QFI of the joint state $Q_{jt}$ by optimizing the pre-selected
state. Subsequently, the post-selected state is optimized to make
$Q_{\text{WVA}}$ attain the maximum $Q_{jt}$. As a demonstration, we employ
the zero-mean Gaussian superposition state $|\Phi\rangle$ from Eq. (1) as the
MS and $|\psi_{i}\rangle$ ($|\psi_{f}\rangle$) serving as the pre- (post-)
selected state in the QS. By considering the interaction Hamiltonian
$\hat{H}=g\delta(t-t_{0})\hat{A}\hat{P}$, we find that
$Q_{jt}=\langle\psi_{i}|\hat{A}^{2}|\psi_{i}\rangle/\sigma^{2}$. When AAV
approximate condition Eq. (5) is satisfied, the QFI of WVA is approximated as
$Q_{\text{WVA}}\approx|\langle\psi_{i}|\hat{A}|\psi_{f}\rangle|^{2}/\sigma^{2}$.
The maximal QFI of WVA $Q_{\text{WVA}}^{\text{max}}$ can achieve $Q_{jt}$
provided an optimal post-selected state
$|\psi_{f}^{\text{opt}}\rangle=\hat{A}|\psi_{i}\rangle/\langle\hat{A}^{2}\rangle^{1/2}$.
Consequently, the post-selected MS with a probability of
$p_{f}=|\langle\psi_{f}^{\text{opt}}|\psi_{i}\rangle|$ can capture all the
metrological information of the joint state. However, the probability of post-
selection cannot be extremely small due to the AAV approximate condition. When
$|\langle\psi_{f}^{\text{opt}}|\psi_{i}\rangle|$ becomes comparable to
$g/\sigma$, $Q_{\text{WVA}}$ begins to decrease. For the case of
$g|\langle\hat{A}\rangle_{w}|/\sigma\gg 1$, we find that
$Q_{\text{WVA}}^{\text{opt}}\approx 0$ and $F_{p}^{\text{opt}}\approx Q_{jt}$.
In this condition, all the metrological information about $g$ can be extracted
from the statistics of the post-selection. The ratios of
$Q_{\text{WVA}}/Q_{jt}$ and $F_{p}/Q_{jt}$ with respect to the angle
$\theta_{i}$ of the pre-selected state $|\psi_{i}\rangle$ and the optimal
post-selected state $|\psi_{f}^{\text{opt}}\rangle$ are depicted in Fig 6 (b).

In the preceding discussion, we focused on condensing all the QFI of the joint
state into that of WVA, which we denote as $Q_{\text{WVA}}=Q_{jt}$. In the
following, we investigate appropriate measurement strategies that extract the
full QFI $Q_{f}$ about $g$ to the classical FI $F_{f}$ of the measured
probability distributions. For a general weak value in standard WVA, we have
derived the optimal observable $\hat{S}_{\theta}$ in section 2.1, which yields
maximal shifts of the post-selected MS. It has been proven that this
observable is also optimal for obtaining the maximal FI
$Q_{f}=F_{f}^{(s_{\theta})}$ Knee and Briggs (2018). Here, we take two
prevalent WVA schemes, namely, the real and imaginary WVA, for comparison. For
the real (imaginary) WVA, the optimal parameters of the QS satisfying
$\theta_{i}=-\theta_{f}$, $\phi=\phi_{i}-\phi_{f}=0$
($\theta_{i}=-\theta_{f}=\pi/2$, $\phi\rightarrow 0$) yields
$Q_{\text{WVA}}\approx Q_{jt}$. In the real (imaginary) WVA, we have the
optimal observable $\hat{S}_{\theta}=\hat{Q}$ ($\hat{S}_{\theta}=\hat{P}$),
from which we obtain the classical FI $p_{f}F_{f}^{(q)}=1/\text{Var}(\hat{Q})$
($p_{f}F_{f}^{(p)}=4\text{Var}(\hat{P})$). Since the Gaussian MS has the
minimum uncertainty, i.e.,
$\text{Var}_{m}(\hat{Q})\text{Var}_{m}(\hat{P})=\sigma^{2}\cdot
1/(2\sigma)^{2}=1/4$, the maximal FI of both the real and the imaginary WVA
are equal to the QFI of WVA
$p_{f}F_{f}^{(q)}=p_{f}F_{f}^{(p)}=Q_{\text{WVA}}$. The difference lies in
that the maximal FI of the real WVA can be attained with large range of weak
values while the imaginary WVA achieves the maximal FI with only large modulus
of weak values. In Fig. 6 (c) and (d), we compare the maximal FI of the real
and imaginary WVA for each probability of post-selection $p_{f}$. The results
indicate that the maximal FI of the real WVA $p_{f}F_{f}^{(q)}$ monotonically
increases with a larger $p_{f}$, and $p_{f}=1$ corresponds to the FI of the CM
$F_{\text{CM}}=Q_{\text{WVA}}$. In contrast, the maximal FI of the imaginary
WVA $p_{f}F_{f}^{(p)}$ approaches $Q_{\text{WVA}}$ only with a small $p_{f}$.
The adjustable success probability of post-selection without discarding the
metrological information, provides the real WVA with the potential advantages
in resisting noise and saturation of the detectors.

The real WVA has demonstrated advantages over the imaginary WVA in the
adjustable range of $p_{f}$ for achieving the maximal FI. However, both the FI
of the imaginary WVA $p_{f}F_{f}^{(p)}$ and $Q_{jt}$ are proportional to the
variance of the observable $\hat{P}$, indicating the significant potential of
the imaginary WVA. In certain scenarios, the uncertainty of the initial MS,
denoted by the subscript $m^{\prime}$, is not at its minimum due to factors
such as the generation of the MS (e.g., the squeezed state) or experimental
noise, leading to
$\text{Var}_{m^{\prime}}(\hat{Q})\text{Var}_{m^{\prime}}(\hat{P})>1/4$. The
influence of noise on the ultimate precision of WVA will be discussed in
sections 4 and 5. Here, we specifically consider the squeezed state with the
observables $\hat{P}$ and $\hat{Q}$ referring to the quadratures in phase
space. In cases where
$\text{Var}_{m^{\prime}}(\hat{Q})<\text{Var}_{m}(\hat{Q})$ and
$\text{Var}_{m^{\prime}}(\hat{P})>\text{Var}_{m}(\hat{P})$, both the maximal
FI of the real WVA and the CM increase to
$1/\text{Var}_{m^{\prime}}(\hat{Q})$. Notably, the imaginary WVA can leverage
the variance of the observable $\hat{P}$ to achieve a larger FI of
$4\text{Var}_{m^{\prime}}(\hat{P})$. Considering that $Q_{jt}\geq
4\Delta^{2}_{m^{\prime}}\hat{P}$, the imaginary WVA offers an optimal
measurement strategy for extracting the full QFI of the joint state. In
contrast, the FI of the real WVA is smaller than that of the imaginary WVA due
to suboptimal pre- and post-selection as well as the measurement on the final
MS. This property of the imaginary WVA presents an avenue for enhancing the
precision of WVA by introducing squeezing, mixture and even noise into the MS,
which will be further discussed in the subsequent two sections.

##  IV 4\. Weak-value amplification with Heisenberg-limited precision

Figure 7: (a) The quantum circuit for implementing the standard WVA. (b) The
quantum circuit for realizing the quantum-enhanced WVA Pang  _et al._ (2014).
(c) WVA with iterative interactions Kim  _et al._ (2022). (d) Realization of
entanglement-assisted WVA with single-photon simulation Chen  _et al._
(2019b). (e) Two-photon entangled state demonstration for realizing the
entanglement-assisted WVA Stárek  _et al._ (2020). (f) Experimental setup for
iterative-interaction WVA Kim  _et al._ (2022).

Considerable research has been conducted regarding the ultimate precision of
standard WVA, which has prompted researchers to investigate its potential to
surpass the SQL , and achieve Heisenberg-limited (HL) precision. In recent
years, various enhanced WVA schemes capable of achieving Heisenberg-scaling
precision have been reported. These advancements are realized through two
approaches: the first involves exploiting entangled states or iterative
interactions, drawing inspiration from the phase estimation scheme in standard
quantum metrology. The second approach relies on the phase-space interaction
scenario, which enables the attainment of Heisenberg-scaling precision using
solely classical resources, in a way similar to nonlinear quantum metrology
Boixo  _et al._ (2007, 2008).

###  IV.1 4.1 Heisenberg-limited weak-value amplification in linear optical
system

We first introduce a phase-estimation WVA scheme with its corresponding
schematic diagram shown in Fig. 7 (a). The pre-selected QS and the initial MS
are both qubit states
$|+\rangle_{s}=|+\rangle_{m}=(|0\rangle+|1\rangle)/\sqrt{2}$, with the control
gate
$R_{z}(2\varphi)=\exp(-i\varphi\hat{\sigma}_{z}^{(s)}\otimes\hat{\sigma}_{z}^{(m)}/2)$
coupling them, resulting in a joint state of
$|\Psi_{b}\rangle=R_{z}(2\varphi)|+\rangle_{s}|+\rangle_{m}$. Subsequent post-
selection on the QS by
$R_{z}(2\epsilon)|-\rangle=(e^{-i\epsilon}|0\rangle-e^{i\epsilon}|1\rangle)/\sqrt{2}$
amplifies the parameter $\varphi$ by the imaginary weak value
$\langle\hat{A}\rangle_{w}=i/\epsilon$ ($\epsilon\ll 1$), leading to a shift
of the observable $\hat{\sigma}_{z}$ on the final MS, given by
$\langle\hat{\sigma}_{z}\rangle_{f}=\varphi\text{Im}\langle\hat{A}\rangle_{w}$.
Evaluating the ultimate precision of $\varphi$ using the FI metric, we find
that the QFI about $\varphi$ of the joint state is
$Q_{b}=4\text{Var}_{+}(\hat{A})\text{Var}_{+}(\hat{\sigma}_{z})=4$.
Considering the probability of post-selection $p_{f}\approx\epsilon^{2}$, the
classical FI of measuring the observable $\hat{\sigma}_{z}$ on
$|+^{\prime}\rangle$ is $F_{\text{WVA}}=p_{f}F_{f}=4$. This result shows that
both the post-selected state and the measurement observable on the MS are
optimal for concentrating the full QFI into a fraction of post-selected MS.
For $N$ independent trials of WVA, the average number of successful post-
selection is $n=1-(1-p_{f})^{N}\approx N\epsilon^{2}$, which scales linearly
with $N$. Therefore, the total FI of all $N$ independent measurements is given
by $F_{\text{WVA}}^{(N)}=nF_{f}=4N$, showing a precision in SQL.

In 2014, Pang et. al. proposed an entanglement-assisted WVA scheme that
improves the measurement precision of $\varphi$ to the HL Pang  _et al._
(2014). The quantum circuit for this scheme is depicted in Fig. 7 (b), where
$N$ quantum-correlated particles form the initial state of the QS
$|\Psi_{i}\rangle$. The interaction between the QS and the MS is described by
the unitary transformation
$\hat{U}=\exp(-i\varphi\hat{A}^{(N)}\hat{\sigma}_{z}^{(m)}/2)$, in which the
observable of the QS becomes $\hat{A}^{(N)}=\sum_{k}\hat{A}_{k}$ with
$\hat{A}_{k}=\hat{I}\otimes\cdots\hat{\sigma}_{z}^{(s)}\otimes\cdots\otimes\hat{I}$
referring to the observable $\hat{\sigma}^{(s)}_{z}$ acting on $k$th state.
The interaction results in the joint state $|\Psi_{jt}^{(N)}\rangle$. Assuming
that $|\Psi_{i}\rangle$ is a balanced state
($\langle\Psi_{i}|\hat{A}|\Psi_{i}\rangle_{s}=0$), the QFI of
$|\Psi_{jt}^{(N)}\rangle$ about the parameter $\varphi$ is given by
$Q_{jt}^{(N)}=4\text{Var}_{|\Psi_{i}\rangle}[\hat{A}^{(N)}]\text{Var}_{+}(\hat{\sigma}^{(m)}_{z})=4\text{Var}_{|\Psi_{i}\rangle}[\hat{A}^{(N)}]$.
Thus, the maximal QFI
$[Q_{jt}^{(N)}]_{\text{max}}=4\\{\text{Var}_{|\Psi_{i}\rangle}[\hat{A}^{(n)}]\\}_{\text{max}}=4N^{2}$
can be achieved by preparing the total pre-selected state $|\Psi_{i}\rangle$
as the maximally entangled form

$|\Psi_{i}^{(e)}\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
N}+e^{i\theta}|1\rangle^{\otimes N})$ (27)

with $\theta$ an arbitrary relative phase. In this scheme, the QFI in the
entanglement-assisted WVA $Q_{jt}^{(N)}$ indicates the Heisenberg-scaling
precision and shows an improvement over that of the standard WVA by a factor
of $N$.

In order to fully extract the QFI of the joint state, Pang et al. proposed two
optimizations for the post-selected state: (i) maximizing the probability of
post-selection $p_{f}^{(N)}$,while keeping the weak value
$\langle\hat{A}^{(N)}\rangle_{w}$ fixed; (ii) maximizing the modulus of the
weak value $|\langle\hat{A}^{(N)}\rangle_{w}|$ while fixing $p_{f}^{(N)}$. In
the former case, the weak value is set as
$\langle\hat{A^{(N)}}\rangle_{w}=i/\epsilon$. By optimizing the post-selected
state, the probability of post-selection can be maximized to
$p_{f}^{(N)}=N^{2}\epsilon^{2}$, which is achieved by the following
expression:

$|\Psi_{f}^{(N)}\rangle_{p}\propto(N+\langle\hat{A}^{(N)}\rangle_{w}^{*})|0\rangle^{\otimes
N}+(N-\langle\hat{A}^{(N)}\rangle_{w}^{*})|1\rangle^{\otimes N}\propto
e^{-in\epsilon}|0\rangle^{\otimes N}-e^{in\epsilon}|1\rangle^{\otimes N}.$
(28)

In the latter case, the probability of post-selection is set as
$p_{f}^{(N)}=N\epsilon^{2}$, which is equivalent to $Np_{f}$ in the standard
WVA. The post-selected state can be decomposed as follows:

$|\Psi_{f}^{(N)}\rangle=\sqrt{p_{f}^{(N)}}|\Psi_{i}\rangle+\sqrt{1-p_{f}^{(N)}}|\Psi_{i}^{\perp}\rangle,$
(29)

where $|\Psi_{i}^{\perp}\rangle$ is orthogonal to $|\Psi_{i}\rangle$. The
maximal modulus of the imaginary weak value can be approximated as:

$\text{max}|\langle\hat{A}^{(N)}\rangle_{w}|\approx\sqrt{\text{Var}_{|\Psi_{i}^{(e)}\rangle}[\hat{A}^{(N)}]/p_{f}^{(N)}}=\sqrt{N}/\epsilon.$
(30)

This maximum value is achieved when $|\Psi_{i}^{\perp}\rangle$ is parallel to
the component of $\hat{A}^{(N)}|\Psi_{i}\rangle$ in the complementary subspace
orthogonal to $|\Psi_{i}\rangle$. Therefore, the optimal post-selected state
for maximizing the weak value is given by

$|\Psi_{f}\rangle_{w}\propto e^{-i\sqrt{N}\epsilon}|0\rangle^{\otimes
N}-e^{i\sqrt{N}\epsilon}|1\rangle^{\otimes N}.$ (31)

Since the QFI of WVA $Q_{\text{WVA}}^{(N)}$ is equal to $Q_{jt}^{(N)}$ in both
cases, both the post-selected states $|\Psi_{f}^{(N)}\rangle_{p}$ and
$|\Psi_{f}^{(N)}\rangle_{w}$ are optimal for concentrating the full QFI
$Q_{jt}^{(N)}$ into the post-selected MS. Furthermore, the QFI of
entanglement-assisted WVA can be extracted by measuring the optimal observable
$\hat{\sigma}_{z}^{(m)}$ on the final MS.

Theoretical analysis has shown that the entanglement-assisted WVA scheme is
advantageous in addressing the inaccuracies in measurement results and the
loss of FI caused by readout errors Pang and Brun (2015b). In 2019, Chen et
al. have conducted an experimental demonstration of the enhanced weak values
or the post-selection probabilities obtained through entanglement-assisted WVA
using the high-dimensional DOF of single photons shown in Fig. 7 (d)Chen  _et
al._ (2019b). Furthermore, in 2020, a two-photon entanglement-assisted WVA was
realized with the experimental setup illustrated in Fig. 7 (e) Stárek  _et
al._ (2020). The experimental findings revealed that the precision improvement
resulting from entanglement is highly sensitive to the purity of the QS state.
Due to limitations in current photonic technology, achieving the theoretical
Heisenberg scaling of the entanglement-assisted WVA remains a challenge.

To circumvent this technical obstacle, Kim et al. proposed a WVA scheme based
on iterative interactions, which achieves the HL without the use of entangled
resources Kim  _et al._ (2022). The schematic diagram of the iterative WVA and
its experimental implementation are illustrated in Fig. 7 (c) and (f),
respectively. In this scheme, the QS is represented by the spin of photons
with a pre-selected state
$|\psi\rangle_{s}=(|+y\rangle_{s}+|-y\rangle_{s})/\sqrt{2}$, where
$|+y\rangle_{s}$ ($|+y\rangle_{s}$) refers to the right (left) circular
polarization state. The optical path state serves as the MS and is initialized
as the superposition state
$|+x\rangle_{m}=(|+z\rangle_{m}+|-z\rangle_{m})/\sqrt{2}$, where
$|+z\rangle_{m}$ and $|-z\rangle_{m}$ represent the upper and lower path
states, respectively. The unitary transformation for $N$ iterative
interactions is described by
$\hat{U}=e^{iN\gamma\hat{A}\otimes\hat{\sigma}_{z}}$, where $\gamma$ is the
coupling strength and $\hat{A}$ ($\hat{\sigma}_{z}$) represents the observable
of the QS (MS). The post-selected states have the same probability amplitudes
as those in $|\Psi_{f}^{(N)}\rangle_{p}$ and $|\Psi_{f}^{(N)}\rangle_{p}$, but
the base states $|0\rangle^{\otimes N}$ and $|1\rangle^{\otimes N}$ are
replaced by $|+y\rangle_{s}$ and $|-y\rangle_{s}$, respectively.
Correspondingly, we denote the post-selected states as
$|\psi_{f}^{(iter)}\rangle_{p}\propto
e^{-iN\epsilon}|+y\rangle_{s}+e^{iN\epsilon}|-y\rangle$ and
$|\psi_{f}^{(iter)}\rangle_{w}\propto
e^{-i\sqrt{N}\epsilon}|+y\rangle_{s}+e^{i\sqrt{N}\epsilon}|-y\rangle$.
Similarly to entanglement-assisted WVA, the post-selected state
$|\psi_{f}^{(iter)}\rangle_{p}$ ($|\psi_{f}^{(iter)}\rangle_{w}$) improves the
probabilities of post-selection $p_{f}^{(iter)}$ (weak value $\langle
N\hat{\sigma}_{y}\rangle_{w}$) by a factor of $N$ ($N$) compared to standard
WVA. In both cases, the QFI of the iterative WVA
$Q_{\text{WVA}}^{(iter)}=N^{2}$, achieves Heisenberg-scaling precision.
Compared to entanglement-assisted WVA, the iterative scheme does not suffer
from the fragility and the preparation of entangled resources, thus providing
a practical pathway for Heisenberg-scaling WVA. However, the interaction time
of the iterative WVA is $N$ times larger than that of the entanglement-
assisted WVA for achieving the same level of precision.

###  IV.2 4.2 weak-value amplification with phase-space interaction

Figure 8: (a) The schematic diagram for the WVA of phase-space interaction.
(b) The experimental setup to implement the phase-space WVA through the Kerr
effect Chen  _et al._ (2018b). (c) Uncertainty of the coherent state
$|\alpha\rangle$ to estimate the coupling strength $g$ Chen  _et al._ (2021b).
(d) The precision of $g$ with the average photon number of coherent state $N$
Chen  _et al._ (2018a). (e) Uncertainty of the mixed coherent state to
estimate the coupling strength $g$ Chen  _et al._ (2021b). (f) The precision
of $g$ with the average photon number of mixed coherent state $n$ Chen  _et
al._ (2018b).

The phase-space distribution provides a comprehensive representation of the
quantum statistical characteristics of photonic states. The interaction in
phase space can alter the photon-number distribution. In 2015, Zhang et al.
considered the phase-space WVA scheme with the schematic diagram depicted in
Fig. 8 (a) Zhang  _et al._ (2015). The which-path information (up and down) of
single photons in the Mach-Zehnder interferometer is referred to as the QS.
The pre-selected state of the QS is parametrized as
$|\psi_{i}\rangle=\cos(\theta_{i}/2)|0\rangle+\sin(\theta_{i}/2)|1\rangle$,
where $|0\rangle$ and $|1\rangle$ denote the number of photons at the downward
arm participating in the interaction. For simplicity, we consider a photonic
state $|\Phi_{p}\rangle$ as the MS. The cross-phase modulation (XPM) between
the QS and the MS is described by the Hamiltonian
$\hat{H}_{p}=g\delta(t-t_{0})\hat{C}\hat{n}_{p}$, where
$\hat{C}=|1\rangle\langle 1|$ is the observable of the QS and $\hat{n}_{p}$ is
the photon-number operator of the MS. The small interaction strength $g$
specifies the third-order nonlinear coefficient, which is the unknown
parameter to be measured. The interaction produces a weakly entangled joint
state, given by

$|\Psi_{jt}^{(p)}\rangle=\cos(\theta_{i}/2)|0\rangle|\Phi_{p}\rangle+\sin(\theta_{i}/2)|1\rangle
e^{ig\hat{n}_{p}}|\Phi_{p}\rangle.$ (32)

According to Eq. (24), the QFI of the joint state about $g$ is given by
$Q_{jt}(|\Phi_{p}\rangle)=4[\langle\hat{C}^{2}\rangle_{s}\text{Var}_{p}(\hat{n})+\text{Var}_{s}(\hat{C})\langle\hat{n}\rangle_{p}^{2}]$.
By setting $\theta_{i}=\pi/2$, the QFI can be simplified as
$Q_{jt}(|\Phi_{p}\rangle)=2\text{Var}_{m}(\hat{n})+\langle\hat{n}\rangle_{m}^{2}$.
Therefore, the photon-number statistics of the MS affects how the QFI scales
with the mean photon numbers. This feature offers possible approaches to
surpass the SQL or achieve the HL with classical photonic states in phase-
space WVA.

In 2015, Zhang et al. proposed a scheme to use the coherent state
$|\alpha\rangle$ as the meter state Zhang  _et al._ (2015). When $|\alpha|\gg
1$, the term associated with the mean photon number
$\langle\hat{n}\rangle_{m}^{2}$ dominates $Q_{jt}$ and exhibits the
Heisenberg-scaling of the ultimate precision. In the work, a comprehensive
analysis of the FI distribution during the post-selection process is
established. As shown in Fig. 6, the post-selection divides the $Q_{jt}$ into
three components $Q_{jt}=p_{f}Q_{f}+p_{r}Q_{r}+F_{p}$. In weak-measurement
regime ($g\ll 1$), we can approximate the results as follows:
$p_{f}Q_{f}\approx(1-\epsilon^{2}/4)N$, $p_{r}Q_{r}\approx\epsilon^{2}N/4$,
$F_{p}\approx N^{2}$, and the success probability
$p_{f}=[1-\cos(2gN+\epsilon)]/2$. These results suggest that the Heisenberg-
scaling precision lying in $F_{p}$ can be achieved by observing the
probability distribution $\\{p_{f},p_{r}\\}$. Jordan et. al. also provided a
simple interpretation of the Heisenberg-scaling precision and an approach to
extracting the information from the perspective of state discrimination Jordan
_et al._ (2015). As Fig. 8 (f) shows, Chen et. al. have achieved practical
Heisenberg-scaling precision, with $\Delta g\approx 1.2/n$ up to $10^{-10}$
rad, with the mean photon number over $10^{6}$.

Next, we consider the metrological contribution of the variance term
$\text{Var}_{p}(\hat{n})$ to the total QFI $Q_{jt}$. With a coherent MS
$|\alpha\rangle$, the measurement of the post-selected MS obtains the FI
$p_{f}Q_{f}\approx(1-\epsilon^{2}/4)N$, only achieving the ultimate precision
of $g$ in SQL. However, the variance $\text{Var}_{p}(\hat{n})$ can be easily
increased through classical operations. Moreover, both the phase-estimation
experiment using white light and the theoretical derivation by Kedem have
indicated that the variance of incoherent MS or due to classical fluctuations
equivalently contributes to the amplification factors multiplied by the
imaginary weak values Xu  _et al._ (2013); Kedem (2012). Motivated by this
observation, Chen et al. utilized a mixture of coherent states as the initial
MS: $\rho_{p}=\sum_{j}p_{j}|\alpha_{j}\rangle\langle\alpha_{j}|$, with the
phase-space distribution shown in Fig. 8 (e) Chen  _et al._ (2018a). Given the
pre- and post-selected states $|\psi_{i}\rangle$ and $|\psi_{f}\rangle$, along
with the interaction Hamiltonian $\hat{H}_{p}$, the final MS evolves according
to

$\rho_{p}^{\prime}=\frac{1}{\sqrt{\mathcal{N}_{p}}}\langle\psi_{f}|\exp(-ig\hat{C}\hat{n}_{p})|\psi_{i}\rangle\langle\psi_{i}|\otimes\rho_{p}\exp(ig\hat{C}\hat{n}_{p})|\psi_{f}\rangle,$
(33)

where the normalization coefficient $\mathcal{N}_{p}$ represents the success
probability of post-selection. Thus, the change in the average photon number
of the final MS can be approximated as:

$\text{Tr}(\rho_{p}^{\prime}\hat{n}_{p})-\text{Tr}(\rho_{p}\hat{n}_{p})\approx
2g\text{Im}\langle\hat{C}\rangle_{w}\text{Var}_{p}(\hat{n}_{p}),$ (34)

where $\langle\hat{C}\rangle_{w}$ denotes the weak value and
$\text{Var}_{p}(\hat{n}_{p})=\text{Tr}(\rho_{p}\hat{n}_{p}^{2})-(\text{Tr}(\rho_{p}\hat{n}_{p}))^{2}$.
The overall classical FI about $g$ in phase-space WVA can be obtained by
substituting the photon-number statistics $f(n,g)=\langle
n|\rho_{p}^{\prime}|n\rangle$ into Eq. (16):

$F_{\text{WVA}}^{(ps)}=4\mathcal{N}_{p}\text{Im}\langle\hat{C}\rangle_{w}^{2}\text{Var}_{p}(\hat{n}_{p}).$
(35)

In their experiment, the pre- and post-selected states were chosen as
$|\psi_{i}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and
$|\psi_{i}\rangle=(|1\rangle-e^{-i\epsilon}|0\rangle)/\sqrt{2}$ with
$\epsilon=0.1$. The mixture of coherent states maintains
$\text{Var}_{p}(\hat{n}_{p})=N^{2}/4$ to achieve Heisenberg-scaling precision
for values of $N$ ranging from $3\times 10^{4}$ to $5\times 10^{6}$. According
to the CRB, the ultimate precision of $g$ for $\nu=2.2\times 10^{5}$
measurements is given by $\text{Var}(g)=1/(\nu F_{\text{WVA}}^{(ps)})\approx
4\times 10^{-7}/N^{2}$. As depicted in Fig. 8 (d), the experimental precision
of $g$ maintains Heisenberg-scaling precision up to $N=10^{5}$ photons and
achieves an unprecedented precision $\Delta g=\sqrt{\text{Var}(g)}=3.6\times
10^{-10}\text{rad}$.

##  V 5\. Advantages of weak-value amplification in the presence of practical
imperfections

In section 3 and 4, we have reviewed how quantum noise determines the ultimate
precision of WVA and introduced approaches to surpass the SQL. However,
experimental imperfections such as technical noise and nonideal detectors are
inevitable factors that prevent measurement protocols from achieving the
quantum-limited precision in practice. The amplification effect resulting from
post-selection in WVA has been shown to be beneficial in resisting various
types of technical noise. This noise tolerance has contributed to
unprecedented measurement sensitivity in many experiments without requiring
special noise-isolation operations. In this section, we review recent
discussions about the metrological advantages of WVA over CM with the
ubiquitous noise in optical systems. We try to construct a unified noise model
for each case and analyze the metrics of SNR and FI. Our aim is to elucidate
the essential mechanism that offers the WVA potential advantages under certain
conditions.

###  V.1 5.1 Weak-value amplification against time-correlated noise

Figure 9: Comparison between WVA and CM in the presence of additive noise. (a)
Measuring the displacement of a mirror $g$ with a random displacement $x_{k}$
in CM. (b)Measuring the displacement of a mirror $g$ inside the Sagnac
interferometer with a random displacement $x_{k}$ in real WVA. (c) The data
set used for estimating the parameter in CM and WVA is schematically
illustrated.

In 2011, Feizpour, Xing and Steinberg theoretically demonstrated the
advantages of WVA over the CM in the presence of additive, time-correlated,
Gaussian random noise with the SNR metric Feizpour  _et al._ (2011). In 2014,
Ferrie and Combes employed the FI metric to compare the precision of WVA and
CM with the time-correlated noise. They found that the post-selection in WVA
discarded the metrological information. Thus, WVA is suboptimal in precision
compared to the CM with the optimal estimator Ferrie and Combes (2014).
Subsequently, Jordan et al. compared the FI metric with the SNR metric from a
practical standpoint Jordan  _et al._ (2014). In 2017, Sinclair et al.
conducted comprehensive analysis on realistic models of time-correlated noise
to clarify the conflicting claims about the optimal estimator and the
usefulness of WVA Sinclair  _et al._ (2017). We adopt a commonly used scenario
for time-correlated noise to summarize the previous findings and highlight the
metrological advantages of WVA.

Fig. 9 (a) and (b) illustrate practical setups to measure the displacement of
a mirror using CM and WVA, respectively. The polarization and transverse
spatial modes of photons are referred to as the QS and the MS, respectively.
The additive time-correlated technical noise is modelled by the random
displacement of the first mirror. In CM, measurements are typically repeated
sequentially under the same circumstances to obtain a data set $\\{s_{k}\\}$,
where $s_{k}=g+x_{k}$ refers to the $k$th measurement result and $x_{k}$
represents the zero-mean Gaussian noise term described by a covariance matrix
$C_{k,l}=\langle x_{k}x_{l}\rangle$. As an experimentally relevant example,
the covariance matrix can be modelled by

$C_{k,l}=a\delta_{k,l}+c\exp(-|k-l|\delta t/\tau),$ (36)

where $a\delta_{k,l}$ represents a white-noise floor. In the time-correlated
term $c\exp(-|k-l|\delta t/\tau)$, $\delta t$ and $\tau$ denote the time
interval between two measurements and the correlation time of noise,
respectively. We consider two limiting cases: when $\tau/\delta t\ll 1$,
$C_{k,l}=(a+c)\delta_{k,l}$ represents the white-noise limit; in the opposite
case, as $\tau/\delta t\gg 1$, $C_{k,l}=a\delta_{k,l}+c$ indicates the slow-
noise limit.

Table 1: The metrological information of WVA and CM with the SNR and FI metrics in different noise conditions Metrological Information | General |  White-noise limit ($\tau\ll\delta t$) |  Slow-noise limit 1 ($\delta t\ll\tau$, $p_{f}\ll\delta t/\tau$) |  Slow-noise limit 2 ($\delta t\ll\tau$, $p_{f}\gg\delta t/\tau$)
---|---|---|---|---
$\mathcal{I}_{\text{CM}}$ | $\frac{1}{V}$ | $\frac{N}{a+c}$ | $\frac{N}{a+Nc}$
$\mathcal{F}_{\text{CM}}$ | $\sum_{i,j}^{N}[C^{-1}]_{i,j}$ | $\frac{N}{a+c}$ | $\frac{N}{a}$
$\mathcal{I}_{\text{WVA}}$ | $\frac{1}{V^{\prime}}$ | $\frac{N}{a+c}$ | $\frac{N}{a+c}$ | $\langle\hat{A}\rangle_{w}^{2}\frac{p_{f}N}{a+p_{f}Nc}$
$\mathcal{F}_{\text{WVA}}$ | $\sum_{i,j}^{Np_{f}}[C^{\prime-1}]_{i,j}$ | $\frac{N}{a+c}$ | $\frac{N}{a+c}$ | $\frac{N}{a}$

In the following, we first examine the metrological performance of CM using
both SNR and FI metrics. As discussed in section 2.1, the SNR metric
corresponds to the estimator of AMR, denoted as
$\hat{g}_{\text{AMR}}^{(\text{CM})}=\sum_{k=1}^{N}s_{k}/N$. The corresponding
variance of AMR estimator is given by $V=\sum_{k,l}^{N}\langle
s_{k}s_{l}\rangle/N^{2}$. Analogous to the FI, we define the metrological
information of SNR as the inverse of the variance of AMR estimator, i.e.,
$\mathcal{I}_{\text{CM}}=1/V$. Specifically, in the white-noise limit ($\delta
t\gg\tau$), $\mathcal{I}_{\text{CM}}=N/(a+c)$. In contrast, in the slow-noise
limit ($\delta t\ll\tau$), $\mathcal{I}_{\text{CM}}=N/(a+Nc)$. As $N$ tends to
infinity, the information $\mathcal{I}_{\text{CM}}$ saturates at $1/c$ for
typical positive correlations ($c>0$). This result is equivalent to an unknown
offset ($c$) which cannot be reduced by large number of measurements.

To facilitate the derivation of the FI of $g$ in CM, we transform the
measurement results $\\{s_{k}\\}$ into the vector form, denoted as $\vec{s}$.
Consequently, we obtain the joint probability distribution of the measurement
results $\vec{s}$ as a high-dimensional Gaussian distribution:

$P_{s}(\vec{s})=\frac{1}{\sqrt{(2\pi)^{N}\text{det}C}}\exp\Big{[}-\frac{(\vec{s}-\mu)C^{-1}(\vec{s}-\mu)}{2}\Big{]},$
(37)

where $C^{-1}$ denotes the inverse of the covariance matrix $C$. According to
Eq. (16), the FI regarding the parameter $g$ is given by

$\mathcal{F}_{\text{CM}}=\sum_{k,l=1}^{N}[C^{-1}]_{k,l}.$ (38)

Consequently, the ultimate precision determined by the CRB can be
asymptotically achieved with the maximum likelihood estimator

$\hat{g}_{\text{MLE}}^{(\text{CM})}=\frac{\sum_{k,l=1}^{N}[C^{-1}]_{k,l}s_{k}}{\sum_{k^{\prime},l^{\prime}=1}^{N}[C^{-1}]_{k^{\prime}l^{\prime}}}=\sum_{k}f_{k}s_{k}.$
(39)

Here, this estimator can be interpreted as the weighted average of each
measurement result $s_{j}$ with the factor
$f_{k}=\sum_{l=1}^{N}[C^{-1}]_{k,l}/\sum_{k^{\prime},l^{\prime}=1}^{N}[C^{-1}]_{k^{\prime}l^{\prime}}$.
In comparison, the SNR method treats every data point equally with a weighting
factor $f_{k}=1/N$. In the white-noise limit ($\delta t\gg\tau$), the maximum
likelihood estimator $\hat{g}_{\text{MLE}}^{(\text{CM})}$ is equivalent to
$\hat{g}_{\text{AMR}}^{(\text{CM})}$, resulting in
$F_{\text{CM}}=\mathcal{I}_{\text{CM}}=N/(a+c)$. In the slow-noise limit
($\delta t\ll\tau$), the FI is approximated as
$F_{\text{CM}}=\sum_{k,l}C^{-1}_{k,l}\approx\sum_{k,l}[(a+cN)\delta_{kl}-c]/(a^{2}+Nac)\approx
N/a$ for large $N$.

We denote the measurement results in WVA as
$s_{k}^{\prime}=g\langle\hat{A}\rangle_{w}+x_{k}^{\prime}$, where the
parameter $g$ is amplified by the weak value $\langle\hat{A}\rangle_{w}$ and
$x_{k}^{\prime}$ represents the post-selected noise. Given the success
probability of post-selection in WVA as $p_{f}$, the size of the post-selected
signals is reduced to $N^{\prime}=Np_{f}$. Assuming a real weak value
$\langle\hat{A}\rangle_{w}$, the optimal precision of real WVA satisfies the
trade-off relation $p_{f}=\langle\hat{A}\rangle_{w}^{2}$. We denote the
variance of the post-selected noise as
$V^{\prime}=(1/{N^{\prime}}^{2})\sum_{k,l}^{N^{\prime}}\langle
x_{k}^{\prime}x_{l}^{\prime}\rangle$. Therefore, the metrological information
of the parameter $g$ in WVA can be generally expressed by
$\mathcal{I}_{\text{WVA}}=\langle\hat{A}\rangle_{w}^{2}/V^{\prime}$. In the
white-noise limit ($\delta t\gg\tau$), the information simplifies to
$\mathcal{I}_{\text{WVA}}=N/(a+c)$, which is equal to
$\mathcal{I}_{\text{CM}}$. Hence, WVA does not exhibit advantages over CM in
the presence of additional white noise. As post-selection significantly alters
the time correlation of the noise, we further divide the slow-noise limit into
two cases based on the relationship between the post-selection probability
$p_{f}$ and the ratio $\delta t/\tau$. When $p_{f}\ll\delta t/\tau$, the time
correlation of post-selected measurements is close to white noise. In this
situation, the metrological information is approximated as
$\mathcal{I}_{\text{WVA}}=N/(a+c)$. In contrast, when $p_{f}\gg\delta t/\tau$,
the noise in the post-selected results is still fully time-correlated, leading
to the metrological information
$\mathcal{I}_{\text{WVA}}=\langle\hat{A}\rangle_{w}^{2}p_{f}N/(a+p_{f}Nc)\approx
1/(p_{f}c)$ for a large $N$. Compared to the metrological information of CM,
the amplification effect in WVA improves the information from
$\mathcal{I}_{\text{CM}}=1/c$ to $\mathcal{I}_{\text{WVA}}=1/(p_{f}c)$, which
aligns with the findings in Ref. Feizpour  _et al._ (2011).

In the table 1, we have summarized the metrological information of WVA and CM
with both the SNR and FI metrics in various noise situations. In white-noise
(uncorrelated) limit, the metrological information of AMR estimator is
identical to the FI. The CM and WVA have identical FI, demonstrating an
equivalent performance. However, in slow-noise (fully correlated) limit, the
FI of the CM monotonically increases with a larger $N$ while the information
of AMR estimator is bounded by $1/c$. Extracting the FI comes at the cost of
knowing the covariance matrix of the correlated noise and performing MLE to
estimate the parameter of interest. Due to the post-selection in WVA, the
slow-noise condition can be further divided into two cases according to
whether the noise of successfully post-selected data is correlated or not. In
slow-noise limit 1, post-selection in WVA eliminates the correlation of noise.
Thus, AMR estimator is effective to achieve larger metrological information
than that of the CM. In slow-noise limit 2, though post-selected data are
still correlated, the amplification effect of the parameter is beneficial to
mitigate the unknown off-set $c$. For a large $N$, the metrological
information of AMR estimator $\mathcal{I}_{\text{WVA}}$ is $1/p_{f}$ times
larger than $\mathcal{I}_{\text{CM}}$.

###  V.2 5.2 Imaginary weak-value amplification against technical noise

In the imaginary WVA, the parameter is measured in the Fourier conjugate
space, which provides the potential to mitigate the technical noise. In 2012,
Kedem demonstrated that a specific type of technical noise is beneficial to
improve the SNR via imaginary WVA Kedem (2012). In 2014, Jordan et al. focused
on the beam-deflection measurements in the presence of combined noise types,
such as detector transverse jitter, angular beam jitter of the initial MS, and
turbulence Jordan  _et al._ (2014). Since both the quantum measurements and
the AMR estimator are optimal to estimate the parameter in these cases, the
SNR and FI metrics are equivalent to evaluate the precision. In this
subsection, we summarize the the metrological advantages of imaginary WVA in
the measurement of beam deflection using FI metric.

Figure 10: Comparison between the imaginary WVA and CM in the presence of
dynamical noise. (a) Measurement of the mirror deflection $k$ in CM. (b)
Measurement of the mirror deflection inside the Sagnac interferometer $k$ in
the imaginary WVA. The noise (i) and (ii) is simulated by the random
deflection and displacement $b_{0}$ and $q_{0}$ of the front mirror,
respectively. The noise (iii) is modelled by the random lateral displacement
$d_{0}$ of detector. BS, beam splitter; SBC, Soleil-Babinet compensator, PSD,
position-sensitive detector.

Fig. 10 (a) depicts the measurement of the mirror’s deflection $k$ in CM. The
implementation of the imaginary WVA to measure the mirror’s deflection inside
the Sagnac interferometer is shown in Fig. 10 (b). To investigate the effects
of dynamical noise in the CM and WVA, we summarize three cases: (i) the noise
of the initial MS is modelled by the deflection $b_{0}$ of the first mirror,
in which $b_{0}$ follows a Gaussian distribution with zero mean and variance
$B_{0}^{2}$; (ii) the noise of the initial MS is modelled by the displacement
$q_{0}$ of the first mirror, in which $q_{0}$ follows a Gaussian distribution
with zero mean and variance $Q_{0}^{2}$; (iii) the jitter noise of the
detectors is modelled by the lateral displacement $d_{0}$, which follows a
Gaussian distribution with zero mean and variance $D_{0}^{2}$.

In case (i), the initial Gaussian MS $|\Phi\rangle$ evolves sequentially with
the following operations: a random momentum kick
$\hat{U}_{b_{0}}=\exp(ib_{0}\hat{x})$, free propagation over a distance
$l_{1}$: $\hat{U}_{l_{1}}=\exp(-i\hat{p}^{2}l_{1}/2k_{0})$, momentum kick
$\hat{U}_{k}=\exp(ik\hat{x})$, quadratic phase modulation by a lens
$\hat{U}_{f}=\exp(-ik_{0}\hat{x}^{2}/2f)$, and free propagation over a
distance $l_{2}$: $\hat{U}_{l2}$, leading to the final spatial distribution on
the detector

$P_{d}(x,p_{0})=|\langle
x|\hat{U}_{l_{2}}\hat{U}_{f}\hat{U}_{k}\hat{U}_{l_{1}}\hat{U}_{p_{0}}|\Phi\rangle|^{2},$
(40)

where $k_{0}$ is the wave number of the light. After averaging the random
variable $b_{0}$, we obtain a Gaussian distribution with an average $\langle
x\rangle=-kf/k_{0}$ and a variance $\langle(x-\langle
x\rangle)^{2}\rangle=f^{2}[1/(2\sigma k_{0})^{2}+P_{0}^{2}/k_{0}^{2}]$. Thus,
the FI about the parameter $k$ is given by

$F_{\text{CM}}=\frac{4N\sigma^{2}}{1+(2\sigma B_{0})^{2}},$ (41)

where $N$ is the number of photons detected and $\sigma$ is the width of the
Gaussian MS. Since the angular jitter noise $b_{0}$ directly adds to the
signal deflection $k$, leading to the reduction of the FI by a factor
$1/[1+(2\sigma B_{0})^{2}]$.

In the imaginary WVA, the QS is encoded on the which-way information of the
Sagnac interferometer with $|0\rangle$ ($|1\rangle$) referring to the
clockwise (counterclockwise) photonic states. Given the observable of the QS
$\hat{W}=|0\rangle\langle 0|-|1\rangle\langle 1|$, the momentum kick operator
$\hat{U}_{k}$ in the CM is substituted by the weak coupling unitary process of
the imaginary WVA represented as $\hat{U}_{kW}=\exp(ik\hat{W}\hat{x})$. The
pre- and post-selected states are prepared as
$|\psi_{i}\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}$ and
$|\psi_{f}\rangle=(|0\rangle-i|1\rangle)/\sqrt{2}$, respectively. The relative
phase shift that is induced by a SBC inside the Sagnac interferometer is
represented by the operator $\hat{U}_{\phi}=\exp(i\phi\hat{W}/2)$.
Consequently, the resulting probability distribution at the detectors is given
by

$P_{\text{WVA}}(x,q)=|\langle
x|\langle\psi_{f}|\hat{U}_{l_{2}}\hat{U}_{kW}\hat{U}_{\phi}\hat{U}_{l_{1}}\hat{U}_{b_{0}}|\psi_{i}\rangle|\Phi\rangle|^{2}.$
(42)

This distribution reveals an average shift of the post-selected MS

$\langle
x\rangle\approx\frac{4k\sigma^{2}}{\phi}+\frac{kl_{1}(l_{1}+l_{2})}{k_{0}^{2}\phi\sigma^{2}}+\frac{b_{0}(l_{1}+l_{2})}{k_{0}},$
(43)

and the variance is given by

$\langle(x-\langle
x\rangle)^{2}\rangle=\sigma^{2}+\Big{(}\frac{l_{1}+l_{2}}{2k_{0}\sigma}\Big{)}^{2}.$
(44)

Assuming that the distribution can be approximated as a Gaussian with the
aforementioned mean and variance, the FI about the parameter $k$ in the
imaginary WVA is calculated as

$F_{\text{WVA}}=\frac{4N\sigma^{2}}{1+(\frac{l_{1}+l_{2}}{2k_{0}\sigma^{2}})^{2}[1+(2\sigma
B_{0})^{2}]}.$ (45)

From the average shift of the MS in Eq. (43), the angular jitter noise $p_{0}$
results in a small correction to the signal deflection due to the beam
diffraction. Thus, compared to the FI of the CM in Eq. (41), the reduction of
the FI caused by the noise $p_{0}$ is mitigated by a diffraction factor of
$(l_{1}+l_{2})/(2k_{0}\sigma^{2})$ in the imaginary WVA. The factor is
typically much smaller than 1, indicating the metrological advantages of the
imaginary WVA over the CM in the presence of angular jitter noise.

In case (ii), a random displacement $q_{0}$ occurs at the first mirror leading
to the initial MS $|\Phi(q-q_{0})\rangle$ with the probability
$\text{Pr}(q_{0})=1/[(2\pi)^{1/2}Q_{0}]\exp(-q_{0}^{2}/(2Q_{0}^{2}))$, where
$Q_{0}^{2}$ is the variance of the noise. The interaction process of measuring
the deflection is still $\hat{U}_{kW}$. The relative phase shift
$\hat{U}_{\phi}$ can be combined to the pre- and post-selected states
resulting in $|\psi_{i}\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}$ and
$|\psi_{f}\rangle=(e^{-i\phi/2}|0\rangle-ie^{i\phi/2}|1\rangle)/\sqrt{2}$.
Thus, the imaginary weak value is given by
$\langle\hat{A}\rangle_{w}=-i2/\phi$. After the interaction and post-
selection, the post-selected MS before the detectors is given by

$|\Phi\rangle_{f}=\mathcal{N}_{q_{0}}(\frac{2\sigma^{2}}{\pi})^{\frac{1}{4}}\exp[-\sigma^{2}(p-q_{0})^{2}-2kp/\phi],$
(46)

where $\mathcal{N}_{q_{0}}$ is the normalization factor. Measurements of the
observable $\hat{P}=|p\rangle\langle p|$ in $|\Phi\rangle_{f}$ yield the
average shift and the variance

$\displaystyle\langle\hat{P}\rangle_{f}$ $\displaystyle=$ $\displaystyle
q_{0}-\frac{k}{\phi\sigma^{2}}{}$ $\displaystyle\langle\hat{P}^{2}\rangle_{f}$
$\displaystyle=$
$\displaystyle\frac{1}{4\sigma^{2}}+(q_{0}-\frac{k}{\phi\sigma^{2}})^{2}.$
(47)

The success probability of post-selection is given by

$\text{Pr}(|\Phi\rangle_{f}|q_{0})=|\langle\psi_{f}|\psi_{i}\rangle|^{2}\mathcal{N}_{q_{0}}^{-2}.$
(48)

When the post-selected MS $|\Phi\rangle_{f}$ is successfully obtained, the
displacement noise follows the distribution:

$\displaystyle\text{Pr}(Q_{0}||\Phi\rangle_{f})$ $\displaystyle=$
$\displaystyle\frac{\text{Pr}(q_{0})\text{Pr}(|\Phi\rangle|q_{0})}{\text{Pr}(|\Phi\rangle)}{}$
(49) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}Q_{0}}\exp[-\frac{q_{0}+2kQ_{0}^{2}/\phi}{2Q_{0}^{2}}].$

By averaging the noise term $q_{0}$, the final average shift and the variance
of the post-selected MS on the detectors are given by

$\displaystyle\overline{\langle\hat{P}\rangle_{f}}$ $\displaystyle=$
$\displaystyle\frac{k}{2\phi}(\frac{1}{(2\sigma)^{2}}+2Q_{0}^{2}){}$
$\displaystyle\overline{\langle\hat{P}^{2}\rangle_{f}}-\overline{\langle\hat{P}\rangle_{f}}^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{\sigma^{2}}+Q_{0}^{2}.$ (50)

Therefore, the FI about the parameter $k$ in WVA is given by

$F_{\text{WVA}}=4N(\sigma^{2}+Q_{0}^{2}).$ (51)

We can find that the incoherent spread in the Fourier space of the initial MS
is beneficial for the imaginary WVA to improve the precision.

In case (iii), the transverse position of detectors shifts with a random
variable $d_{0}$, which subjects to zeros-mean Gaussian distribution with the
variance $D_{0}^{2}$. In the CM, the transverse detector jitter simply
increases the average beam waist to $\sqrt{\sigma^{2}+D_{0}^{2}}$, leading to
the new FI

$F_{\text{CM}}=\frac{N(f/k_{0})^{2}}{\sigma_{f}^{2}+D_{0}^{2}}=\frac{4N\sigma^{2}}{1+(\frac{2k_{0}\sigma
D_{0}}{f})^{2}},$ (52)

where $\sigma_{f}$ is the focused beam waist given by
$\sigma_{f}=f/(2k_{0}\sigma)$. The result indicates that the noise term
$(\frac{2k_{0}\sigma D_{0}}{f})^{2}$ in the denominator can be suppressed by a
large focal length $f\geq 2k_{0}\sigma D_{0}$. This is equivalent to generate
a large focal spot $\sigma_{f}$ as well as induce a large displacement
$fk/k_{0}$, which intuitively mitigates the reduction of FI due to the
detector jitter.

In the imaginary WVA, the detector jitter results in the new FI of

$F_{\text{WVA}}=\frac{4N\sigma^{2}}{1+D_{0}^{2}/\sigma^{2}}.$ (53)

Therefore, a large initial beam waist $\sigma$ not only increases the original
FI but also helps to suppresses noise. Therefore, the practical advantages of
WVA over the CM mainly lie in two aspects: (i) The widely used PSD such as
split detectors have a separation between two photoelectric detectors, which
establishes a minimum beam diameter and consequently restricts its propagation
through horizontal turbulence in CM. (ii) In WVA, the PSD can be placed
immediately after the BS to shorten the optical path length, which mitigates
the possible noise caused by the air turbulence.

###  V.3 5.3 Weak-value amplification with imperfect detectors

In the preceding two subsections, we have showcased the metrological
advantages of WVA in the presence of dynamical noise. Furthermore, the
imperfections of photodetectors also play a crucial role in influencing the
measurement precision in an optical system. Pixelation, classical electrical
noise and saturation effect are common imperfections found in an arrayed
photodetectors such as change-coupled devices (CCDs) and complementary metal
oxide semiconductor (CMOS). In the following, we focus on discussing whether
WVA can provide advantages over the CM in the presence of these imperfections.

Figure 11: WVA with imperfect detectors. (a) The basic procedure of the
standard WVA scheme and the post-selected MS is detected by arrayed detectors
with the pixel size $r$ Knee and Gauger (2014). (b) The experimental setup to
implement the real WVA scheme to measure a small displacement of a mirror
inside the Sagnac interferometer Xu  _et al._ (2020). (c) Numerically obtained
relationships between relative information $\alpha$ and inverse resolution $R$
Knee and Gauger (2014). (d) and (e) illustrate the response matrix of the CCD
before and after the saturation, respectively Xu  _et al._ (2020). (f)
Comparison of the beam profiles between the CM and the WVA in the presence of
detector saturation Harris  _et al._ (2017). (g) Comparison between the
precision of the CM and that of the WVA Xu  _et al._ (2020). (h) The precision
of the real WVA with different success probability of post-selection Xu  _et
al._ (2020).

In 2014, George et al. conducted the initial analysis on how the pixelation
affects the ultimate precision to estimate a small lateral displacement of a
Gaussian beam Knee and Gauger (2014). As depicted in Fig. 11 (a), the
pixelation of arrayed photodetectors transforms the probability density
function $P(q)$ into a discrete probability distribution, given by

$\text{Pr}(n)=\int_{(n-1/2)r}^{(n+1/2)r}P(q)dx,$ (54)

where $r$ represents the width of the pixel. The pixel number $n$ is defined
as the Integer part of $q/r$, denoted as $n=\lfloor q/r\rceil$. Generally, we
assume that the wave function is shifted by $\nu g$, leading to the final
probability distribution $P(q-\nu g)$. When the pixel size tends towards
infinitely small, the coefficient $\nu$ can be separated from the pixelation
effect in the FI about the parameter $g$

$F[Pr(n^{\prime})]=\nu^{2}F[Pr(n)],$ (55)

where $n^{\prime}=\lfloor(q-\nu g)/r\rceil$. However, as the pixel size
increases, the misalignment between the centroid of $\text{Pr}(n^{\prime})$
and the pixel boundaries become significant. This relative alignment is
described by $h=\nu g-\mu\text{mod}r$ with $\mu$ a controllable quantity.
Consequently, the FI about $g$ with pixelated detectors can be expressed as

$F_{g}[\text{Pr}(\lfloor(q-\nu
g)/r\rceil)]=\nu^{2}F_{g}[(\lfloor(q-h)/r\rceil)]$ (56)

As the relation between $h$ and $\nu$ is somewhat arbitrary, a larger $\nu$
does not necessarily imply a better choice of $h$ to increase $F_{g}$. In
comparing the FI between the real WVA and the CM, we take the identical $h$,
resulting in

$\frac{p_{f}F_{g}[\text{Pr}(\lfloor(q-\text{Re}(A_{w})g)/r\rceil)]}{F_{g}[\text{Pr}(\lfloor(q-\lambda_{\max}g)/r\rceil)]}=\frac{Re(\langle\psi_{f}|\hat{A}|\psi_{i}\rangle)^{2}}{\lambda_{\max}^{2}}\leq
1.$ (57)

Therefore, the pixelation effect is cancelled for both the real WVA and the
CM. The amplification effect cannot compensate for the reduction of
information due to pixelation. Similarly, the ratio of FI between the
imaginary WVA and CM is given by

$\frac{p_{f}F_{g}[\text{Pr}(\lfloor(p-\text{Im}(A_{w})g)/r\rceil)]}{F_{g}[\text{Pr}(\lfloor(q-\lambda_{\max}g)/r\rceil)]}=\frac{\alpha_{p}}{\alpha_{q}},$
(58)

where the coefficients $\alpha_{p}$ and $\alpha_{q}$ represent the ratio of FI
between the pixlated detection and the ideal detection. The numerical
relationship between the information ratio $\alpha$ and $h$ is illustrated in
Fig. 11 (c). When $R$ approaches $0$, the information ratio tends towards $1$
which is quite robust against the misalignment $h$. However, as $R$ becomes
larger, $h=0.5$ is the best choice for the detection. Since $\alpha_{p}$ and
$\alpha_{q}$ approximately approach unity to the first order of the relative
resolution $R=r/\Delta q$, we have $\alpha_{p}/\alpha_{q}\approx 1$. Thus, the
imaginary WVA cannot offer metrological advantages over the CM. This
conclusion appears to contradict the findings of Brunner et al., where the
imaginary WVA combined with frequency-domain detection has the potential to
outperform the CM in measuring small longitudinal phase shifts. In Brunner’s
paper, the main experimental constraints are the alignment errors of the
equipment and the resolution limit of the detectors. We need to emphasize that
the resolution limit is not equivalent to the pixel size of the arrayed
detectors. It can be interpreted as the ultimate sensitivity of detecting the
shift, incorporating the effect of pixelation and other photo-electrical
noise.

When $R$ tends to infinity, it corresponds to the case of split detectors,
which is popular in monitoring the lateral displacement of a light beam. Once
the optimal misalignment parameter $h=0.5$ is taken, only about a third
information is lost during the detection. In 2015, Knee et. al. have studied
both the SNR and FI metrics in estimating a Gaussian beam displacement with a
split detector Knee and Munro (2015). The use of the split detectors is
optimized to obtain the most FI. The study implies that the amplification
effect cannot bring improvement of FI with split detection in the absence of
other technical noise, the real WVA is considered to be strictly worse than
the CM. Nevertheless, the split detection may maintain the robustness of WVA
scheme to certain technical noise and therefore is an effective detection
method in WVA.

By exploiting noise-robust measurement schemes, we can effectively improve the
precision of estimating an parameter in the practical situations. However, the
precision of estimation is ultimately limited by the SNL due to the quantum
fluctuations of light in the optical metrology. Although the use of quantum
resources, such as entangled and squeezed states, can push the measurement
precision to the HL, the preparation and control of these quantum states are
currently complicated, limiting their practical applications. An alternative
approach is to increase the average photon number of coherent states, which
improves the measurement precision along the SNL. In optical systems, a
frequent constraint of this approach is the saturation of photodetectors. The
distortion of detecting profiles caused by saturation may lead to the
deviation of the precision improvement from the SNL. Therefore, how to enhance
the dynamic range of the measurement systems and mitigate the information loss
due to the detector saturation is crucial in the practical metrology tasks. In
section 3.3, theoretical analysis has indicated that a small fraction of post-
selected photons in WVA are able to carry as much metrological information as
almost all the input photons. This provides WVA a potential to avoid the loss
of the information due to the detector saturation and equivalently extend the
dynamic range of detectors.

In 2017, Harris et. al. proposed the theoretical framework to compare the
measurement precision of CM and WVA by carefully considering the pixelated,
digitized, and noisy response detector with the saturation threshold Harris
_et al._ (2017). In their model, the beam profiles of CM and WVA are
illustrated in Fig. 11 (f). Their numerical results indicated that WVA shows
metrological advantages over the CM with the saturation effect in combination
with the pixel or digitization noise. In 2020, Xu et al. optimized the WVA
scheme by maximizing the FI over the probability of post-selection to fully
investigate the capability of the standard WVA in the resistance against
detector saturation Xu  _et al._ (2020). As introduced in section 3.3, the
precision of the real WVA scheme is approximately the same as that of CM and
maintains almost unchanged along with the reduction of the post-selection
probability within a certain range. Given the number of input photons $N_{j}$
and the readout $k_{j}$ on the $j$th pixel of the CCD, the response matrix
$\mathcal{k_{j}|N_{j}}$ of the photodetectors are carefully calibrated and
illustrated in Figs. 11 (d) and (e) before and after the saturation,
respectively. The readout of the pixel is bounded by the saturation threshold
$k_{s}$. Assume that the average number of photons inputting on the $j$th
pixel is $\bar{n}_{j}$, the probability distribution of readouts on the $j$th
pixel is given by

$P(k_{j}|g)=\sum_{N_{j}}\mathcal{R}(k_{j}|N_{j})P(N_{j}|\eta\bar{n}_{j},g),$
(59)

where $\eta$ is the detection efficiency of the detector and $g$ is the
coupling strength to be estimated. This expression is applicable to both the
WVA and the CM, with the difference lying in the $\bar{n}_{j}$. If the input
light is in coherent state and the statistical distribution of photon numbers
subjects to Poisson distribution, the FI about the parameter $g$ can be
simplified as

$F(g)=\sum_{j}\frac{\eta}{\bar{n}_{j}}\left(\frac{d\bar{n}_{j}}{dg}\right)^{2}\Gamma(\mathcal{R},\bar{n}_{j}),$
(60)

in which $\Gamma(\mathcal{R},\bar{n}_{j})$ can be interpreted as an equivalent
SNR at $j$th pixel. As illustrated in Fig. 11 (b), the displacement of mirror
inside the Sagnac interferometer is measured with the real WVA and the CM,
respectively. The parameter is estimated with the MLE. The experimental
results in Fig. 11 (g) has demonstrated a distinct advantage of WVA over the
CM in the presence of detector saturation. Moreover, the optimal precision WVA
over a range of success probabilities of post-selection $p_{f}$ allows to
minimize the overall detector imperfections and maximize the precision. As
Fig. 11 (h) shows, the variances of the optimal precision WVA maintains about
1.19 times larger than the SNL when the input photons saturates more and more
certain pixels.

##  VI 6\. Various modifications of weak-value amplification schemes

###  VI.1 6.1 Inverse weak-value amplification

Figure 12: (a) The relative phase caused by half-wave plate (HWP) and Soleil-
Babinet compensator (SBC) between two paths inside the Sagnac interferometer
is inversely proportional to the transverse momentum shift and can be
precisely measured by monitoring the beam profile in the dark port Starling
_et al._ (2010b). (b) Schematic diagram for the inverse WVA. (c) The relative
phase due to the mirror’s tilt in a modified Sagnac interferometer is
amplified by the tilt of another mirror in the perpendicular direction
Martínez-Rincón  _et al._ (2017a). (d) The inverse WVA is realized with phase
front tilters in an on-chip Mach-Zehnder interferometer Song  _et al._ (2021).

The inverse WVA was first experimentally studied by Starling et al. to amplify
the relative phase between two optical paths with the experimental setup
depicted in Fig. 12 (a) Starling  _et al._ (2010b). The post-selection results
in a transverse bimodal pattern at the dark port and the small phase can be
inferred from the mean of the distribution. In 2012, Kofman et al. provided a
theoretical description of the inverse WVA from the perspective of quantum
theory Kofman  _et al._ (2012). In Fig. 12 (b), we demonstrate the inverse WVA
scheme from the joint state $|\Psi_{jt}\rangle$ which originates from the
interaction between the QS pre-selected by the state
$|\psi_{i}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and a Gaussian MS
$|\Phi\rangle$ in Eq. (1) under the Hamiltonian
$\hat{H}=g\delta(t-t_{0})\hat{A}\hat{P}$. It is worth noting that in the
inverse WVA, the coupling strength $g$ is set to measure the unknown
parameters $\theta_{I}$ and $phi_{I}$ encoded in the QS. For the instance, the
post-selected state involves the unknown parameter, given by
$|\psi_{f}\rangle=\cos(\pi/4-\theta_{I}/2)|0\rangle-\sin(\pi/4-\theta_{I}/2)e^{i\phi_{I}}|1\rangle$.
The post-selection on the QS leads to the final MS $|\Phi_{f}\rangle$. The
inverse WVA works under the condition that the overlap between the pre- and
post-selection is much smaller than the coupling strength, i.e.,
$|\langle\psi_{f}|\psi_{i}\rangle|\ll g/\sigma\ll 1$. Thus, both parameters in
$|\psi_{f}\rangle$, namely $\theta_{I}$ and $\phi_{I}$ tend towards 0. The
measurement results of the final MS exhibit a bimodal pattern instead of the
Gaussian distribution in standard WVA. For simplicity, we consider two cases
where the weak value
$\langle\hat{A}\rangle_{w}=\cot(\theta_{I}/2)-i\cot(\phi_{I}/2)$ is purely
real ($\phi_{I}=0$) or imaginary ($\theta_{I}=0$). According to Eq. (12), the
approximate average shifts in the observables $\hat{P}$ and its conjugate
$\hat{Q}$ in the final MS are given by

$\langle\hat{Q}\rangle_{f}\approx\frac{4\sigma^{2}}{g[\langle\hat{A}\rangle_{w}]_{r}}=\frac{2\theta_{I}\sigma^{2}}{g}$
(61)

for the real weak value $[\langle\hat{A}\rangle_{w}]_{r}\approx 2/\theta_{I}$
and

$\langle\hat{Q}\rangle_{f}\approx\frac{2}{g[\langle\hat{A}\rangle_{w}]_{i}}=-\frac{\phi_{I}}{g}$
(62)

for the imaginary weak value
$[\langle\hat{A}\rangle_{w}]_{r}\approx-2i/\phi_{I}$. Correspondingly, the
success probabilities of post-selection for the real and imaginary weak values
are given by $p_{f}=g^{2}/(4\sigma^{2})$ and $p_{f}=\phi_{I}^{2}/4$,
respectively. It can also be proved that the inverse WVA is capable of
recovering the complete FI about the parameter $\theta_{I}$ or $\phi_{I}$
through observing the post-selected MS. This explains why the sensitivity of
phase measurement in Starling  _et al._ (2010b) is similar to balanced
homodyne detectors through monitoring only the dark port of the
interferometer.

In 2017, Martínez-Rincón et al. applied the inverse WVA to measure the tilt of
a mirror in a modified Sagnac interferometer, as shown in Fig. 12 (c)
Martínez-Rincón  _et al._ (2017a). The mirror’s tilt induces a phase
difference between the counter-propagating light, and the tilt in the
orthogonal plane couples the QS and the MS to amplify the phase. The
experiment achieved a shot-noise-limited sensitivity of 56
$\text{frad}/\sqrt{\text{Hz}}$ using 1.2 mW continuous-wave laser, displaying
excellent performance at low frequencies. In 2018, Lyons et al. demonstrated
that the inverse WVA exhibits similar robustness to specific noise sources,
such as additive Gaussian white noise and angular jitter noise, compared to
standard WVA Lyons  _et al._ (2018). In 2021, the inverse WVA was implemented
on an integrated photonic platform utilizing a multi-mode interferometer, as
shown in Fig. 12 (d) Song  _et al._ (2021). This setup showed a 7 dB
enhancement in phase measurement compared to a standard Mach-Zehnder
interferometer.

###  VI.2 6.2 Almost-balanced weak-value amplification

Figure 13: (a) The schematic of measuring ultra-small time delays of light by
joint weak measurement Strübi and Bruder (2013). (b) The schematic diagram of
the almost-balanced WVA. (c) Experimental setup for measuring small angular
velocities $\omega_{0}$ of the half-wave plate (HWP) without post-selection
Martínez-Rincón  _et al._ (2016). (d) Experimental setup for measuring the
rotation of a mirror via ABWVA Liu  _et al._ (2017). (e) Experimental setup
for measuring linear velocities of a piezo-mounted mirror Martínez-Rincón  _et
al._ (2017b).

In section 3.2, we have discussed the distribution of metrological information
during the post-selection process in WVA. Both the successful and failure
modes of the MS generally contain the FI of the coupling strength $g$. The
optimization of the standard WVA aims to concentrate all information about $g$
into the successful mode, which exhibits distinct advantages in suppressing
certain technical noise and avoiding saturation. However, the significant
reduction in signal intensity may lead to a terrible SNR due to the stray
light and electrical noise of detectors. To resolve these problems, Strubi and
Bruder proposed a joint measurement protocol that exploits the full
information of the correlations between the QS and the MS Strübi and Bruder
(2013). The joint scheme is illustrated in Fig. 13 (a) to measure a small
relative time delay $\tau$ between two paths of a Mach-Zehnder interferometer.
The QS (path DOF) and the MS (spectrum DOF $p_{0}(\omega)$) interact under the
Hamiltonian $\hat{H}=-\tau\hat{\sigma}_{y}\Omega$, where the Pauli operator
$\hat{\sigma}_{y}$ is the observable of the QS and $\hat{\Omega}$ is the
angular frequency observable of the MS with the eigenvalue $\omega$. The QS is
pre-selected by $|\psi_{i}\rangle=(|-\rangle+|+\rangle)/\sqrt{2}$ and the
post-selection is performed with two orthogonal states, written as
$|\psi_{f}^{(q)}\rangle=(|-\rangle+qe^{i\phi}|+\rangle)/\sqrt{2}$ with $q=-1$
or $1$. Subsequently, the post-selected MSs are individually detected in the
frequency domain. Taking into account the Gaussian fluctuations of the
alignment parameter $\phi$ subject to the noise kernel

$\xi(\phi,\phi^{\prime})=\frac{1}{\sqrt{2\pi}\epsilon}\exp\Big{[}-\frac{(\phi-\phi^{\prime})^{2}}{2\epsilon^{2}}\Big{]},$
(63)

the probability distribution on detectors ($q=1,-1$) is given by

$P_{q}(\omega;\tau,\phi)=\frac{1}{2}P_{0}(\omega)[1+q\exp(-\epsilon^{2}/2)\cos(\phi-\omega\tau)].$
(64)

Based on the joint probabilities, both the coupling parameter $\tau$ and the
parameter $\phi$ of the experimental setup can be estimated by maximizing the
log-likelihood function $\tau_{est},\phi_{est}=\max
l(\tau,\phi)=\max\sum_{q=\pm 1}\int d\omega f_{q}(\omega)\log
P_{q}(\omega;\tau,\phi)$. Considering that there is a zero-mean Gaussian white
noise with the variance $\Omega^{2}$ for the frequency detection, the
approximate estimation of $\tau$ in the presence of the fluctuation $\epsilon$
and detection noise $\Omega$ is

$\tau=\tau_{0}\Big{[}1+\frac{1}{2}\big{(}\frac{\epsilon}{\sin\phi}\big{)}^{2}+\frac{1}{2}\big{(}\frac{\Omega}{\Delta\omega}\big{)}^{2}\Big{]},$
(65)

where $\tau_{0}$ denotes the estimated $\tau$ without noise. The balanced
post-selection, i.e., $\sin\phi\approx 1$ minimizes the effect of
fluctuations. The detector noise $\omega$ can be effectively reduced with a
wide frequency spread $\Delta\omega$, which coincides with the analysis in
section 4.3. Compared to the standard WVA, the joint weak measurement scheme
has the advantages in removing the systematic errors and the fluctuations in
optical alignment during the parameter estimation. In 2016, Fang et al.
developed a general theory for the arbitrary post-selection in weak
measurement, which includes the standard WVA and the joint weak measurement as
two special cases Fang  _et al._ (2016). With the experimental setup depicted
in Fig. 13 (c), they demonstrated the significant improvement in precision of
joint weak measurement over the standard WVA when measuring ultra-small time
delays. The joint weak measurement exhibited robustness against misalignment
errors and imperfections in wavelength-dependent optical components.

Martínez-Rincón et al. further extended the joint weak measurement protocol
and obtained a WVA-like response by subtracting the two post-selected readouts
of detectors Martínez-Rincón  _et al._ (2016). In Fig. 13 (d), they employed
polarization DOF and Gaussian temporal pulse as the QS and the MS,
respectively, to measure the angular velocity of the rotating HWP. The
experimental results confirmed that compared to the conventional measurement
schemes, the joint weak measurement achieved improved sensitivity and
precision. This technique was termed almost-balanced WVA (ABWVA) since the
amplification effect stemmed from the differential signals of two almost-
balanced post-selected results. Fig. 13 (b) illustrates the basic procedure to
implement the ABWVA. We start from the joint state $|\Psi_{jt}\rangle$ after
the interaction between the pre-selected state
$|\psi_{i}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and the Gaussian MS
$|\Phi\rangle$ under the Hamiltonian $\hat{H}=g\hat{A}\hat{P}\delta(t-t_{0})$.
Two orthogonal states $|\psi_{f}^{(1,2)}\rangle=(|0\rangle\pm
ie^{i\epsilon}|1\rangle)/\sqrt{2}$ perform post-selection on the QS, leading
to two final MSs. Two probability distributions
$P_{1,2}(p;g)=[1\pm\sin(\epsilon+2gp)]P_{0}(p)/2$ are obtained by measuring
the final MSs, where $P_{0}(p)=|\Phi(q)|^{2}$. The distribution of the initial
MS $P_{0}(p)$ can be recovered by $\tilde{P}_{+}(p)=P_{1}+P_{2}=P_{0}(p)$,
while the difference of the distribution gives rise to

$\tilde{P}_{-}(p;g)=P_{2}-P_{1}\approx\sin(\epsilon)P(p-\frac{g}{2\sigma^{2}}\cot\epsilon).$
(66)

In ABWVA, the signal strength of $\tilde{P}_{-}(p;g)$ is linearly proportional
to $\sin(\epsilon)\approx\epsilon$, which is improved by a factor of
$4/\epsilon$ compared to the standard WVA. Thus, the ABWVA allows for larger
amplification (smaller $\epsilon$) in practical situations limited by
technical noise than the standard WVA protocol. This advantage has been
experimentally demonstrated by Liu et al. The experimental setup is shown in
Fig. 13 (d). They have measured the angular velocity of a mirror $\omega_{0}$
inside the interferemeter using both the standard WVA and ABWVA Liu  _et al._
(2017). The minimal detectable value of the parameter $\phi$ is $\sim
4\mu\text{rad}$ and $\sim 83\text{nrad}$ for the standard WVA and ABWVA,
respectively. The experimental results indicate that the maximum amplification
factors of ABWVA is 24 times larger than the optimal standard WVA. In Fig. 13
(e), Martínez-Rincón et al. demonstrated the practical advantages of ABWVA
over the standard WVA in larger SNR and longer integration time due to slow
drifts. The experiment has reported an improvement of velocity measurements
with the sensitivity of 60 fm/s using ABWVA Martínez-Rincón  _et al._ (2017b).
Importantly, the idea of balanced detection in the ABWVA protocol can also be
combined with other generalized forms of WVA schemes to further improve its
performance. The similar experimental results are obtained in Luo  _et al._
(2020b).

###  VI.3 6.3 Biased weak-value amplification

Figure 14: (a) The basic procedure for biased WVA scheme. (b) The probability
density of the post-selected MS in the CDI regime Zhang  _et al._ (2016). (c)
The experimental setup for the biased WVA to measure the magnetic strength Yin
_et al._ (2021).

In 2016, Zhang et. al. proposed a biased WVA scheme by introducing a bias
phase to the initial joint state of the QS and the MS before the standard WVA
Zhang  _et al._ (2016). This bias allows for a much larger amplification
factor than the standard WVA, with a correspondingly much lower success
probability of post-selection. Fig. 14 (a) illustrates the basic procedure of
the biased WVA scheme. The initial joint state of the pre-selected state and
the MS can be expressed as
$|\psi_{i}\rangle\otimes|\Phi\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}\otimes\int
f(\omega)d\omega|\omega\rangle$, which evolves under a pre-coupling unitary
process $\hat{U}_{pre}=\exp(-i\beta\hat{A}\hat{\Omega})$, where
$\hat{A}=|0\rangle\langle 0|-|1\rangle\langle 1|$ and $\hat{\Omega}$ are the
observables of the QS and MS, respectively. The pre-coupling leads to an
entangled state, given by

$|\Psi\rangle_{Ei}=\int
d\omega\frac{1}{\sqrt{2}}f(\omega)[e^{i\omega\beta}|0\rangle+ie^{-i\omega\beta}|1\rangle]|\omega\rangle.$
(67)

The longitude phase change (LPC) $\omega\tau$ is encoded to the entangled
initial state $|\Psi\rangle_{Ei}$ with the unitary transformation
$\hat{U}_{\tau}=\exp(-i\tau\hat{A}\hat{\Omega})$. Subsequently, the joint
state is post-selected by
$|\psi_{f}\rangle=(ie^{i\epsilon}|0\rangle+e^{-i\epsilon}|1\rangle)$ giving
the final MS

$|\Phi_{f}\rangle=\frac{i}{2\sqrt{p_{f}}}\int d\omega
f(\omega)\Big{\\{}\exp\big{[}i\omega(\beta+\tau)-i\epsilon\big{]}-\exp\big{[}-i\omega(\beta+\tau)+i\epsilon\big{]}\Big{\\}}|\omega\rangle$
(68)

with the success probability of post-selection

$p_{f}=\frac{1}{2}\Big{\\{}1-\exp(-\delta^{2}\beta^{2})\cos\big{[}2(\omega_{0}\beta+\omega_{0}\tau-\epsilon)\big{]}\Big{\\}}.$
(69)

The probability distribution of $|\Phi_{f}\rangle$ in the frequency domain is

$S(\omega)=\sin^{2}\big{[}\omega(\beta+\tau)-\epsilon\big{]}|f(\omega)|^{2}.$
(70)

Compared to the standard WVA ($\beta=0$) with the probability distribution
$S_{WVA}(\omega)\approx\epsilon^{2}|f(\omega)|^{2}$, the extra bias phase
induces the conjugate destructive interference (CDI) and reshapes the
frequency spectrum with an ultra-small $\tau$ around $\beta_{s}$ satisfying
$\omega_{0}\beta_{s}-\epsilon\approx 0$. The CDI in both time and frequency
domain is illustrated in Fig. 14 (b). The spectrum shift $\Delta\omega=\int
S(\omega)\omega d\omega/\int S(\omega)d\omega-\omega_{0}$ caused by the LPC
can be approximated as

$\Delta\omega\approx\frac{d\Delta\omega}{d\tau}|_{\beta_{s}}\tau\approx
2\frac{\omega_{0}^{2}}{\epsilon}\tau,$ (71)

with the corresponding success probability
$p_{f}\approx\delta^{2}\epsilon^{2}/(2\omega_{0}^{2})$.

Since the central frequency of light $\omega_{0}$ is much larger than its
frequency spread $\delta_{\omega}$, the amplification factor of the biased WVA