#  Gaussian Formalism:
Concrete Realization of Joint Measurement
for Heisenberg’s Uncertainty Relation for Errors

Kin-ya Oda  and Naoya Ogawa

∗Department of Information and Mathematical Sciences,
Tokyo Woman’s Christian University, Tokyo 167-8585, Japan
†Department of Complex Systems Science, Graduate School of Informatics,
Nagoya University, Nagoya 464-8601, Japan  E-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
###### Abstract

We point out that the Gaussian wave-packet formalism can serve as a concrete
realization of the joint measurement of position and momentum, which is an
essential element in understanding Heisenberg’s original philosophy of the
uncertainty principle, in line with the universal framework of error,
disturbance, and their uncertainty relations developed by Lee and Tsutsui. We
show that our joint measurement in the Gaussian phase space, being a Positive
Operator-Valued Measure (POVM) measurement, smoothly interpolates between the
projective measurements of position and momentum. We, for the first time, have
obtained the Lee-Tsutsui (LT) error and the refined Lee error for the
position-momentum measurement. We find that the LT uncertainty relation
becomes trivial, $0=0$, in the limiting case of projective measurement of
either position or momentum. Remarkably, in contrast to the LT relation, the
refined Lee uncertainty relation, which assesses errors for local
representability, provides a constant lower bound unaffected by these limits
and is invariably saturated, for a pure Gaussian initial state. The obtained
lower bound is in agreement with Heisenberg’s value.

##  1 Introduction

Heisenberg’s Uncertainty Principle, introduced in 1927, is a cornerstone of
quantum mechanics, fundamentally altering our understanding of measurement
precision and predictability at the quantum level [1]. Originally, this
principle concerned the errors and disturbances in the measurement of physical
observables. This pivotal concept was then mathematically refined by Kennard,
enhancing the principle’s precision [2], and later generalized by Robertson to
apply to any pair of observables [3]. The result of these developments is now
known as the Kennard-Robertson (KR) inequality; see also Ref. [4] for
Schrödinger’s subsequent contribution, now known as the Schrödinger
inequality. Unlike the original Heisenberg’s principle, the KR and Schrödinger
inequalities exclusively concern the fluctuations of physical observables, and
are completely decoupled from the errors and disturbances arising from the
measurements.

Building upon this foundational research, the contributions of Arthurs, Kelly,
and Goodman led to the formulation of what is now known as the AKG inequality,
significantly advancing our understanding of errors and the costs of
measurements in quantum mechanics [5, 6]. Ozawa further expanded upon these
concepts, resulting in the development of the Ozawa inequality through his
studies on error and error-disturbance relations [7, 8]. Estimation theory, as
highlighted in the works of Yuen-Lax [9] and further developed by Watanabe-
Sagawa-Ueda, culminated in the WSU inequality [10, 11], offering alternative
perspectives. These collective advancements have significantly enhanced our
comprehension of quantum uncertainty.

Lee and Tsutsui have provided an operationally tangible uncertainty relation,
which we call the Lee-Tsutsui (LT) inequality, marking a significant
advancement in the field of quantum uncertainty. The LT inequality offers a
universal and geometric perspective on quantum measurement errors and
disturbances, unifying the KR, Schrödinger, AKG, Ozawa, and WSU inequalities
into a comprehensive, operationally interpretable framework [12]. This
framework was subsequently applied to a concrete two-state system [13]. Lee
further expanded this approach by introducing the concepts of local
representability and joint measurability, leading to the establishment of what
we call the Lee inequality [14, 15]. However, a challenge of this elegant
framework is the lack of a concrete realization for the abstractly constructed
mathematical objects, beyond the simple two-state system presented in Ref.
[13]. The current paper aims to fill this gap, by providing a concrete
realization of the key object in Lee’s inequality, the representability for a
joint measurement, in the case of Heisenberg’s position-momentum uncertainty.

Recent developments in the Gaussian wave-packet formalism have opened up new
possibilities in quantum field theory [16, 17, 18, 19, 20]. Particularly
notable is the rigorous proof of the emergence of time-boundary effect,
initially claimed in Ref. [21], which has been recently established [22].
Gaussian wave packets are significant as they form an (over)complete set that
spans the position-momentum phase space, including that of the free one-
particle subspace, which underpins ordinary non-relativistic quantum mechanics
[16]; see also Refs. [23, 24, 25] for a viewpoint that the Gaussian basis can
be regarded as a complete set of coherent states in position-momentum space.
Furthermore, the Gaussian basis can be generalized to a complete set of
Lorentz-invariant wave-packet basis [23, 24, 25], and further to Lorentz-
covariant wave-packet basis for spinor fields, incorporating spin degrees of
freedom [26].

The importance of the phase space is also emphasized in Refs. [27, 28, 29],
particularly in the context of providing a rigorous examination and proof of
Heisenberg-type inequalities, with a focus on the precision-disturbance trade-
off in quantum measurements.

In this paper, we posit that a Positive Operator-Valued Measure (POVM)
measurement onto the Gaussian wave-packet basis naturally serves as the joint
measurement for position and momentum, a key element in the discussion on
Heisenberg’s uncertainty in Ref. [15]. We provide concrete expressions for the
pullback and pushforward, which remained abstract in the original construction
[12, 13, 15]. We will demonstrate that the Gaussian basis naturally
interpolates between the position-space basis and the momentum-space basis in
the limits of infinitely narrow and wide widths, respectively.

The organization of this paper is as follows: In Sec. 2, we briefly review the
Lee-Tsutsui (LT) formalism to clarify our notation, focusing on the parts
relevant to our discussion. In Sec. 3, we review the Gaussian formalism and
show that the Gaussian wave packet smoothly interpolates between the position
and momentum bases. In Sec. 4, we study the separate measurement of position
or momentum of a Gaussian initial state in the LT formalism. We show the
concrete forms of the pullback and pushforward for the position and momentum,
which were previously only abstract constructs. In Sec. 5, we present our main
findings: that the POVM measurement onto the Gaussian basis can be regarded as
the joint measurement of position and momentum. Again we show the concrete
form of the pullback and pushforward for the position and momentum for the
joint measurement of the Gaussian initial state. In Sec. 6, we summarize our
results and discuss future directions.

##  2 Lee-Tsutsui formalism

We briefly review the formalism of Lee and Tsutsui [12, 13] and of Lee [14,
15], which we collectively call the LT formalism hereafter, focusing on the
part relevant to our discussion.

| Observables | States
---|---|---
Quantum | ${\color[rgb]{1,0.2,0}S\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}$ |  $\ni{\color[rgb]{1,0.2,0}\widehat{A}}$ (self-adjoint operator) | ${\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$ |  $\ni{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ (density operator)
Classical | ${\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$ |  $\ni{\color[rgb]{0,0.5,1}f}$ (real function) | ${\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$ |  $\ni{\color[rgb]{0.2,1,1}\mathsfit{p}}$ (PDF)
Table 1: Reminder table for the spaces. A typical element is also given for
each space.

###  2.1 Basic quantum mechanics

A physical _observable_ is represented by a _self-adjoint operator_
${\color[rgb]{1,0.2,0}\widehat{A}}$ that satisfies
${\color[rgb]{1,0.2,0}\widehat{A}^{\dagger}}={\color[rgb]{1,0.2,0}\widehat{A}}$,
where ${\color[rgb]{1,0.2,0}\widehat{A}^{\dagger}}$ is the adjoint of
${\color[rgb]{1,0.2,0}\widehat{A}}$, on a Hilbert space $\mathscr{H}$. We
write the linear space of all the self-adjoint operators as
${\color[rgb]{1,0.2,0}S\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}$;
see Table 1. A physical _state_ is represented by a _density operator_
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ that is self-adjoint,
${\color[rgb]{0.9,0.3,1}\mathsfit{\rho^{\dagger}}}={\color[rgb]{0.9,0.3,1}\mathsfit{\rho}}$,
has a trace of unity,
$\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\right]=1$,
and has a spectrum that is entirely non-negative,
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\geq 0$. We write the state
space
${\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$;
see Table 1. Typically in our discussion, one may assume the state to be a
_pure state_ given as a _projection_ operator
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}=\Ket{\psi}\Bra{\psi}$,
where $\Ket{\psi}\in\mathscr{H}$ (and $\Bra{\psi}$ is its dual). With this
case in mind, we also call a vector $\Ket{\psi}\in\mathscr{H}$ the state.

An expectation value of (not necessarily self-adjoint) operator
${\color[rgb]{1,0.2,0}\widehat{A}}$ on
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ is given by

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{A}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle:=\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\right],$
(1)

and a semi-inner product of any (not necessarily self-adjoint) operators
${\color[rgb]{1,0.2,0}\widehat{A}}$ and ${\color[rgb]{1,0.2,0}\widehat{B}}$ by

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{B}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle:=\Braket{\left\\{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{B}}\right\\}\over
2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}},$ (2)

where the commutator and anticommutator are
$[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}}]:={\color[rgb]{1,0.2,0}\widehat{A}}{\color[rgb]{1,0.2,0}\widehat{B}}-{\color[rgb]{1,0.2,0}\widehat{B}}{\color[rgb]{1,0.2,0}\widehat{A}}$
and
$\\{{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}}\\}:={\color[rgb]{1,0.2,0}\widehat{A}}{\color[rgb]{1,0.2,0}\widehat{B}}+{\color[rgb]{1,0.2,0}\widehat{B}}{\color[rgb]{1,0.2,0}\widehat{A}}$,
respectively. The semi-inner product naturally induces the semi-norm and the
standard deviation for any (not necessarily self-adjoint) operator
${\color[rgb]{1,0.2,0}\widehat{A}}$ on
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$:111 In this paper, we
regard a map from a space to $\mathbb{R}$ (or $\mathbb{C}$) as a _functional_
, and place its argument into square brackets rather than into parentheses,
conforming to physics community notation.

$\displaystyle\left\|{\color[rgb]{1,0.2,0}\widehat{A}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle:=\sqrt{\Braket{{\color[rgb]{1,0.2,0}\widehat{A}^{\dagger}},\,{\color[rgb]{1,0.2,0}\widehat{A}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}},$
$\displaystyle\sigma_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}}\right]$
$\displaystyle:=\sqrt{\left\|{\color[rgb]{1,0.2,0}\widehat{A}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}-\Braket{{\color[rgb]{1,0.2,0}\widehat{A}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}}.$
(3)

If ${\color[rgb]{1,0.2,0}\widehat{A}}$ is self-adjoint, the standard deviation
reduces to
$\sigma_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\bigl{[}{\color[rgb]{1,0.2,0}\widehat{A}}\bigr{]}=\braket{{\color[rgb]{1,0.2,0}\widehat{A}^{2}}}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-\braket{{\color[rgb]{1,0.2,0}\widehat{A}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}$.
Further if ${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ is a pure state
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}=\Ket{\psi}\Bra{\psi}$, the
semi-norm reduces to the norm of the state after the operation,
$\|{\color[rgb]{1,0.2,0}\widehat{A}}\|_{\Ket{\psi}\Bra{\psi}}^{2}=\|{\color[rgb]{1,0.2,0}\widehat{A}}\Ket{\psi}\|^{2}$,
and accordingly the expectation value (1) reduces to
$\braket{{\color[rgb]{1,0.2,0}\widehat{A}}missing}_{\Ket{\psi}\Bra{\psi}}=\Bra{\psi}{\color[rgb]{1,0.2,0}\widehat{A}}\Ket{\psi}$.

###  2.2 Measurement process

A _measurement_ $\mathsf{M}$ is an _affine map_
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\mapsto{\color[rgb]{0.2,1,1}\mathsfit{p}}$
in the sense that, for any states
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{1}}},{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{2}}}$
and $\forall\lambda\in[0,1]$,

$\displaystyle\mathsf{M}\bigl{(}\lambda{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{1}}}+\left(1-\lambda\right){\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{2}}}\bigr{)}$
$\displaystyle=\lambda\mathsf{M}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{1}}}+\left(1-\lambda\right)\mathsf{M}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{2}}},$
(4)

where ${\color[rgb]{0.2,1,1}\mathsfit{p}}$ is a _probability density function_
(PDF) that satisfies
$\int_{\Omega}\text{d}\omega\,{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}=1$
and, $\forall\omega\in\Omega$,
${\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\geq
0$, in which $\Omega$ is a _sample space_ and $\text{d}\omega$ its measure. We
write the convex set of all the PDFs as
${\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$;
see Table 1. Hereafter, we write
${\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}:=\mathsf{M}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$
as a reminder to reinforce that
${\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}$ is a PDF. As a whole,
$\mathsf{M}\colon{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}\to{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$
is defined.

Two comments are in order: First, the affine-ness (4) is the least requirement
to allow the interpretation of probability mixture but is powerful enough to
derive most of the results in the LT formalism. An affine map between any pair
of state spaces, in the right of Table 1, is in general called a _process_.
The LT formalism can handle all the processes on equal foot. Especially when
applied to a _quantum process_
$\mathsf{\Theta}\colon{\color[rgb]{0.9,0.3,1}\mathsfit{Z_{1}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}_{1}$}}\right)}}\to{\color[rgb]{0.9,0.3,1}\mathsfit{Z_{2}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}_{2}$}}\right)}}$,
its _disturbance_ can be described in quite a parallel manner to various
$\mathsf{M}$’s _errors_ given below. For notational simplicity, we do not
exploit this beauty of the formalism, and will stick to a measurement
$\mathsf{M}$ as a process (except in Sec. 2.6). Second, for most of our
purpose, one may assume a _projective measurement_ that gives,
$\forall{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\in{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$,222
Here, the term “projective” should be understood in a broader sense; see
footnote 10.

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}=\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\bigl{[}\Ket{\omega}\Bra{\omega}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\bigr{]}.$
(5)

On the sample space $\Omega$, we define the _classical_ expectation value,
semi-inner product, semi-norm, and standard deviation in accordance with their
quantum counterparts:

$\displaystyle\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}$
$\displaystyle:=\int_{\Omega}\text{d}\omega\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}f},\,{\color[rgb]{0,0.5,1}g}}_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}$
$\displaystyle:=\int_{\Omega}\text{d}\omega\,{\color[rgb]{0,0.5,1}f^{*}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}{\color[rgb]{0,0.5,1}g\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)},$
(6)
$\displaystyle\left\|{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}$
$\displaystyle:=\sqrt{\Braket{{\color[rgb]{0,0.5,1}f},\,{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}}=\sqrt{\int_{\Omega}\text{d}\omega\left|{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\right|^{2}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}},$
$\displaystyle\sigma_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0,0.5,1}f}\right]$
$\displaystyle:=\sqrt{\left\|{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}^{2}-\left|\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}\right|^{2}},$
(7)

where ${\color[rgb]{0,0.5,1}f},{\color[rgb]{0,0.5,1}g}$ are arbitrary real
functions on $\Omega$ (but we have retained the complex conjugate, denoted by
$*$ per physics community notation, to accommodate possible generalizations).
More concretely for the real function ${\color[rgb]{0,0.5,1}f}$, we have
$\sigma_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0,0.5,1}f}\right]=\int_{\Omega}\text{d}\omega{\color[rgb]{0,0.5,1}f^{2}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}-\left(\int_{\Omega}\text{d}\omega{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\right)^{2}$.

###  2.3 LT adjoint, pullback, and pushforward

Given a measurement $\mathsf{M}$, we define (what we call) the _Lee-Tsutsui
(LT) adjoint_ operator ${\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}$ of a real
function ${\color[rgb]{0,0.5,1}f}$ by,
$\forall{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\in{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$,

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}.$
(8)

We also call this the _pullback_ operator of ${\color[rgb]{0,0.5,1}f}$.333 In
the original construction [12, 13, 15], it was crucial to define the pullback
(and pushforward given below) _locally_ in the sense that, everything is
partitioned into equivalence classes as
$[{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}]_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$,
$\left[{\color[rgb]{0,0.5,1}f}\right]_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}$,
etc., by the equivalences
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{A}}\stackrel{{\scriptstyle{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}}{{\sim}}{\color[rgb]{1,0.2,0}\widehat{B}}$
$\displaystyle\Longleftrightarrow\left\|{\color[rgb]{1,0.2,0}\widehat{A}}-{\color[rgb]{1,0.2,0}\widehat{B}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}=0,$
$\displaystyle{\color[rgb]{0,0.5,1}f}\stackrel{{\scriptstyle{\color[rgb]{0.2,1,1}\mathsfit{p}}}}{{\sim}}{\color[rgb]{0,0.5,1}g}$
$\displaystyle\Longleftrightarrow\left\|{\color[rgb]{0,0.5,1}f}-{\color[rgb]{0,0.5,1}g}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}=0.$
For our purpose, this step is not necessary and we neglect it, though can be
important for other purposes.  As a whole, we define the LT adjoint
$\mathsf{M}^{\star}\colon{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}\to{\color[rgb]{1,0.2,0}S\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}$,
where
${\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$
is the space of all the real functions on $\Omega$; see Table 1.

It is important that the following inequality follows from the Kadison-Schwarz
inequality:

$\displaystyle\left\|{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}\geq\left\|{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}.$
(9)

By definitions (3), (7), and (8), we may rewrite inequality (9) into that of
the standard deviations:

$\displaystyle\sigma_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0,0.5,1}f}\right]$
$\displaystyle\geq\sigma_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right].$
(10)

To quote [15], _the operational cost of acquiring the expectation value of a
quantum observable through measurements can never break the quantum limit
imposed by the said observable._

Once the LT adjoint is obtained, we define the _pushforward_ function
${\color[rgb]{0,0.5,1}M_{\star}\widehat{A}}$ of a given self-adjoint operator
${\color[rgb]{1,0.2,0}\widehat{A}}$ (over a state
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$) by,
$\forall{\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$,

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}M_{\star}\widehat{A}},\,{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}},$
(11)

or more concretely by,
$\forall{\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$,

$\displaystyle\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{\left\\{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\\}\over
2}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\right]$
$\displaystyle=\int_{\Omega}\text{d}\omega\,{\color[rgb]{0,0.5,1}\left[M_{\star}\widehat{A}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{\left[M\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}.$
(12)

By construction, the pushforward function is obtained _locally_ , namely
obtained for each given ${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$,
and should be more properly written as
${\color[rgb]{0,0.5,1}M_{\widehat{\rho}\star}\widehat{A}}$. However, this
dependence on ${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ is trivially
understood for most of our purposes, and we will use the shorthand notation to
avoid clutter. As a whole, we define the pushforward
$\mathsf{M}_{\star}\colon{\color[rgb]{1,0.2,0}S\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}\to{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$
(for each given ${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ as
mentioned).

Finally, we note that for the case of projective measurement (5), its LT
adjoint also becomes projective, in the sense that given any function
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$,

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}=\int_{\Omega}\text{d}\omega{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\Ket{\omega}\Bra{\omega}.$
(13)

Also, the pushforward for the projective measurement of
${\color[rgb]{1,0.2,0}\widehat{A}}$ becomes the identity
${\color[rgb]{0,0.5,1}M_{\star}\widehat{A}}={\color[rgb]{0,0.5,1}\text{id}}$;
see Eq. (50) below for a concrete realization.

###  2.4 LT error

We define (what we call) the LT error [12]:

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M}\right]$
$\displaystyle:=\sqrt{\left\|{\color[rgb]{1,0.2,0}\widehat{A}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-\left\|{\color[rgb]{0,0.5,1}M_{\star}\widehat{A}}\right\|^{2}_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}}=\sqrt{\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0.9,0.3,1}\mathsfit{\rho}}{\color[rgb]{1,0.2,0}\widehat{A}^{2}}\right]-\int_{\Omega}\text{d}\omega\,{\color[rgb]{0,0.5,1}\left[M_{\star}\widehat{A}\right]^{2}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{\left[M\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}},$
(14)

where
${\color[rgb]{0,0.5,1}\bigl{[}M_{\star}\widehat{A}\bigr{]}^{2}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}:=\bigl{(}{\color[rgb]{0,0.5,1}\bigl{[}M_{\star}\widehat{A}\bigr{]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\bigr{)}^{2}$
as above. The meaning of LT error is well explained in Ref. [12], which we
will briefly review hereafter. Given an estimator function
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$,
we introduce an error with respect to ${\color[rgb]{0,0.5,1}f}$ (_abbr._
${\color[rgb]{0,0.5,1}f}$-error):

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M},{\color[rgb]{0,0.5,1}f}\right]$
$\displaystyle:=\sqrt{\left\|{\color[rgb]{1,0.2,0}\widehat{A}}-{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}+\left(\left\|{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}^{2}-\left\|{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\right)}.$
(15)

The first term in the square root is the goodness of the fit of
${\color[rgb]{1,0.2,0}\widehat{A}}$ by the pullback operator of the estimator
${\color[rgb]{0,0.5,1}f}$, while the second is the increase in the variance
from that of the quantum pullback operator
${\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}$ to that of the classical
estimator ${\color[rgb]{0,0.5,1}f}$:

$\displaystyle\left\|{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}^{2}-\left\|{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}$
$\displaystyle=\left(\left\|{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}^{2}-\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}^{2}\right)-\left(\left\|{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}-\Braket{{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\right)$
$\displaystyle=\sigma_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0,0.5,1}f}\right]-\sigma_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right],$
(16)

where we used the definition of LT adjoint (8).

The first term in the square root in Eq. (15) reads

$\displaystyle\left\|{\color[rgb]{1,0.2,0}\widehat{A}}-{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{A}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-2\Braket{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}+\Braket{{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\left\|{\color[rgb]{1,0.2,0}\widehat{A}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}-2\Braket{{\color[rgb]{0,0.5,1}M_{\star}\widehat{A}},\,{\color[rgb]{0,0.5,1}f}}_{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}+\left\|{\color[rgb]{1,0.2,0}\widehat{M^{\star}f}}\right\|_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2},$
(17)

where we used the definition of pushforward (11). Putting Eq. (17) into Eq.
(15), we obtain

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M},{\color[rgb]{0,0.5,1}f}\right]$
$\displaystyle=\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M}\right]+\left\|{\color[rgb]{0,0.5,1}M_{\star}\widehat{A}}-{\color[rgb]{0,0.5,1}f}\right\|_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}^{2}.$
(18)

The ${\color[rgb]{0,0.5,1}f}$-error is minimized by choosing
${\color[rgb]{0,0.5,1}f}$ to be the optimal pushforward function
${\color[rgb]{0,0.5,1}M_{\star}\widehat{A}}$, and the LT error (14) gives the
smallest error achieved by this choice.

The LT error satisfies (what we call) the LT inequality:

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M}\right]\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]\geq\sqrt{\mathcal{I}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]+\mathcal{R}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]},$
(19)

where

$\displaystyle\mathcal{I}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]$
$\displaystyle:=\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{B}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{M^{\star}M_{\star}\widehat{A}}},\,{\color[rgb]{1,0.2,0}\widehat{B}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}M_{\star}\widehat{B}}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}},$ (20)
$\displaystyle\mathcal{R}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]$
$\displaystyle:=\Braket{\left\\{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{B}}\right\\}\over
2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-\Braket{{\color[rgb]{0,0.5,1}M_{\star}\widehat{A}},\,{\color[rgb]{0,0.5,1}M_{\star}\widehat{B}}}_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}.$
(21)

The imaginary parts
$\mathcal{I}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\bigl{[}{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\bigr{]}$
represents the quantum contribution to the lower bound, while the real parts
$\mathcal{R}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\bigl{[}{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\bigr{]}$
the semi-classical contribution.

###  2.5 Lee error

Following Ref. [15], we define the _error for local representability_ , which
we call the _Lee error_ hereafter:

$\displaystyle\widetilde{\varepsilon}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M}\right]$
$\displaystyle:=\sqrt{\left\|{\color[rgb]{0,0.5,1}M^{\star-1}\widehat{A}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}-\left\|{\color[rgb]{1,0.2,0}\widehat{A}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}},$
(22)

where
${\color[rgb]{0,0.5,1}M^{\star-1}\widehat{A}}:=\left(\mathsf{M}^{\star}\right)^{-1}{\color[rgb]{1,0.2,0}\widehat{A}}$.444
When the inverse cannot be defined, a _partial inverse_ is involved in Ref.
[15]. For our purpose, this is an unnecessary detour, and we will neglect it,
though can be important for other purposes.  More explicitly,

$\displaystyle\widetilde{\varepsilon}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M}\right]$
$\displaystyle=\int_{\Omega}\text{d}\omega\,{\color[rgb]{0,0.5,1}\left[M^{\star-1}\widehat{A}\right]^{2}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{\left[M\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega$}}\right)}-\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}^{2}}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\right].$
(23)

The Lee inequality reads

$\displaystyle\widetilde{\varepsilon}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}};\mathsf{M}\right]\widetilde{\varepsilon}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]$
$\displaystyle\geq\sqrt{\mathcal{I}_{0{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}}\right]+\widetilde{\mathcal{R}}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]},$
(24)

where the contributors to the lower bound are

$\displaystyle\mathcal{I}_{0{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}}\right]$
$\displaystyle:=\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{B}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}},$ (25)
$\displaystyle\widetilde{\mathcal{R}}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{A}},{\color[rgb]{1,0.2,0}\widehat{B}};\mathsf{M}\right]$
$\displaystyle:=\Braket{\left\\{{\color[rgb]{1,0.2,0}\widehat{A}},\,{\color[rgb]{1,0.2,0}\widehat{B}}\right\\}\over
2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}-\Braket{{\color[rgb]{0,0.5,1}M^{\star-1}\widehat{A}},\,{\color[rgb]{0,0.5,1}M^{\star-1}\widehat{B}}}_{{\color[rgb]{0.2,1,1}\mathsfit{M\widehat{\rho}}}}.$
(26)

Again the imaginary and real parts represent quantum and semi-classical
contributions, respectively.

###  2.6 Joint measurement and classical projection of marginalization

Following Lee’s construction [15], we consider a joint measurement
$\mathsf{J}\colon{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}\to{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$,
whose sample space (also called the outcome space) is separated into
$\Omega=\Omega_{1}\times\Omega_{2}$ so that the outcome PDF is a function of
$\omega_{i}\in\Omega_{i}$ ($i=1,2$), namely,
${\color[rgb]{0.2,1,1}\mathsfit{p}}\in{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega$}}\right)}$
is written as
${\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1},\omega_{2}$}}\right)}$.
Now we introduce the following _classical projection processes_
$\bm{\pi}_{i}\colon{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{1}\times\Omega_{2}$}}\right)}\to{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{i}$}}\right)}$
($i=1,2$), defined by the projection to marginal:

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{1}p\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1}$}}\right)}$
$\displaystyle:=\int_{\Omega_{2}}\text{d}\omega_{2}\,{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1},\omega_{2}$}}\right)},$
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{2}p\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{2}$}}\right)}$
$\displaystyle:=\int_{\Omega_{1}}\text{d}\omega_{1}\,{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1},\omega_{2}$}}\right)}.$
(27)

If two measurements
$\mathsf{M}_{i}\colon{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}\to{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{i}$}}\right)}$
($i=1,2$) can be written as $\mathsf{M}_{i}=\bm{\pi}_{i}\circ\mathsf{J}$, they
are said to admit the joint measurement $\mathsf{J}$.

The LT adjoint $\bm{\pi}^{\star}_{i}$ is given in parallel to that of the
measurement (8): Given a function
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{i}$}}\right)}$,
its LT adjoint function
${\color[rgb]{0,0.5,1}\pi^{\star}_{i}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{1}\times\Omega_{2}$}}\right)}$
is defined by,
$\forall{\color[rgb]{0.2,1,1}\mathsfit{p}}\in{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{1}\times\Omega_{2}$}}\right)}$,

$\displaystyle\Braket{{\color[rgb]{0,0.5,1}\pi^{\star}_{i}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{p}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{\pi_{i}p}}},\qquad(i=1,2)$
(28)

or more concretely,
$\forall{\color[rgb]{0.2,1,1}\mathsfit{p}}\in{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\Omega_{1}\times\Omega_{2}$}}\right)}$,

$\displaystyle\int_{\Omega_{1}}\text{d}\omega_{1}\int_{\Omega_{2}}\text{d}\omega_{2}\,{\color[rgb]{0,0.5,1}\left[\pi^{\star}_{i}f\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1},\omega_{2}$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1},\omega_{2}$}}\right)}$
$\displaystyle=\int_{\Omega_{i}}\text{d}\omega_{i}\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{i}$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{i}p\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{i}$}}\right)}.\qquad(i=1,2)$
(29)

We see that, by the definition (27), the LT adjoint of
${\color[rgb]{0,0.5,1}f}$ is trivially

$\displaystyle{\color[rgb]{0,0.5,1}\left[\pi^{\star}_{i}f\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{1},\omega_{2}$}}\right)}$
$\displaystyle={\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\omega_{i}$}}\right)}.\qquad(i=1,2)$
(30)

##  3 Gaussian formalism

We review the Gaussian wave-packet formalism within a free one-particle
subspace, decoupled from the spin degrees of freedom. In this paper, we
restrict our attention to the 1D position-momentum space since the
generalization to higher dimensions is straightforward. Throughout the paper,
we restrict ourselves to the consideration of a fixed time slice and do not
consider the time evolution/translation; see Refs. [16, 17, 18, 19] for a
Gaussian formalism applied to a relativistic particle, Refs. [23, 24, 25] for
insights into the free one-particle subspace of the relativistic particle and
generalization to a Lorentz-invariant complete basis, and Ref. [26] for an
account on the transformation of the space-like hyperplane under the Lorentz
transformation and for a generalization to include the relativistic spin
degrees of freedom into a Lorentz-covariant complete basis.

###  3.1 Plane-wave basis

Let $\Set{\Ket{p}}_{p\in\mathbb{R}}$ be the momentum-space basis that spans
the 1D free one-particle subspace, which we write $\mathscr{H}$ hereafter,
without spin degrees of freedom. We employ the normalization

$\displaystyle\Braket{p}{p^{\prime}}$
$\displaystyle=\delta\mathchoice{\\!}{\\!}{}{}\left(p-p^{\prime}\right),$ (31)

where the right-hand side is the Dirac delta function (distribution),
resulting in the standard normalization for the completeness relation
(resolution of identity):

$\displaystyle\int_{\mathbb{R}}\text{d}p\Ket{p}\Bra{p}$
$\displaystyle={\color[rgb]{1,0.2,0}\widehat{1}},$ (32)

in which ${\color[rgb]{1,0.2,0}\widehat{1}}$ is the identity operator on
$\mathscr{H}$. Here and hereafter, the integral region of momentum (and
position) is always from $-\infty$ to $\infty$ unless otherwise stated.555
Usually one would write $\int_{-\infty}^{\infty}\text{d}x$ or $\int\text{d}x$
in short. Here, we conform to the language of Eq. (6).  The dual position-
space basis $\Set{\Ket{x}}_{x\in\mathbb{R}}$ is defined by

$\displaystyle\Braket{x}{p}$ $\displaystyle:={e^{ipx}\over\sqrt{2\pi}}.$ (33)

Here and hereafter, we employ the natural unit $\hbar=1$, unless otherwise
stated, so that momentum and wavenumber have the same dimension (unit). The
definition (33) leads to the completeness in the position space:

$\displaystyle\int_{\mathbb{R}}\text{d}x\Ket{x}\Bra{x}$
$\displaystyle={\color[rgb]{1,0.2,0}\widehat{1}}.$ (34)

We identify this position space as an arbitrarily chosen equal-time
hyperplane. It is worth reiterating that the position-space basis can be used
to span any fixed time slice, even for a relativistic particle, both in
quantum mechanics (see e.g. Ref. [25]) and in quantum field theory [16, 17,
18] (this viewpoint is particularly emphasized in Refs. [25, 26]).

On $\mathscr{H}$, we define the position and momentum operators:666 See Ref.
[25] for a detailed discussion on the position operator in the relativistic
one-particle subspace.

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{x}}\Ket{x}$
$\displaystyle=x\Ket{x},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{p}}\Ket{p}$
$\displaystyle=p\Ket{p}.$ (35)

It follows that

$\displaystyle\Bra{x}{\color[rgb]{1,0.2,0}\widehat{p}}$
$\displaystyle=-i{\partial\over\partial x}\Bra{x},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{p}}\Ket{x}$
$\displaystyle=\Ket{x}i{\overleftarrow{\partial}\over\partial x},$
$\displaystyle\Bra{p}{\color[rgb]{1,0.2,0}\widehat{x}}$
$\displaystyle=i{\partial\over\partial p}\Bra{p},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{x}}\Ket{p}$
$\displaystyle=\Ket{p}\left(-i{\overleftarrow{\partial}\over\partial
p}\right),$ (36)

and

$\displaystyle\left[{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right]$
$\displaystyle=i{\color[rgb]{1,0.2,0}\widehat{1}},$
$\displaystyle\left[{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{x}}\right]=\left[{\color[rgb]{1,0.2,0}\widehat{p}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right]=0.$
(37)

The projective measurement for position $\mathsf{M}_{\text{pos}}$ and that for
momentum $\mathsf{M}_{\text{mom}}$ are defined by,
$\forall{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\in{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$,

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\bigl{[}\Ket{x}\Bra{x}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\bigr{]},$
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{mom}}\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\bigl{[}\Ket{p}\Bra{p}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\bigr{]}.$
(38)

Especially when ${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$ is a pure
state
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}=\Ket{\psi}\Bra{\psi}$, we
recover the familiar PDFs:

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\bigl{[}M_{\text{pos}}\Ket{\psi}\Bra{\psi}\bigr{]}}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=\left|\Braket{x}{\psi}\right|^{2},$
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\bigl{[}M_{\text{mom}}\Ket{\psi}\Bra{\psi}\bigr{]}}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=\left|\Braket{p}{\psi}\right|^{2}.$ (39)

###  3.2 Gaussian basis

We define the Gaussian wave-packet state $\Ket{X,P;\sigma}$ on the momentum-
space basis [16, 17, 18] (see also Ref. [25] for a historical account):

$\displaystyle\Braket{p}{X,P;\sigma}$
$\displaystyle:=\left(\sigma\over\pi\right)^{1\over
4}\exp\mathchoice{\\!}{\\!}{}{}\left(-ipX-{\sigma\over
2}\left(p-P\right)^{2}\right),$ (40)

where we have normalized such that $\bigl{\|}\Ket{X,P;\sigma}\bigr{\|}^{2}=1$.
Physically, $P$ and $X$ give the centers of the wave packet in the position
and momentum spaces, respectively, and $\sigma$ ($\sigma^{-1}$) roughly gives
the spatial (momentum) width-squared. The corresponding wave function becomes

$\displaystyle\Braket{x}{X,P;\sigma}$
$\displaystyle=\int_{\mathbb{R}}\text{d}p\Braket{x}{p}\Braket{p}{X,P;\sigma}={1\over\left(\pi\sigma\right)^{1\over
4}}\exp\mathchoice{\\!}{\\!}{}{}\left(iP\left(x-X\right)-{\left(x-X\right)^{2}\over
2\sigma}\right).$ (41)

It is important that the Gaussian wave-packet states form a complete basis
that spans $\mathscr{H}$: For any fixed $\sigma$,

$\displaystyle\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}\Ket{X,P;\sigma}\Bra{X,P;\sigma}$
$\displaystyle={\color[rgb]{1,0.2,0}\widehat{1}},$ (42)

which can be verified by sandwiching both-hand sides by $\Bra{p}$ and
$\Ket{p^{\prime}}$ (or equally by $\Bra{x}$ and $\Ket{x^{\prime}}$).777 Here,
it is meant that $\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}:={1\over
2\pi}\int_{-\infty}^{\infty}\text{d}X\int_{-\infty}^{\infty}\text{d}P$; see
footnote 5.  It is somewhat beautiful that the phase space measure
${\text{d}X\,\text{d}P\over 2\pi}$ emerges automatically. Note that the
spatial width-squared $\sigma$ is fixed and is not summed in the completeness
(42).

It is also noteworthy that the Gaussian basis states are not mutually
orthogonal:

$\displaystyle\Braket{X,P;\sigma}{X^{\prime},P^{\prime};\sigma}$
$\displaystyle=\int_{\mathbb{R}}\text{d}p\Braket{X,P;\sigma}{p}\Braket{p}{X^{\prime},P^{\prime};\sigma}$
$\displaystyle=\exp\mathchoice{\\!}{\\!}{}{}\left(i{P+P^{\prime}\over
2}\left(X-X^{\prime}\right)-{\left(X-X^{\prime}\right)^{2}\over
4\sigma}-{\sigma\over 4}\left(P-P^{\prime}\right)^{2}\right).$ (43)

In this sense, the relation (42) is sometimes called the overcompleteness. The
relation (42) tells that the Gaussian basis provides a non-projective POVM:
$\Set{\Ket{X,P;\sigma}\Bra{X,P;\sigma}}_{\left(X,P\right)\in\mathbb{R}^{2}}$
for any given fixed $\sigma$.

##  4 Separate measurement of position or momentum

For the rest of this paper (except in Eq. (86)), we will exclusively focus on
measurements of an initial Gaussian pure state:

$\displaystyle{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}$
$\displaystyle=\Ket{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\Bra{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}.$
(44)

In this section, we _separately_ measure either its position by
$\mathsf{M}_{\text{pos}}$ or its momentum by $\mathsf{M}_{\text{mom}}$, as
defined in Eq. (38). Though we list both the results of position and momentum
measurements hereafter for conciseness, it should be kept in mind that either
one of the measurements is performed, independently from the other.

Given the initial state (44), its measured PDFs in the position and momentum
spaces are

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=\left|\Braket{x}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}={1\over\sqrt{\pi\sigma_{\text{in}}}}\exp\mathchoice{\\!}{\\!}{}{}\left(-{\left(x-X_{\text{in}}\right)^{2}\over\sigma_{\text{in}}}\right),$
(45)
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{mom}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=\left|\Braket{p}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}=\sqrt{\sigma_{\text{in}}\over\pi}\exp\mathchoice{\\!}{\\!}{}{}\left(-\sigma_{\text{in}}\left(p-P_{\text{in}}\right)^{2}\right).$
(46)

They are centered around $x=X_{\text{in}}$ and $p=P_{\text{in}}$, with the
widths-squared roughly $\sigma_{\text{in}}$ and $\sigma_{\text{in}}^{-1}$,
respectively.

###  4.1 Expectation and variance for separate measurements

We obtain the expectation value, the squared semi-norm, and the variance
(given in Eqs. (1), (3), (6), and (7)) for $\mathsf{M}_{\text{pos}}$ of
${\color[rgb]{1,0.2,0}\widehat{x}}$ (left) and for $\mathsf{M}_{\text{mom}}$
of ${\color[rgb]{1,0.2,0}\widehat{p}}$ (right):

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{x}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}\text{id}}missing}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}=X_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{p}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}\text{id}}missing}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}_{\text{in}}}}}=P_{\text{in}},$
(47)
$\displaystyle\left\|{\color[rgb]{1,0.2,0}\widehat{x}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=\left\|{\color[rgb]{0,0.5,1}\text{id}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}=X_{\text{in}}^{2}+{\sigma_{\text{in}}\over
2},$
$\displaystyle\left\|{\color[rgb]{1,0.2,0}\widehat{p}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=\left\|{\color[rgb]{0,0.5,1}\text{id}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}_{\text{in}}}}=P_{\text{in}}^{2}+{1\over
2\sigma_{\text{in}}},$ (48)
$\displaystyle\sigma^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}}\right]$
$\displaystyle=\sigma^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0,0.5,1}\text{id}}\right]={\sigma_{\text{in}}\over
2},$
$\displaystyle\sigma^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}}\right]$
$\displaystyle=\sigma^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}_{\text{in}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{0,0.5,1}\text{id}}\right]={1\over
2\sigma_{\text{in}}},$ (49)

where id is the identity function

$\displaystyle{\color[rgb]{0,0.5,1}\text{id}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=x,$
$\displaystyle{\color[rgb]{0,0.5,1}\text{id}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=p;$ (50)

recall that
$\left\|{\color[rgb]{1,0.2,0}\widehat{x}}\right\|^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}:=\Braket{{\color[rgb]{1,0.2,0}\widehat{x}^{2}}}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$,
$\left\|{\color[rgb]{0,0.5,1}\text{id}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}:=\Braket{{\color[rgb]{0,0.5,1}\text{id}^{2}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}$,
etc.

We also have the following results, which are independent of measurements:

$\displaystyle\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle={1\over 2},$
$\displaystyle\Braket{\left\\{{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right\\}\over
2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=X_{\text{in}}P_{\text{in}}.$ (51)

For some readers’ ease, we list some matrix elements as well:

$\displaystyle\Braket{p}{{\color[rgb]{1,0.2,0}\widehat{x}}missing}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$
$\displaystyle=\bigl{(}X_{\text{in}}-i\sigma_{\text{in}}\left(p-P_{\text{in}}\right)\bigr{)}\Braket{p}{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}},$
(52)
$\displaystyle\Braket{x}{{\color[rgb]{1,0.2,0}\widehat{p}}missing}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$
$\displaystyle=\left(P_{\text{in}}+i{x-X_{\text{in}}\over\sigma_{\text{in}}}\right)\Braket{x}{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}},$
(53)

which follow from the derivative representation (36) and may be useful to
derive expressions below.

###  4.2 Pullback and pushforward for separate measurements

Having completed our review, we now turn to new developments from here on.

Given a function
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}$}}\right)}$,
we find its LT adjoint:

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}f}}$
$\displaystyle=\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\Ket{x}\Bra{x},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{mom}}f}}$
$\displaystyle=\int_{\mathbb{R}}\text{d}p\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}\Ket{p}\Bra{p},$
(54)

It is straightforward to check that they satisfy,
$\forall{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\in{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$,

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}}}},$
$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{mom}}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}}}},$
(55)

by expanding the general state as

$\displaystyle{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$
$\displaystyle=\int_{\mathbb{R}}\text{d}x\int_{\mathbb{R}}\text{d}x^{\prime}\,{\color[rgb]{0.9,0.3,1}\mathsfit{\rho\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x,x^{\prime}$}}\right)}}\Ket{x}\Bra{x^{\prime}}=\int_{\mathbb{R}}\text{d}p\int_{\mathbb{R}}\text{d}p^{\prime}\,{\color[rgb]{0.9,0.3,1}\mathsfit{\widetilde{\rho}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p,p^{\prime}$}}\right)}}\Ket{p}\Bra{p^{\prime}},$
(56)

where
${\color[rgb]{0.9,0.3,1}\mathsfit{\rho\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x,x^{\prime}$}}\right)}}:=\Bra{x}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\Ket{x^{\prime}}$,
with
${\color[rgb]{0.9,0.3,1}\mathsfit{\rho^{*}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x,x^{\prime}$}}\right)}}={\color[rgb]{0.9,0.3,1}\mathsfit{\rho\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x^{\prime},x$}}\right)}}$
and
$\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0.9,0.3,1}\mathsfit{\rho\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x,x$}}\right)}}=1$
(in particular, it follows that
${\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}={\color[rgb]{0.9,0.3,1}\mathsfit{\rho\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x,x$}}\right)}}$);
and similarly for the momentum space. The result (54) is indeed trivial
because of the generality (13), but we spelled it out anyway for concreteness.

Equipped with the definite representation (40), yielding the matrix elements
(52) and (53), we can explicitly compute the pushforward of
${\color[rgb]{1,0.2,0}\widehat{x}}$,

$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{pos}\star}\widehat{x}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=x,$
$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{mom}\star}\widehat{x}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=X_{\text{in}},$ (57)

and the pushforward of ${\color[rgb]{1,0.2,0}\widehat{p}}$,

$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{pos}\star}\widehat{p}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=P_{\text{in}},$
$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{mom}\star}\widehat{p}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=p.$ (58)

One may verify these pushforwards to satisfy the defining relation (11), or
more concretely (12): As an illustration, we show the derivation of
${\color[rgb]{0,0.5,1}M_{\text{pos}\star}\widehat{p}}$ given in the left of
Eq. (58). All other computations in this paper can be performed quite
similarly. The defining relation (11) reads,
$\forall{\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}$}}\right)}$,

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{p}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}f}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}=\Braket{{\color[rgb]{0,0.5,1}M_{\text{pos}\star}\widehat{p}},\,{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}},$
(59)

or more concretely from Eq. (12),
$\forall{\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}$}}\right)}$,

$\displaystyle\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{\left\\{{\color[rgb]{1,0.2,0}\widehat{p}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}f}}\right\\}\over
2}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}\right]$
$\displaystyle=\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0,0.5,1}\left[M_{\text{pos}\star}\widehat{p}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}.$
(60)

Putting the pullback (54) into the left-hand side, we obtain

l.h.s.
$\displaystyle=\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\left[{{\color[rgb]{1,0.2,0}\widehat{p}}\Ket{x}\Bra{x}+\Ket{x}\Bra{x}{\color[rgb]{1,0.2,0}\widehat{p}}\over
2}\Ket{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}}\Bra{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}}\right]$
$\displaystyle=\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\,\Re\Bigg{(}\Braket{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}}{x}\underbrace{\Braket{x}{{\color[rgb]{1,0.2,0}\widehat{p}}missing}{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}}}_{\left(P_{\text{in}}+i{x-X_{\text{in}}\over\sigma_{\text{in}}}\right)\Braket{x}{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}}}\Bigg{)}$
$\displaystyle=\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\,P_{\text{in}}\,\underbrace{\bigl{|}\Braket{x}{{X_{\text{in}}},P_{\text{in}};\sigma_{\text{in}}}\bigr{|}^{2}}_{{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}},$
(61)

where $\Re$ denotes the real part and we used Eq. (53) in the second last
line. Comparing with the right-hand side of Eq. (60), we obtain the constant
solution
${\color[rgb]{0,0.5,1}\left[M_{\text{pos}\star}\widehat{p}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}=P_{\text{in}}$.

###  4.3 LT error for separate measurements

Having obtained the pushforward, we can now compute the classical semi-norm
for ${\color[rgb]{1,0.2,0}\widehat{x}}$,

$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{pos}\star}\widehat{x}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}^{2}+{\sigma_{\text{in}}\over 2},$
$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{mom}\star}\widehat{x}}\right\|^{2}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=X_{\text{in}}^{2},$ (62)

and the classical semi-norm for ${\color[rgb]{1,0.2,0}\widehat{p}}$,

$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{pos}\star}\widehat{p}}\right\|^{2}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{mom}\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=P_{\text{in}}^{2},$
$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{mom}\star}\widehat{p}}\right\|^{2}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{mom}\widehat{\rho}_{\text{in}}}}}=P_{\text{in}}^{2}+{1\over
2\sigma_{\text{in}}}.$ (63)

Combining with the quantum semi-norm (48), the resultant LT error (14) is, for
${\color[rgb]{1,0.2,0}\widehat{x}}$,

$\displaystyle\varepsilon^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{pos}}\right]$
$\displaystyle=0,$
$\displaystyle\varepsilon^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{mom}}\right]$
$\displaystyle={\sigma_{\text{in}}\over 2},$ (64)

and, for ${\color[rgb]{1,0.2,0}\widehat{p}}$,

$\displaystyle\varepsilon^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{pos}}\right]$
$\displaystyle={1\over 2\sigma_{\text{in}}},$
$\displaystyle\varepsilon^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{mom}}\right]$
$\displaystyle=0.$ (65)

We see that for both the position and momentum measurements, the left-hand
side of the LT inequality (19) vanishes:

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{pos}}\right]\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{pos}}\right]$
$\displaystyle=0,$
$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{mom}}\right]\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{mom}}\right]$
$\displaystyle=0.$ (66)

Let us see if the right-hand side vanishes consistently.

###  4.4 LT inequality for separate measurements

The pullback of pushforward is, for ${\color[rgb]{1,0.2,0}\widehat{x}}$,

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}M_{\text{pos}\star}\widehat{x}}}$
$\displaystyle={\color[rgb]{1,0.2,0}\widehat{x}},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{mom}}M_{\text{mom}\star}\widehat{x}}}$
$\displaystyle=X_{\text{in}}{\color[rgb]{1,0.2,0}\widehat{1}},$ (67)

and for ${\color[rgb]{1,0.2,0}\widehat{p}}$,

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}M_{\text{pos}\star}\widehat{p}}}$
$\displaystyle=P_{\text{in}}{\color[rgb]{1,0.2,0}\widehat{1}},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{mom}}M_{\text{mom}\star}\widehat{p}}}$
$\displaystyle={\color[rgb]{1,0.2,0}\widehat{p}}.$ (68)

Therefore for the position measurement,

$\displaystyle\left[{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}M_{\text{pos}\star}\widehat{x}}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right]$
$\displaystyle=i{\color[rgb]{1,0.2,0}\widehat{1}},$
$\displaystyle\left[{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{pos}}M_{\text{pos}\star}\widehat{p}}}\right]$
$\displaystyle=0,$ (69)

and for the momentum measurement,

$\displaystyle\left[{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{mom}}M_{\text{mom}\star}\widehat{x}}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right]$
$\displaystyle=0,$
$\displaystyle\left[{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{mom}}M_{\text{mom}\star}\widehat{p}}}\right]$
$\displaystyle=i{\color[rgb]{1,0.2,0}\widehat{1}}.$ (70)

Combining with the measurement-independent part (51), we see that the quantum
contribution to the LT lower bound (20) vanishes:

$\displaystyle\mathcal{I}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{pos}}\right]$
$\displaystyle=0,$
$\displaystyle\mathcal{I}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{mom}}\right]$
$\displaystyle=0.$ (71)

We can compute

$\displaystyle\Braket{{\color[rgb]{0,0.5,1}M_{\text{pos}\star}\widehat{x}},\,{\color[rgb]{0,0.5,1}M_{\text{pos}\star}\widehat{p}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}\text{id}}missing}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}_{\text{in}}}}P_{\text{in}}=X_{\text{in}}P_{\text{in}},$
(72)
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}M_{\text{mom}\star}\widehat{x}},\,{\color[rgb]{0,0.5,1}M_{\text{mom}\star}\widehat{p}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}\Braket{{\color[rgb]{0,0.5,1}\text{id}}missing}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}_{\text{in}}}}=X_{\text{in}}P_{\text{in}},$
(73)

where we used Eq. (47). Combining with the measurement-independent part (51),
we see that the semi-classical contribution to the LT lower bound (21) also
vanishes:

$\displaystyle\mathcal{R}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{pos}}\right]$
$\displaystyle=0,$
$\displaystyle\mathcal{R}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{mom}}\right]$
$\displaystyle=0.$ (74)

The LT inequality turned out to be $0=0$!

###  4.5 Lee error for separate measurement?

To compute the Lee error (22), we need the inverse of the pullback (54). It is
trivial for (so to say) the “diagonal” measurement:

$\displaystyle{\color[rgb]{0,0.5,1}M_{\text{pos}}^{\star-1}\widehat{x}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$
$\displaystyle=x,$
$\displaystyle{\color[rgb]{0,0.5,1}M_{\text{mom}}^{\star-1}\widehat{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
$\displaystyle=p.$ (75)

It is straightforward to check that they satisfy,
$\forall{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\in{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$,

$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{x}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}M_{\text{pos}}^{\star-1}\widehat{x}}}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{pos}}\widehat{\rho}}}},$
$\displaystyle\Braket{{\color[rgb]{1,0.2,0}\widehat{p}}missing}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}$
$\displaystyle=\Braket{{\color[rgb]{0,0.5,1}M_{\text{mom}}^{\star-1}\widehat{p}}}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{mom}}\widehat{\rho}}}},$
(76)

by expanding the general state as in Eq. (56).

However, it is hard to obtain the inverse of pullback for the “off-diagonal”
measurements
${\color[rgb]{0,0.5,1}M_{\text{mom}}^{\star-1}\widehat{x}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$p$}}\right)}$
and
${\color[rgb]{0,0.5,1}M_{\text{pos}}^{\star-1}\widehat{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$.
For example for the latter, we need to find
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}$}}\right)}$
that satisfies
${\color[rgb]{1,0.2,0}\widehat{p}}=\int_{\mathbb{R}}\text{d}x\,{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\,\Ket{x}\Bra{x}$
to match Eq. (54). However, the derivative representation (36) leads to

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{p}}$
$\displaystyle=\int_{\mathbb{R}}\text{d}x\int_{\mathbb{R}}\text{d}x^{\prime}\Ket{x}\Braket{x}{{\color[rgb]{1,0.2,0}\widehat{p}}missing}{x^{\prime}}\Bra{x^{\prime}}=\int_{\mathbb{R}}\text{d}x\int_{\mathbb{R}}\text{d}x^{\prime}\left(-i{\partial\over\partial
x}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime}\right)\right)\Ket{x}\Bra{x^{\prime}}.$
(77)

Formally, the solution (if it exists) can be expressed as
${\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}=\int_{\mathbb{R}}\text{d}x^{\prime}\left(-i{\partial\over\partial
x}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime}\right)\right)$.888 We can
find
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}$}}\right)}$
that satisfies
${\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\Bra{x}=\int_{\mathbb{R}}\text{d}x^{\prime}\left(-i{\partial\over\partial
x}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime}\right)\right)\Bra{x^{\prime}}$
as follows:
$\Longleftrightarrow{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime\prime}\right)=\int_{\mathbb{R}}\text{d}x^{\prime}\left(-i{\partial\over\partial
x}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime}\right)\right)\delta\mathchoice{\\!}{\\!}{}{}\left(x^{\prime}-x^{\prime\prime}\right)=\left(-i{\partial\over\partial
x}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime\prime}\right)\right)\Longleftrightarrow{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}=\int_{\mathbb{R}}\text{d}x^{\prime}\left(-i{\partial\over\partial
x}\delta\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime}\right)\right)$.  This
expression evidently requires regularization due to the derivative of the
Dirac delta function. For example, using the regularization
$\delta_{\sigma}\mathchoice{\\!}{\\!}{}{}\left(x-x^{\prime}\right)=\left(\pi\sigma\right)^{-1/2}\exp\mathchoice{\\!}{\\!}{}{}\bigl{(}-\left(x-x^{\prime}\right)^{2}/\sigma\bigr{)}$
leads to a vanishing result for
${\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$x$}}\right)}$,
indicating the need for further elaboration. Below, we will demonstrate that a
joint measurement in the Gaussian phase space yields a naturally regularized
expression, as will be detailed in Eq. (99).

##  5 Joint measurement in phase space

Below, we first review known facts about the Gaussian matrix element, then
restart with our main new development from Sec. 5.1.

The completeness (42) implies that the Gaussian basis can expand _any_
function, including another Gaussian wave packet with a different width-
squared: $\Ket{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$. The expansion
coefficient can be computed as [18]

$\displaystyle\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$
$\displaystyle=\int_{\mathbb{R}}\text{d}p\Braket{X,P;\sigma}{p}\Braket{p}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$
$\displaystyle=\sqrt{2}\left(\sigma_{\text{red}}\over\sigma_{\text{sum}}\right)^{1\over
4}\exp\mathchoice{\\!}{\\!}{}{}\left(i\overline{P}\left(X-X_{\text{in}}\right)-{\left(X-X_{\text{in}}\right)^{2}\over
2\sigma_{\text{sum}}}-{\sigma_{\text{red}}\over
2}\left(P-P_{\text{in}}\right)^{2}\right),$ (78)

where999 We call the harmonic mean $\sigma_{\text{red}}$ the reduced width-
squared, in analogy to reduced mass.

$\displaystyle\sigma_{\text{sum}}$ $\displaystyle:=\sigma+\sigma_{\text{in}},$
$\displaystyle\sigma_{\text{red}}$
$\displaystyle:={1\over{1\over\sigma}+{1\over\sigma_{\text{in}}}}={\sigma\sigma_{\text{in}}\over\sigma+\sigma_{\text{in}}}.$
(79)

Physically, the inner product (78) can be understood as the transition
amplitude from the initial state
$\Ket{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$ to the final state
$\Ket{X,P;\sigma}$, with the transition probability being

$\displaystyle\left|\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$
$\displaystyle=2\pi\left({1\over\sqrt{\pi\sigma_{\text{sum}}}}e^{-{1\over\sigma_{\text{sum}}}\left(X-X_{\text{in}}\right)^{2}}\right)\left(\sqrt{\sigma_{\text{red}}\over\pi}e^{-\sigma_{\text{red}}\left(P-P_{\text{in}}\right)^{2}}\right).$
(80)

We see that the detection probability of the particle is centered around
$X=X_{\text{in}}$ and $P=P_{\text{in}}$, with the width-squared roughly
$\sigma_{\text{sum}}$ and $\sigma_{\text{red}}^{-1}$, respectively.

When we do not observe the momentum, we integrate the probability (80) over
all the possible momenta:

$\displaystyle\int_{\mathbb{R}}{\text{d}P\over
2\pi}\left|\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$
$\displaystyle={1\over\sqrt{\pi\sigma_{\text{sum}}}}\exp\mathchoice{\\!}{\\!}{}{}\left(-{\left(X-X_{\text{in}}\right)^{2}\over\sigma_{\text{sum}}}\right).$
(81)

In the limit of detector spatial resolution finer than the initial wave packet
size, $\sigma\ll\sigma_{\text{in}}$, we obtain
$\sigma_{\text{sum}}\to\sigma_{\text{in}}$, recovering the position-space PDF:
${\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}=\left|\Braket{X}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$;
see Eq. (45). In the opposite limit $\sigma\gg\sigma_{\text{in}}$, the
probability is governed by the detector spatial resolution,
${1\over\sqrt{\pi\sigma}}\exp(-{1\over\sigma}\left(X-X_{\text{in}}\right)^{2})$,
which reduces to $\delta\mathchoice{\\!}{\\!}{}{}\left(X-X_{\text{in}}\right)$
in the limit $\sigma\to 0$ (while keeping $\sigma\gg\sigma_{\text{in}}$),
whereas in the opposite (physically more reasonable) limit $\sigma\to\infty$,
the position dispersion becomes infinitely large, and the information of
$X_{\text{in}}$ is also lost.

When we do not observe the position, we integrate the probability (80) over
all the possible positions:

$\displaystyle\int_{\mathbb{R}}{\text{d}X\over
2\pi}\left|\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$
$\displaystyle=\sqrt{\sigma_{\text{red}}\over\pi}\exp\mathchoice{\\!}{\\!}{}{}\left(-\sigma_{\text{red}}\left(P-P_{\text{in}}\right)^{2}\right).$
(82)

In the limit of detector spatial resolution coarser than the initial wave
packet size, $\sigma\gg\sigma_{\text{in}}$, we obtain
$\sigma_{\text{red}}\to\sigma_{\text{in}}$, recovering the momentum-space PDF:
${\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{mom}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}=\left|\Braket{P}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$;
see Eq. (46). In the opposite limit $\sigma\ll\sigma_{\text{in}}$, the
probability is governed by the detector resolution,
$\sqrt{\sigma\over\pi}\exp(-\sigma\left(P-P_{\text{in}}\right)^{2})$, which
reduces to $\delta\mathchoice{\\!}{\\!}{}{}\left(P-P_{\text{in}}\right)$ in
the limit $\sigma\to\infty$ (while keeping $\sigma\ll\sigma_{\text{in}}$),
whereas in the opposite (physically more reasonable) limit $\sigma\to 0$, the
momentum dispersion becomes infinitely large, and the information of
$P_{\text{in}}$ is also lost.

For reader’s ease, we list some matrix elements, which may be useful to derive
expressions below: Using the derivative representation (36), we obtain

$\displaystyle\Braket{X,P;\sigma}{{\color[rgb]{1,0.2,0}\widehat{x}}missing}{X_{\text{in}},P_{\text{in}},\sigma_{\text{in}}}$
$\displaystyle=\left(\overline{X}-i\sigma_{\text{red}}\left(P-P_{\text{in}}\right)\right)\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}},\sigma_{\text{in}}},$
(83)
$\displaystyle\Braket{X,P;\sigma}{{\color[rgb]{1,0.2,0}\widehat{p}}missing}{X_{\text{in}},P_{\text{in}},\sigma_{\text{in}}}$
$\displaystyle=\left(\overline{P}+i{X-X_{\text{in}}\over\sigma_{\text{sum}}}\right)\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}},$
(84)

where

$\displaystyle\overline{X}$
$\displaystyle:=\sigma_{\text{red}}\left({X\over\sigma}+{X_{\text{in}}\over\sigma_{\text{in}}}\right),$
$\displaystyle\overline{P}$ $\displaystyle:={\sigma
P+\sigma_{\text{in}}P_{\text{in}}\over\sigma+\sigma_{\text{in}}}.$ (85)

###  5.1 Projective measurement onto phase space

We assert that the transition probability (80) can be regarded in the LT
formalism as the result of the _POVM measurement onto phase space_
$\mathsf{M}_{\text{ph}}$ defined by,
$\forall{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}\in{\color[rgb]{0.9,0.3,1}\mathsfit{Z\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathscr{H}$}}\right)}}$,101010
This measurement $\mathsf{M}_{\text{ph}}$ is non-projective since the states
$\Set{\Ket{X,P;\sigma}}_{\left(X,P\right)\in\mathbb{R}^{2}}$ are not mutually
orthogonal, as demonstrated in Eq. (43).

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{ph}}\widehat{\rho}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=\operatorname{Tr}\mathchoice{\\!}{\\!}{}{}\bigl{[}\Ket{X,P;\sigma}{\Bra{X,P;\sigma}{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}\bigr{]}.$
(86)

Given an initial state $\Ket{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}$,
or more precisely the initial pure state (44), we recover the PDF (80):

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=\left|\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}.$
(87)

For some reader’s ease, we list some results of the integrals:

$\displaystyle\Braket{{\color[rgb]{0,0.5,1}X}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}P}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=P_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}X}{\color[rgb]{0,0.5,1}P}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}P_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}\overline{X}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}\overline{P}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=P_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}\overline{X}}\,{\color[rgb]{0,0.5,1}\overline{P}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}P_{\text{in}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}X^{2}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}^{2}+{\sigma_{\text{sum}}\over 2},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}P^{2}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=P_{\text{in}}^{2}+{1\over 2\sigma_{\text{red}}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}\overline{X}^{2}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}^{2}+{\sigma_{\text{in}}^{2}\over
2\sigma_{\text{sum}}},$
$\displaystyle\Braket{{\color[rgb]{0,0.5,1}\overline{P}^{2}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=P_{\text{in}}^{2}+{\sigma\over
2\sigma_{\text{in}}\sigma_{\text{sum}}},$ (88)

where
${\color[rgb]{0,0.5,1}X\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}:=X$
and
${\color[rgb]{0,0.5,1}P\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}:=P$
as well as
${\color[rgb]{0,0.5,1}\overline{X}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}:=\overline{X}$
and
${\color[rgb]{0,0.5,1}\overline{P}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}:=\overline{P}$,
with $\overline{X}$ and $\overline{P}$ being given in Eq. (85), respectively.

Now we consider the marginalization (27). We define the classical projection
(marginalization) processes $\bm{\pi}_{\text{pos}}$ and
$\bm{\pi}_{\text{mom}}$ by,
$\forall{\color[rgb]{0.2,1,1}\mathsfit{p}}\in{\color[rgb]{0.2,1,1}\mathsfit{W}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}^{2}$}}\right)}$,

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{pos}}p\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}$
$\displaystyle:=\int_{\mathbb{R}}{\text{d}P\over
2\pi}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)},$
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{mom}}p\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}$
$\displaystyle:=\int_{\mathbb{R}}{\text{d}X\over
2\pi}{\color[rgb]{0.2,1,1}\mathsfit{p}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)},$
(89)

with their trivial LT adjoints (30):
$\forall{\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}$}}\right)}$,

$\displaystyle{\color[rgb]{0,0.5,1}\left[\pi^{\star}_{\text{pos}}f\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle={\color[rgb]{0,0.5,1}f}\mathchoice{\\!}{\\!}{}{}\left(X\right),$
$\displaystyle{\color[rgb]{0,0.5,1}\left[\pi^{\star}_{\text{mom}}f\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle={\color[rgb]{0,0.5,1}f}\mathchoice{\\!}{\\!}{}{}\left(P\right).$
(90)

More concretely,

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{pos}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}$
$\displaystyle=\int_{\mathbb{R}}{\text{d}P\over
2\pi}\left|\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}={1\over\sqrt{\pi\sigma_{\text{sum}}}}\exp\mathchoice{\\!}{\\!}{}{}\left(-{\left(X-X_{\text{in}}\right)^{2}\over\sigma_{\text{sum}}}\right),$
(91)
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{mom}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}$
$\displaystyle=\int_{\mathbb{R}}{\text{d}X\over
2\pi}\left|\Braket{X,P;\sigma}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}=\sqrt{\sigma_{\text{red}}\over\pi}\exp\mathchoice{\\!}{\\!}{}{}\left(-\sigma_{\text{red}}\left(P-P_{\text{in}}\right)^{2}\right).$
(92)

For the Gaussian pure state (44), the joint PDF (87) has become separable:

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=2\pi\,{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{pos}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}\,{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{mom}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}.$
(93)

We see that the Gaussian measurement smoothly interpolates the two limiting
measurements in the momentum space and in the position space:

  * •

In the limit of detector spatial resolution finer than the original wave-
packet size, $\sigma\ll\sigma_{\text{in}}$, we obtain
$\sigma_{\text{sum}}\to\sigma_{\text{in}}$ and $\sigma_{\text{red}}\to\sigma$,
and hence

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{pos}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}$
$\displaystyle\to{1\over\sqrt{\pi\sigma_{\text{in}}}}\exp\mathchoice{\\!}{\\!}{}{}\left(-{\left(X-X_{\text{in}}\right)^{2}\over\sigma_{\text{in}}}\right)={\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{pos}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)},$
(94)
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{mom}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}$
$\displaystyle\to\sqrt{\sigma\over\pi}\exp\mathchoice{\\!}{\\!}{}{}\left(-\sigma\left(P-P_{\text{in}}\right)^{2}\right).$
(95)

We have recovered the original PDF in the position space:
$\left|\Braket{X}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$
(see Eq. (45)), whereas the information of the original wave-packet size is
lost in the momentum space. Further limit of spatially coarse detector
$\sigma\to\infty$ (while keeping $\sigma\ll\sigma_{\text{in}}$) gives
${\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{mom}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}\to\delta\mathchoice{\\!}{\\!}{}{}\left(P-P_{\text{in}}\right)$,
whereas in the opposite (physically more reasonable) limit of spatially fine
detector $\sigma\to 0$ makes the momentum dispersion infinitely large, leading
to the loss of information of $P_{\text{in}}$.

  * •

In the limit of detector spatial resolution coarser than the original wave-
packet size, $\sigma\gg\sigma_{\text{in}}$, we obtain
$\sigma_{\text{sum}}\to\sigma$ and $\sigma_{\text{red}}\to\sigma_{\text{in}}$,
and hence

$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{pos}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}$
$\displaystyle\to{1\over\sqrt{\pi\sigma}}\exp\mathchoice{\\!}{\\!}{}{}\left(-{1\over\sigma}\left(X-X_{\text{in}}\right)^{2}\right),$
(96)
$\displaystyle{\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{mom}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}$
$\displaystyle\to\sqrt{\sigma_{\text{in}}\over\pi}\exp\mathchoice{\\!}{\\!}{}{}\left(-\sigma_{\text{in}}\left(P-P_{\text{in}}\right)^{2}\right)={\color[rgb]{0.2,1,1}\mathsfit{\left[M_{\text{mom}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$P$}}\right)}.$
(97)

We have recovered the original PDF in the momentum space
$\left|\Braket{P}{X_{\text{in}},P_{\text{in}};\sigma_{\text{in}}}\right|^{2}$
(see Eq. (46)), whereas the information of the original wave-packet size is
lost in the position space. Further limit of spatially fine detector
$\sigma\to 0$ (while keeping $\sigma\gg\sigma_{\text{in}}$) gives
${\color[rgb]{0.2,1,1}\mathsfit{\left[\pi_{\text{pos}}M_{\text{ph}}\widehat{\rho}_{\text{in}}\right]}\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X$}}\right)}\to\delta\mathchoice{\\!}{\\!}{}{}\left(X-X_{\text{in}}\right)$,
whereas in the opposite (physically more reasonable) limit of spatially coarse
detector $\sigma\to\infty$ makes the position dispersion infinitely large,
leading to the loss of information of $X_{\text{in}}$.

###  5.2 Pullback and pushforward for joint measurement

One might find the LT adjoint $\mathsf{M}^{\star}_{\text{ph}}$ slightly less
trivial because of the non-orthogonality of the basis (43); see also footnote
10. However, we find that the result is diagonal too: For a given function
${\color[rgb]{0,0.5,1}f}\in{\color[rgb]{0,0.5,1}R\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$\mathbb{R}^{2}$}}\right)}$,
its LT adjoint is

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{ph}}f}}$
$\displaystyle=\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}{\color[rgb]{0,0.5,1}f\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}\Ket{X,P;\sigma}\Bra{X,P;\sigma},$
(98)

which can be verified by inserting the expansion (56) into the defining
relation
$\langle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{ph}}f}}\rangle_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}}=\Braket{{\color[rgb]{0,0.5,1}f}}_{{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}}}}$.
This shows that the LT adjoint maintains its diagonal nature even in the
context of the POVM measurement (86).

The position and momentum operators have the following decomposition in the
Gaussian phase space:

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{x}}$
$\displaystyle=\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}X\Ket{X,P;\sigma}\Bra{X,P;\sigma},$
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{p}}$
$\displaystyle=\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}P\Ket{X,P;\sigma}\Bra{X,P;\sigma},$ (99)

which can be verified by sandwiching with $\Bra{x}$ and $\Ket{x^{\prime}}$ for
${\color[rgb]{1,0.2,0}\widehat{x}}$, and with $\Bra{p}$ and $\Ket{p^{\prime}}$
for ${\color[rgb]{1,0.2,0}\widehat{p}}$. We can also confirm this by
sandwiching with $\Bra{X^{\prime},P^{\prime};\sigma}$ and
$\Ket{X^{\prime\prime},P^{\prime\prime};\sigma}$, and then using Eq. (43),
(83), and (84). Comparing with Eq. (98) for the general LT adjoint/pullback,
we obtain its inverse:

$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{ph}}^{\star-1}\widehat{x}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=X,$
$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{ph}}^{\star-1}\widehat{p}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=P.$ (100)

From Eq. (88), we obtain their classical semi-norms,

$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{ph}}^{\star-1}\widehat{x}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}^{2}+{\sigma_{\text{sum}}\over 2},$
$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{ph}}^{\star-1}\widehat{p}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=P_{\text{in}}^{2}+{1\over 2\sigma_{\text{red}}},$ (101)

and their classical semi-inner product,

$\displaystyle\Braket{{\color[rgb]{0,0.5,1}M_{\text{ph}}^{\star-1}\widehat{x}},\,{\color[rgb]{0,0.5,1}M_{\text{ph}}^{\star-1}\widehat{p}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}P_{\text{in}}.$ (102)

Similarly, we obtain the pushforward of ${\color[rgb]{1,0.2,0}\widehat{x}}$
and ${\color[rgb]{1,0.2,0}\widehat{p}}$ (over
${\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}}}$):

$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{ph}\star}\widehat{x}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=\overline{X},$
$\displaystyle{\color[rgb]{0,0.5,1}\left[M_{\text{ph}\star}\widehat{p}\right]\mathchoice{\\!}{\\!}{}{}\left({\color[rgb]{0,0,0}\text{$X,P$}}\right)}$
$\displaystyle=\overline{P},$ (103)

where $\overline{X}$ and $\overline{P}$ are given in Eq. (85). From Eq. (88),
we obtain

$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{ph}\star}\widehat{x}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}^{2}+{\sigma_{\text{in}}^{2}\over
2\sigma_{\text{sum}}},$
$\displaystyle\left\|{\color[rgb]{0,0.5,1}M_{\text{ph}\star}\widehat{p}}\right\|^{2}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=P_{\text{in}}^{2}+{\sigma\over
2\sigma_{\text{in}}\sigma_{\text{sum}}},$ (104)

and

$\displaystyle\Braket{{\color[rgb]{0,0.5,1}M_{\text{ph}\star}\widehat{x}},\,{\color[rgb]{0,0.5,1}M_{\text{ph}\star}\widehat{p}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}$
$\displaystyle=X_{\text{in}}P_{\text{in}}.$ (105)

Accordingly, the pullback of pushforward is

$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{ph}}M_{\text{ph}\star}\widehat{x}}}$
$\displaystyle=\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}\overline{X}\Ket{X,P;\sigma}\Bra{X,P;\sigma},$ (106)
$\displaystyle{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{ph}}M_{\text{ph}\star}\widehat{p}}}$
$\displaystyle=\int_{\mathbb{R}^{2}}{\text{d}X\,\text{d}P\over
2\pi}\overline{P}\Ket{X,P;\sigma}\Bra{X,P;\sigma}.$ (107)

###  5.3 LT error for joint measurement

Subtracting the classical semi-norm squared of pushforward (104) from the
quantum one of the original operator (48), we obtain the resultant LT errors
(14) for ${\color[rgb]{1,0.2,0}\widehat{x}}$ and
${\color[rgb]{1,0.2,0}\widehat{p}}$:

$\displaystyle\varepsilon^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle={\sigma_{\text{red}}\over 2},$
$\displaystyle\varepsilon^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle={1\over 2\sigma_{\text{sum}}}.$ (108)

We see that the left-hand side of the LT inequality (19) becomes

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{ph}}\right]\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle={1\over 2}\sqrt{\sigma_{\text{red}}\over\sigma_{\text{sum}}}.$
(109)

In the limits of spatially fine and coarse detectors $\sigma\to 0$ and
$\infty$, respectively, we obtain

$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle\to\begin{cases}0&(\sigma\to 0),\\\ {\sigma_{\text{in}}\over
2}&(\sigma\to\infty),\end{cases}$
$\displaystyle\varepsilon_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}^{2}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle\to\begin{cases}{1\over 2\sigma_{\text{in}}}&(\sigma\to 0),\\\
0&(\sigma\to\infty).\end{cases}$ (110)

The product of LT errors (109) goes to zero in both the limits $\sigma\to 0$
and $\infty$, while it takes the maximum value $1/4$ when
$\sigma=\sigma_{\text{in}}$.

###  5.4 LT inequality for joint measurement

Using Eqs. (106) and (107), and then Eqs. (83) and (84), we obtain

$\displaystyle\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{ph}}M_{\text{ph}\star}\widehat{x}}},\,{\color[rgb]{1,0.2,0}\widehat{p}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=\Im\Braket{{\color[rgb]{0,0.5,1}\overline{X}}\left({\color[rgb]{0,0.5,1}\overline{P}}+i{{\color[rgb]{0,0.5,1}X}-X_{\text{in}}\over\sigma_{\text{sum}}}\right)}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}={\sigma_{\text{in}}\over
2\sigma_{\text{sum}}},$ (111)
$\displaystyle\Braket{\left[{\color[rgb]{1,0.2,0}\widehat{x}},\,{\color[rgb]{1,0.2,0}\widehat{M^{\star}_{\text{ph}}M_{\text{ph}\star}\widehat{p}}}\right]\over
2i}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}$
$\displaystyle=\Im\Braket{\left({\color[rgb]{0,0.5,1}\overline{X}}-i\sigma_{\text{red}}\left({\color[rgb]{0,0.5,1}P}-P_{\text{in}}\right)\right)^{*}{\color[rgb]{0,0.5,1}\overline{P}}}_{\color[rgb]{0.2,1,1}\mathsfit{M_{\text{ph}}\widehat{\rho}_{\text{in}}}}={\sigma\over
2\sigma_{\text{sum}}},$ (112)

where $\Im$ denotes the imaginary part and we used Eq. (88) in the last step
of each. Subtracting the above two from the measurement-independent part in
the former of Eq. (51), we see that the quantum contribution to the LT lower
bound (20) vanishes:

$\displaystyle\mathcal{I}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle=0.$ (113)

For the semi-classical contribution to the LT lower bound (21), we subtract
the measurement-dependent part (102) from the independent part in the latter
of Eq. (51). As a result, we find it to vanish:

$\displaystyle\mathcal{R}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle=0.$ (114)

It is important that the LT inequality gives no lower bound for the joint
measurement of position and momentum. The LT inequality (19) now becomes

$\displaystyle{1\over 2}\sqrt{\sigma_{\text{red}}\over\sigma_{\text{sum}}}$
$\displaystyle\geq 0,$ (115)

which is saturated in the limits $\sigma\to 0$ and $\infty$. The left-hand
side takes the maximum value $1/4$ at $\sigma=\sigma_{\text{in}}$.

###  5.5 Lee error for joint measurement

Subtracting the quantum semi-norm (48) from the classical (101), we obtain the
Lee errors (22) for ${\color[rgb]{1,0.2,0}\widehat{x}}$ and
${\color[rgb]{1,0.2,0}\widehat{p}}$:

$\displaystyle\widetilde{\varepsilon}^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle={\sigma\over 2},$
$\displaystyle\widetilde{\varepsilon}^{2}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle={1\over 2\sigma}.$ (116)

It is remarkable that, unlike the LT error, the Lee error is solely determined
from the detector resolution, independent of the width of the measured state.

Their product gives the left-hand side of the Lee inequality:

$\displaystyle\widetilde{\varepsilon}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}};\mathsf{M}_{\text{ph}}\right]\widetilde{\varepsilon}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle={1\over 2}.$ (117)

###  5.6 Lee inequality for joint measurement

The quantum contribution to the Lee lower bound (25) is identical to that of
Heisenberg’s, arising from the measurement-independent part, in the former of
Eq. (51):

$\displaystyle\mathcal{I}_{0{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}}\right]$
$\displaystyle={1\over 2}.$ (118)

On the other hand, subtracting the measurement-dependent part (88) from the
independent (51), we find that the semi-classical contribution to the Lee
lower bound (26) vanishes:

$\displaystyle\widetilde{\mathcal{R}}_{{\color[rgb]{0.9,0.3,1}\mathsfit{\widehat{\rho}_{\text{in}}}}}\mathchoice{\\!}{\\!}{}{}\left[{\color[rgb]{1,0.2,0}\widehat{x}},{\color[rgb]{1,0.2,0}\widehat{p}};\mathsf{M}_{\text{ph}}\right]$
$\displaystyle=0.$ (119)

Unlike the LT inequality, the Lee inequality provides the meaningful lower
bound, which has turned out to be identical to Heisenberg’s, and is invariably
saturated for the pure Gaussian initial state.

##  6 Summary and discussion

We have demonstrated that the Gaussian wave-packet formalism can be a
practical realization of the joint measurement of position and momentum,
shedding light on Heisenberg’s original philosophy of the uncertainty
principle within the framework of Lee and Tsutsui’s error, disturbance, and
their uncertainty relations. We have successfully obtained the Lee-Tsutsui
(LT) error and the refined Lee error in the context of position-momentum
measurement for the first time. Our findings indicate that the LT uncertainty
relation, in the limiting case of projective measurement of either position or
momentum, becomes trivial: $0=0$.

In contrast, the refined Lee uncertainty relation, focusing on errors for
local representability, provides a constant lower bound that remains
unaffected by such limits and is invariably saturated for a pure Gaussian
initial state. This lower bound aligns with Heisenberg’s value. Our study
delves into the Gaussian measurement process in the phase space, demonstrating
its capability to interpolate smoothly between the projective measurements of
position and momentum.

Building on the foundational research of Heisenberg; Kennard and Robertson
(KR); and Schrödinger, and the advancements by Arthurs, Kelly, and Goodman
(AKG); Ozawa; and Watanabe, Sagawa, and Ueda (WSU), our research deepens the
understanding of quantum uncertainty. We leverage the Gaussian wave-packet
formalism to provide a comprehensive and tangible approach to the abstract
constructs of the LT formalism. This paper advocates for the POVM measurement
onto the Gaussian wave-packet basis as a natural method for joint position and
momentum measurement, offering concrete expressions for previously abstract
concepts like pullback and pushforward within the LT formalism. Our work
bridges the theoretical and practical aspects of quantum mechanics, setting
the stage for further research and applications in quantum uncertainty.

It has been shown that the Gaussian basis used in our current study on a fixed
time slice can be extended to the whole spacetime [16, 17, 18, 22]; see Ref.
[25] for the complete basis of Lorentz invariant wave packets and Ref. [26]
for that of Lorentz covariant ones including the spin degrees of freedom. It
would be fascinating to explore the time-energy uncertainty relation on this
ground. It is also interesting to show the LT and Lee inequalities for the
standard Clauser-Horne-Shimony-Holt (CHSH) inequality as a concrete
realization in the finite-dimensional space. These developments will be
presented in subsequent publications.

### Acknowledgement

N.O. is indebted to Sho Machiyama for assistance in understanding the LT
formalism and to Shogo Tanimura for fostering a welcoming research environment
in his group. We are grateful to Jaeha Lee for providing the talk slides, to
Kenji Nishiwaki for helpful conversation, and to Juntaro Wada for useful
comments. The work of K.O. is in part supported by the JSPS KAKENHI Grant Nos.
JP19H01899 and JP21H01107.

## References

  * [1] W. Heisenberg, _Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik_ , _Z. Phys._ 43 (1927) 172.
  * [2] E. H. Kennard, _Zur Quantenmechanik einfacher Bewegungstypen_ , _Z. Phys._ 44 (1927) 326.
  * [3] H. P. Robertson, _The Uncertainty Principle_ , _Phys. Rev._ 34 (1929) 163.
  * [4] E. Schrodinger, _About Heisenberg uncertainty relation_ , _Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. )_ 19 (1930) 296 [quant-ph/9903100].
  * [5] E. Arthurs and J. L. Kelly, _B.S.T.J. briefs: On the simultaneous measurement of a pair of conjugate observables_ , _Bell Syst. Tech. J._ 44 (1965) 725.
  * [6] E. Arthurs and M. S. Goodman, _Quantum correlations: A generalized Heisenberg uncertainty relation_ , _Phys. Rev. Lett._ 60 (1988) 2447.
  * [7] M. Ozawa, _Uncertainty relations for joint measurements of noncommuting observables_ , _Physics Letters A_ 320 (2004) 367.
  * [8] M. Ozawa, _Universally valid reformulation of the heisenberg uncertainty principle on noise and disturbance in measurement_ , _Phys. Rev. A_ 67 (2003) 042105.
  * [9] H. Yuen and M. Lax, _Multiple-parameter quantum estimation and measurement of nonselfadjoint observables_ , _IEEE Trans. Info. Theor._ 19 (1973) 740.
  * [10] Y. Watanabe, T. Sagawa and M. Ueda, _Uncertainty relation revisited from quantum estimation theory_ , _Phys. Rev. A_ 84 (2011) 042121.
  * [11] Y. Watanabe and M. Ueda, _Quantum estimation theory of error and disturbance in quantum measurement_ , 1106.2526.
  * [12] J. Lee and I. Tsutsui, _Geometric formulation of universally valid uncertainty relation for error_ , 2002.04008.
  * [13] J. Lee and I. Tsutsui, _Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems_ , _Entropy_ 22 (2020) 1222.
  * [14] J. Lee, _A Universal Formulation of Uncertainty Relation for Errors under Local Representability_ , 2203.08197.
  * [15] J. Lee, _A Universal Formulation of Uncertainty Relation for Error-Disturbance and Local Representability of Quantum Observables_ , 2204.11814.
  * [16] K. Ishikawa and T. Shimomura, _Generalized S-matrix in mixed representations_ , _Prog. Theor. Phys._ 114 (2006) 1201 [hep-ph/0508303].
  * [17] K. Ishikawa and K.-y. Oda, _Particle decay in Gaussian wave-packet formalism revisited_ , _PTEP_ 2018 (2018) 123B01 [1809.04285].
  * [18] K. Ishikawa, K. Nishiwaki and K.-y. Oda, _Scalar scattering amplitude in the Gaussian wave-packet formalism_ , _PTEP_ 2020 (2020) 103B04 [2006.14159].
  * [19] H. Mitani and K.-y. Oda, _Decoherence in Neutrino Oscillation between 3D Gaussian Wave Packets_ , 2307.12230.
  * [20] K. Ishikawa, O. Jinnouchi, K. Nishiwaki and K.-y. Oda, _Wave-Packet Effects: A Solution for Isospin Anomalies in Vector-Meson Decay_ , 2308.09933.
  * [21] K. Ishikawa and Y. Tobita, _Finite-size corrections to Fermi’s golden rule: I. Decay rates_ , _PTEP_ 2013 (2013) 073B02 [1303.4568].
  * [22] K. Ishikawa, K. Nishiwaki and K.-y. Oda, _New effect in wave-packet scattering of quantum fields_ , _Phys. Rev. D_ 108 (2023) 096013 [2102.12032].
  * [23] G. Kaiser, _Phase Space Approach to Relativistic Quantum Mechanics. 1. Coherent State Representation for Massive Scalar Particles_ , _J. Math. Phys._ 18 (1977) 952.
  * [24] G. R. Kaiser, _Phase Space Approach to Relativistic Quantum Mechanics. 2. Geometrical Aspects_ , _J. Math. Phys._ 19 (1978) 502.
  * [25] K.-y. Oda and J. Wada, _A complete set of Lorentz-invariant wave packets and modified uncertainty relation_ , _Eur. Phys. J. C_ 81 (2021) 751 [2104.01798].
  * [26] K.-y. Oda and J. Wada, _Lorentz-covariant spinor wave packet_ , 2307.05932.
  * [27] P. Busch, P. Lahti and R. F. Werner, _Proof of Heisenberg’s error-disturbance relation_ , _Phys. Rev. Lett._ 111 (2013) 160405 [1306.1565].
  * [28] P. Busch, P. Lahti and R. F. Werner, _Measurement uncertainty relations_ , _J. Math. Phys._ 55 (2014) 042111 [1312.4392].
  * [29] P. Busch, P. Lahti and R. F. Werner, _Colloquium: Quantum root-mean-square error and measurement uncertainty relations_ , _Rev. Mod. Phys._ 86 (2014) 1261.