Using the normalisation condition $\sum_{i}\alpha_{i}^{2}=1$, we arrive at

$\alpha_{j}^{2}\left[1-\mathrm{Re}\left(\omega^{jk}\langle
b_{j}|B_{2}^{k}|b_{j}\rangle\right)\right]\leq\varepsilon.$ (4.117)

Again using the property (4.112),and then the inequality (4.111), we arrive at

$\displaystyle\frac{1}{d}\left[1-\mathrm{Re}\left(\omega^{jk}\langle
b_{j}|B_{2}^{k}|b_{j}\rangle\right)\right]$ $\displaystyle\leq$
$\displaystyle\varepsilon+\sqrt{2\varepsilon}\left[1-\mathrm{Re}\left(\omega^{jk}\langle
b_{j}|B_{2}^{k}|b_{j}\rangle\right)\right]$ (4.118) $\displaystyle\leq$
$\displaystyle\varepsilon+2\sqrt{2\varepsilon}$ $\displaystyle\leq$
$\displaystyle 3\sqrt{2\varepsilon},$

where we used the fact that the maximum of the term
$\left[1-\mathrm{Re}\left(\omega^{jk}\langle
b_{j}|B_{2}^{k}|b_{j}\rangle\right)\right]$ is two. Thus, we can finally
conclude that

$\mathrm{Re}\left(\omega^{jk}\langle b_{j}|B_{2}^{k}|b_{j}\rangle\right)\geq
1-3d\sqrt{2\varepsilon}.$ (4.119)

Recall that the eigenvalues of $B_{2}$ is $\\{1,\omega,\ldots,\omega^{d-1}\\}$
and thus we can consider the eigendecomposition of $B_{2}$ using orthogonal
projections $P_{i}$ as

$B_{2}=\sum_{i=0}^{d-1}\omega^{i}P_{i}.$ (4.120)

Here $P_{i}$ in general might be of rank higher than one. Plugging this
decomposition into the condition (4.119), we have that

$\sum_{i=0}^{d-1}\mathrm{Re}\left(\omega^{(i+j)k}\langle
b_{j}|P_{i}|b_{j}\rangle\right)\geq 1-3d\sqrt{2\varepsilon}\quad\forall k.$
(4.121)

Notice that the above equation holds trivially for $k=0$. Taking a sum over
$k$ gives us

$\langle b_{j}|P_{-j}|b_{j}\rangle\geq 1-3d\sqrt{2\varepsilon},$ (4.122)

which implies that each vector ${|{b_{j}}\rangle}$ is close to the projection
$P_{-j}$ which signifies the subspace corresponding to the $(d-j)-th$ outcome
of $B_{2}$. Now, let us look at the following normalised vectors

$\displaystyle{|{\overline{v}_{j}}\rangle}=\frac{P_{-j}{|{b_{j}}\rangle}}{\|P_{-j}{|{b_{j}}\rangle}\|},$
(4.123)

where we can infer that ${|{\overline{v}_{j}}\rangle}$ are mutually orthogonal
as well, that is,
${\langle{\overline{v}_{i}}|}\overline{v}_{j}\rangle=\delta_{i,j}$. Note that,
from the condition (4.122) the normalisation factor of each of the vector
${|{\overline{v}_{j}}\rangle}$ is close to one, that is,
$\|P_{-j}{|{b_{j}}\rangle}\|\geq 1-2d\sqrt{2\varepsilon}$. Further, using
these vectors we can express $B_{2}$ as

$B_{2}=\sum_{i=0}^{d-1}\omega^{-i}{|{\overline{v}_{i}}\rangle}\\!{\langle{\overline{v}_{i}}|}\oplus
B_{2}^{\prime},$ (4.124)

such that $B_{2}^{\prime}$ is some operator acting on the support orthogonal
to the subspace spanned by the vectors $\\{{|{\overline{v}_{i}}\rangle}\\}$.
Now, there always exist a unitary
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ such that
$U_{B}{|{\overline{v}_{i}}\rangle}={|{i}\rangle}$ and thus,

$U_{B}B_{2}U_{B}^{\dagger}=\sum_{i=0}^{d-1}\omega^{-i}{|{i}\rangle}\\!{\langle{i}|}\oplus
B_{2}^{\prime\prime}.$ (4.125)

Substituting the above form of $B_{2}$ to (4.119) and then taking a sum over
$k$, we can also deduce that

$\mathrm{Re}\left(\langle b_{i}|U_{B}^{\dagger}|i\rangle\right)\geq
1-3d\sqrt{2\varepsilon}.$ (4.126)

Finally, we can compute the distance between the states (4.97) using the above
results as,

$\displaystyle\left\|\mathbb{1}_{A}\otimes
U_{B}{|{{\psi}}\rangle}-{|{\phi^{+}_{d}}\rangle}\right\|$ $\displaystyle=$
$\displaystyle\left\\{2\left[1-\mathrm{Re}(\langle{\psi}|U_{B}^{\dagger}|\phi_{d}^{+}\rangle)\right]\right\\}^{1/2}$
(4.127) $\displaystyle=$
$\displaystyle\left\\{2\left[1-\frac{1}{\sqrt{d}}\sum_{i}\alpha_{i}\,\mathrm{Re}(\langle
b_{i}|U_{B}^{\dagger}|i\rangle)\right]\right\\}^{1/2}.$

Now, using the conditions (4.108) and (4.126), we have

$\displaystyle\sum_{i}\alpha_{i}\,\mathrm{Re}(\langle
b_{i}|U_{B}^{\dagger}|i\rangle)$ $\displaystyle\geq$
$\displaystyle(1-3d\sqrt{2\varepsilon})\sum_{i}\alpha_{i}$ (4.128)
$\displaystyle\geq$
$\displaystyle(1-3d\sqrt{2\varepsilon})\sqrt{d}\sqrt{1-\varepsilon}$
$\displaystyle\geq$
$\displaystyle\sqrt{d}\left[1-(3d+1)\sqrt{2\varepsilon}\right],$

where the inequality in the third line is a consequence of two inequalities,
first, $\sqrt{1-\varepsilon}\geq 1-\varepsilon$ for any $\varepsilon\leq 1$
and second, $\varepsilon\leq\sqrt{2\varepsilon}$. Using the inequality
(4.128), we finally obtain the robustness of the state (4.97) as

$\displaystyle\left\|\left(\mathbb{1}_{A}\otimes
U_{B}\right){|{{\psi}}\rangle}-{|{\phi^{+}_{d}}\rangle}\right\|\leq\sqrt{2(3d+1)}\sqrt[4]{2\varepsilon}.$
(4.129)

Let us now prove the inequalities (4.98) and (4.99). For simplicity, we denote
the ideal observables as $B_{i}^{\prime}$ and
$U_{B}B_{i}U_{B}^{\dagger}=\widetilde{B}_{i}$. We first observe that the
Hilbert Schmidt norm can be written as a vector norm using
${|{\phi_{+}^{d}}\rangle}$ as

$\left\|\widetilde{B}_{i}^{k}-(B^{\prime}_{i})^{k}\right\|_{2}=\sqrt{d}\left\|\left[\widetilde{B}_{i}^{k}-(B^{\prime}_{i})^{k}\right]{|{\phi_{d}^{+}}\rangle}\right\|.$
(4.130)

Then, using the triangle inequality $|A+B|\geq\left||A|-|B|\right|$ and
denoting ${|{\widetilde{\psi}}\rangle}=\mathbb{1}_{A}\otimes
U_{B}{|{\psi}\rangle}$, we have that

$\displaystyle\left\|\widetilde{B}_{i}^{k}{|{\widetilde{\psi}}\rangle}-(B^{\prime}_{i})^{k}{|{\phi_{d}^{+}}\rangle}\right\|$
$\displaystyle=$
$\displaystyle\left\|\widetilde{B}_{i}^{k}{|{\widetilde{\psi}}\rangle}-(B^{\prime}_{i})^{k}{|{\phi_{d}^{+}}\rangle}+(B^{\prime}_{i})^{k}{|{\phi_{d}^{+}}\rangle}-\widetilde{B}_{i}^{k}{|{\phi_{d}^{+}}\rangle}\right\|$
(4.131) $\displaystyle\geq$
$\displaystyle\left\|\left[\widetilde{B}_{i}^{k}-(B^{\prime}_{i})^{k}\right]{|{\phi_{d}^{+}}\rangle}\right\|-\left\|\widetilde{B}_{i}^{k}({|{\widetilde{\psi}}\rangle}-{|{\phi_{d}^{+}}\rangle})\right\|,$

Then using using the inequality (4.130) and the fact that $B_{i}$ is unitary,
leads us to

$\left\|\widetilde{B}_{i}^{k}-(B^{\prime}_{i})^{k}\right\|_{2}\leq\sqrt{d}\left(\left\|\widetilde{B}_{i}^{k}{|{\widetilde{\psi}}\rangle}-(B^{\prime}_{i})^{k}{|{\phi_{d}^{+}}\rangle}\right\|+\left\|{|{\widetilde{\psi}}\rangle}-{|{\phi_{d}^{+}}\rangle}\right\|\right).$
(4.132)

The first term in the right hand side of the above inequality can be computed
using (4.127) by first noting that $A_{i}^{k}\otimes B_{i}^{\prime
k}{|{\phi_{+}^{d}}\rangle}={|{\phi_{+}^{d}}\rangle}$ and then using the fact
that $A_{i}$ is unitary,

$\displaystyle\left\|(A_{i}^{k}\otimes\widetilde{B}_{i}^{k}){|{\widetilde{\psi}}\rangle}-{|{\widetilde{\psi}}\rangle}+{|{\widetilde{\psi}}\rangle}-{|{\phi^{+}_{d}}\rangle}\right\|\leq\left\|(A_{i}^{k}\otimes\widetilde{B}_{i}^{k}){|{\widetilde{\psi}}\rangle}-{|{\widetilde{\psi}}\rangle}\right\|+\left\|{|{\widetilde{\psi}}\rangle}-{|{\phi^{+}_{d}}\rangle}\right\|.$
(4.133)

Finally using the conditions (4.101) we get

$\displaystyle\left\|(A_{i}^{k}\otimes\widetilde{B}_{i}^{k}){|{\widetilde{\psi}}\rangle}-{|{\widetilde{\psi}}\rangle}\right\|$
$\displaystyle=$ $\displaystyle\left\|(A_{i}^{k}\otimes
B_{i}^{k}){|{\psi}\rangle}-{|{\psi}\rangle}\right\|$ (4.134) $\displaystyle=$
$\displaystyle\left\\{2\left[1-\mathrm{Re}\left(\langle\psi|A_{i}^{k}\otimes
B_{i}^{k}|\psi\rangle\right)\right]\right\\}^{1/2}$ $\displaystyle\leq$
$\displaystyle\sqrt{2\varepsilon}.$

Using the above inequality, we get that

$\displaystyle\left\|(A_{i}^{k}\otimes\widetilde{B}_{i}^{k}){|{\widetilde{\psi}}\rangle}-{|{\widetilde{\psi}}\rangle}+{|{\widetilde{\psi}}\rangle}-{|{\phi^{+}_{d}}\rangle}\right\|\leq\sqrt{2\varepsilon}+2\left\|{|{\widetilde{\psi}}\rangle}-{|{\phi^{+}_{d}}\rangle}\right\|,$
(4.135)

and then using the condition (4.97) we obtain (4.98) and (4.99). ∎

###  4.5 Conclusions and discussions

We proposed a one-sided device-independent scheme for a certification of a
large family of incompatible measurements with arbitrary number of outcomes
termed here genuinely incompatible. Our scheme also allows for the
certification of the two-qudit maximally entangled of any arbitrary local
dimension using only two measurements on each side. This is the first
certification of mutually unbiased bases of any dimension using quantum
steering. Unlike the previous approaches in literature, our scheme is more
general in the sense that we do not assume that the state shared between the
parties is pure or the measurements on the untrusted side are projective. This
makes our scheme even significant in cryptographic scenarios where there can
be an external Eve who can have access to the state as well as the untrusted
measurements. We also find a simple technique for robust certification of a
smaller family of genuinely incompatible observables including the complete
mutually unbiased bases of any prime dimension and pair of them for non-prime
dimensions. We also considered the scenario when the measurements are not
genuinely incompatible and observed that only a part of the measurement can be
certified based on the quantum state that realises the quantum bound of the
steering functional.

One of the drawbacks of the 1SDI scheme as compared to fully device-
independent schemes is that one has to assume that one of the parties is
trusted. However, they still posses some of the essential features that would
be useful in certification of quantum technologies. First, if one possesses a
well characterised quantum device, here it is the measuring device of the
trusted party, our scheme provides a way to compare any other untrusted device
with the trusted device using minimal resources. In other words, our scheme
given a trusted measurement device allows one to verify that any other device
performs the desired measurements. Second, our scheme is applicable to every
1SDI scenario where a client wants to verify the state supplied by an
untrusted source along with the untrusted measuring device. Thus, one-sided
quantum key distribution schemes [121, 122] and randomness generation would be
two major applications of our certification scheme. Third, even from a
practical point of view, implementing 1SDI protocols is much easier than the
DI protocols [121]. The reason being that demonstration of quantum steering is
practically more robust to noise and can be observed with detectors of lower
efficiencies than Bell violations [174, 175]. Fourth, there are a very few
analytical methods dedicated to the field of quantum certification, let alone
to higher dimensional quantum certification. Thus, purely from a mathematical
perspective, we introduced techniques that can be relevant for other schemes
that aim to characterise arbitrary dimensional quantum systems. Finally, in
certain scenarios 1SDI schemes can be made device-independent if one can
device-independently characterise the measuring device of the trusted party as
pointed out recently in Ref. [3].

Some interesting follow-up problems arise from our work. The most important
among them would be whether our 1SDI scheme can be made fully device-
independent, that is, the question whether it is possible to design a scheme
for certifying every set of genuinely incompatible observables in a device-
independent way still remains open. Another interesting problem would be to
find 1SDI scheme that can be used for certification of incompatible
observables that are not genuine incompatible. It would also be interesting to
construct steering functionals whose quantum bound is saturated by non-
maximally entangled state of arbitrary local dimension $d$ and thus eventually
finding a scheme allowing for certification of every bipartite entangled
state. This problem is tackled in the next chapter of this thesis. An
ambitious problem would be extend the idea of genuine incompatibility to non-
projective schemes and then find steering functionals for certification of
POVM’s.

##  Chapter 5 Certification of any pure bipartite entangled state and optimal
randomness using quantum steering

###  5.1 Introduction

Generating genuine random outputs inaccessible to hackers is one of the key
steps in any key distribution protocol, be it classical or quantum. As
discussed before in Chapter 2, in classical physics the security of such
protocols relies on the fact that large numbers can not be efficiently
factorized using classical computers. However, using quantum computers such
protocols can be broken in polynomial time. As was shown in [76], violation of
Bell inequalities serve as the most secure way to certify genuine randomness.
However, from a practical point of view, performing a Bell experiment is
extremely challenging as one requires very low levels of noise along with the
detectors being highly efficient. As a consequence, we need to consider
scenarios where one can efficiently generate secure randomness, easier to
implement, require minimal resources and are robust to noise. We showed in
Chapter 3 that one can securely generate $\log_{2}d$ bits of randomness using
a quantum system of dimension $d$. From a foundational point of view, it still
remains an open and highly non-trivial problem whether one can securely
generate the optimal amount of randomness using a quantum system of arbitrary
dimension $d$ which is $2\log_{2}d$ bits.

In this chapter, we aim to solve the above problem by considering the one-
sided device-independent (1SDI) scenario where the required resource is
quantum steering. As a matter of fact, it was shown in [121] that quantum
steering can be observed using detectors with much lower efficiency and more
noise-robust when compared to observing Bell non-locality. This makes 1SDI
setting an ideal scenario to construct protocols for randomness generation
that can be practically implemented. Randomness generation in 1SDI setting has
been recently explored in [132, 176]. Particularly in Ref. [176], the authors
construct a protocol for generating $\log_{2}d$ bits of randomness from a
quantum system of dimension $d$ in a secure way by using $d$-outcome
projective measurements.

To this end, we first construct a family of steering inequalities maximally
violated by any pure entangled state of local dimension $d$ and two
$d-$outcome measurements on each side. Using the maximal violation of the
steering inequalities, we then certify any pure bipartite entangled state and
a pair of mutually unbiased bases of arbitrary dimension on the untrusted
side. This is the first instance, where any pure bipartite entangled state can
be certified using the least number of measurements required to observe
quantum non-locality, that is, using only two measurements on each side.
Additionally, we demonstrate that any rank-one extremal measurement can be
certified using our protocol. Based on these results, we finally demonstrate
the certification of $2\log_{2}d$ bits of randomness using the certified
entangled state of local dimension $d$ and the certified $d^{2}-$outcome
extremal measurement. We further show that for systems of dimension
$d=3,4,5,6$, the optimal amount of randomness can be certified using partially
entangled states. This further strengthens our scheme as one can generate
highest amount of randomness by employing less resource in terms of
entanglement.

###  5.2 Family of steering inequalities

Let us shortly describe again the quantum steering scenario introduced in
Chapter 2 and also in Chapter 4 in an analogous way to Bell scenario. Alice
and Bob are located in spatially separated labs. Both of them receive two
subsystems from a preparation device. Alice is trusted and performs two known
$d-$outcome measurements on the received subsystem labelled by $x=1,2$. In our
case, we consider these measurements in the observable form as $A_{0}=Z_{d}$
and $A_{1}=X_{d}$ which as described in Eq. (2.91) of Chapter 4, constitute a
pair of mutually unbiased bases. Bob also performs two $d-$outcome
measurements on his subsystem labelled by $y=1,2$. They collect enough
statistics to construct the joint probability distribution $\\{p(a,b|x,y)\\}$.
The scenario is depicted in Fig. 4.1 of Chapter 4 such that $N=2$.

We now construct a family of steering inequalities which is expressed in the
observable picture and using a collection of positive non-zero numbers
$\bm{\alpha}=\\{\alpha_{0},\alpha_{1},\ldots,\alpha_{d-1}\\}$ such that
$\sum_{i=0}^{d-1}\alpha_{i}^{2}=1$, as

$I_{2,d,2}(\bm{\alpha})=\sum_{k=1}^{d-1}\left\langle A_{0}^{k}\otimes
B_{k|0}+\gamma(\bm{\alpha})A_{1}^{k}\otimes
B_{k|1}+\delta_{k}(\bm{\alpha})A_{0}^{k}\right\rangle\leq\beta_{L}(\bm{\alpha}),$
(5.1)

where the coefficients $\gamma(\bm{\alpha})$ and $\delta_{k}(\bm{\alpha})$ are
given by

$\displaystyle\gamma(\bm{\alpha})=d\left({\sum_{\begin{subarray}{c}i,j=0\\\
i\neq
j\end{subarray}}^{d-1}\frac{\alpha_{i}}{\alpha_{j}}}\right)^{-1},\qquad\delta_{k}(\bm{\alpha})=-\frac{\gamma(\bm{\alpha})}{d}\sum_{\begin{subarray}{c}i,j=0\\\
i\neq j\end{subarray}}^{d-1}\frac{\alpha_{i}}{\alpha_{j}}\omega^{k(d-j)}.$
(5.2)

Notice that $\gamma(\bm{\alpha})\geq 0$ for any choice of $\bm{\alpha}$ and
$\delta_{k}(\bm{\alpha})$ are in general complex. Alice is trusted (or fully
characterised), and her measurements are expressed in the observable picture
as

$A_{0}=Z_{d}=\sum_{i=0}^{d-1}\omega^{i}{|{i}\rangle}\\!{\langle{i}|},\qquad
A_{1}=X_{d}=\sum_{i=0}^{d-1}{|{i+1}\rangle}\\!{\langle{i}|}.$ (5.3)

Recall that the set of vectors $\\{{|{i}\rangle}\\}_{i=0}^{d-1}$ represents
the computational basis in $\mathbbm{C}^{d}$. It is worth noting here that the
expression $I_{2,d,2}(\bm{\alpha})$ (2.67) is real for any choice of
$\bm{\alpha}$. This is because $B_{d-k|i}=(B_{k|i})^{\dagger}$ for any
generalised observable (see Eq. (2.22) of Chapter 2) and also that the
coefficient $\delta_{d-k}(\bm{\alpha})=\delta_{k}^{*}(\bm{\alpha})$ for any
$k$. Let us now express the steering functional in (2.67) using the joint
probability picture as

$\displaystyle I_{2,d,2}(\bm{\alpha})=d\sum_{a,b=0}^{d-1}c_{a,b}p(a,b|0,0)\\!$
$\displaystyle+$
$\displaystyle\\!\gamma(\bm{\alpha})\left(d\sum_{a,b=0}^{d-1}c_{a,b}p(a,b|1,1)-\sum_{\begin{subarray}{c}i,a=0\\\
i\neq a\end{subarray}}^{d-1}\alpha_{i}\frac{p(a|0)}{\alpha_{a}}\right)$ (5.4)
$\displaystyle-$
$\displaystyle\\!1-\gamma(\bm{\alpha})-\delta_{0}(\bm{\alpha}),$

such that

$\displaystyle c(a,b)=\begin{cases}1\quad\text{if}\quad a\oplus_{d}b=0\\\
0\quad\text{otherwise}\ \ \ \ \ \ ,\end{cases}$ (5.5)

where $a\oplus_{d}b$ represents $a+b$ modulo $d$. To arrive at the above
expression we used the fourier transform to express the expectation values in
terms of the joint probabilities as defined in (4.4). Note from Eq. (5.2) that
$\delta_{0}(\bm{\alpha})=-1$, which implies that

$\displaystyle
I_{2,d,2}(\bm{\alpha})=d\sum_{a,b=0}^{d-1}c_{a,b}p(a,b|0,0)+\gamma(\bm{\alpha})\left(d\sum_{a,b=0}^{d-1}c_{a,b}p(a,b|1,1)-\sum_{\begin{subarray}{c}i=0\end{subarray}}^{d-1}\alpha_{i}\sum_{a=0}^{d-1}\frac{p(a|0)}{\alpha_{a}}\right).$

Using the above expression of the steering functional, let us now compute the
classical bound $\beta_{L}(\bm{\alpha})$ of the above steering inequality.

####  5.2.1 Classical bound

As derived before in Eq. (4.7) of Chapter 4, to find the classical bound of a
steering functional, we need to consider an LHS model such that the joint
probability distribution can be expressed as

$p(a,b|x,y)=\sum_{\lambda}p(\lambda)p(a|x,\rho_{\lambda})p(b|y,\lambda),$
(5.7)

where $\lambda$ denotes some unknown variables the collection of which is
denoted by $\lambda$. These variables occur with a probability distribution
$p(\lambda)$. Here $p(a|x,\rho_{\lambda})$ is the local probability of Alice
obtaining an outcome $a$ given the input $x$ and some quantum state
$\rho_{\lambda}$ acts on $\mathbbm{C}^{d}$ 111This is due to the fact that
Alice is trusted and is known to perform quantum measurments on some quantum
state. and $p(b|y,\lambda)$ is the local probability distribution depending on
some unknown variable $\lambda$. For such a probability distribution (5.7),
the steering functional from Eq. (5.2) can be expressed as,

$\displaystyle I_{2,d,2}(\bm{\alpha})$ $\displaystyle=$ $\displaystyle
d\sum^{d-1}_{\begin{subarray}{c}a=0\end{subarray}}\sum_{\lambda}\
p(\lambda)p(a|0,\rho_{\lambda})p(d-a|0,\lambda)$ $\displaystyle+$
$\displaystyle\gamma(\bm{\alpha})\left(d\sum^{d-1}_{\begin{subarray}{c}a=0\end{subarray}}\sum_{\lambda}\
p(\lambda)p(a|1,\rho_{\lambda})p(d-a|1,\lambda)-\sum_{i}\alpha_{i}\sum_{a=0}^{d-1}\sum_{\lambda}\frac{p(\lambda)p(a|0,\rho_{\lambda})}{\alpha_{a}}\right),$

where to obtain the last term, we used the no-signalling conditions given in
Eq. (2.53) of Chapter 2 such that

$\displaystyle p(a|0)=\sum_{b}p(a,b|0,y)\qquad\forall y.$ (5.9)

Notice that in the above expression we used the fact that
$\sum_{b}p(b|y,\lambda)=1$ for any $\lambda$ and then denoted
$\rho_{A}=\sum_{\lambda}p(\lambda)\rho_{\lambda}$. Let us first consider the
first two terms in Eq. (5.2.1) and find their upper bound in the following
way,

$\displaystyle I_{2,d,2}(\bm{\alpha})$ $\displaystyle\leq$ $\displaystyle
d\sum_{\lambda}\ p(\lambda)\max_{a}p(a|0,\rho_{\lambda})$ $\displaystyle+$
$\displaystyle\gamma(\bm{\alpha})\left(d\sum_{\lambda}\
p(\lambda)\max_{a}p(a|1,\rho_{\lambda})-\sum_{i}\alpha_{i}\sum_{a=0}^{d-1}\sum_{\lambda}\frac{p(\lambda)p(a|0,\rho_{\lambda})}{\alpha_{a}}\right),$

where to obtain the first inequality we used the fact that
$\sum_{a}p(d-a|y,\lambda)=1$ for any $y$ and $\lambda$ and then used the
mathematical identity $\sum_{i}s_{i}w_{i}\leq\max_{i}\\{s_{i}\\}$ whenever
$w_{i}\geq 0$ and $\sum_{i}w_{i}=1$. Now as was done before in Chapter 4 [see
Eq. (4.12)] and using the fact that $\sum_{\lambda}p(\lambda)=1$, we get that

$I_{2,d,2}(\bm{\alpha})\leq\max_{\rho}\left[d\max_{a}\\{p(a|0,\rho)\\}+\gamma(\bm{\alpha})\left(d\max_{a}\\{p(a|1,\rho)\\}-\sum_{i=0}^{d-1}\alpha_{i}\sum_{a=0}^{d-1}\frac{p(a|0,\rho)}{\alpha_{a}}\right)\right].$
(5.11)

Notice the above expression is convex in the state $\rho$. As a consequence,
the maximisation in the above expression can be taken over pure states
${|{\psi}\rangle}\in\mathbbm{C}^{d}$. Now expressing the state
${|{\psi}\rangle}$ in the computational basis of $\mathbbm{C}^{d}$ as
${|{\psi}\rangle}=\sum_{i}\eta_{i}{|{i}\rangle}$, and then plugging in Alice’s
observables, $A_{0}=Z_{d}$ and $A_{1}=X_{d}$, the formula (5.11) can be
rewritten as

$I_{2,d,2}(\bm{\alpha})\leq\max_{\begin{subarray}{c}|\eta_{0}|,\ldots,|\eta_{d-1}|\\\
|\eta_{0}|^{2}+\ldots+|\eta_{d-1}|^{2}=1\end{subarray}}\left\\{d\max_{a}\\{|\eta_{a}|^{2}\\}+\gamma(\bm{\alpha})\left[\left(\sum_{i=0}^{d-1}|\eta_{i}|\right)^{2}-\sum_{i=0}^{d-1}\alpha_{i}\sum_{a=0}^{d-1}\frac{|\eta_{a}|^{2}}{\alpha_{a}}\right]\right\\}.$
(5.12)

Given any arbitrary collection of positive numbers $\bm{\alpha}$ such that the
sum of the squares of those numbers is one, it is not straightforward to find
this bound. However, we can show here that the right hand side of the above
formula is strictly less than $d$ for any $\bm{\alpha}$. We prove this claim
using the technique of contradiction. Let us first consider the term inside
the square brackets of the above expression (5.12) and show that it is always
negative. For this, let us recall the Cauchy–Schwarz inequality for positive
real numbers also known as Sedrakyan’s inequality [177],

$\displaystyle\frac{(\sum_{i}u_{i})^{2}}{\sum_{i}v_{i}}\leq\sum_{i}\frac{u_{i}^{2}}{v_{i}}.$
(5.13)

Now, substituting $u_{i}=|\eta_{i}|$ and $v_{i}=\alpha_{i}$, we can rewrite
the above expression as

$\left(\sum_{i=0}^{d-1}|\eta_{i}|\right)^{2}\leq\sum_{i=0}^{d-1}\alpha_{i}\sum_{j=0}^{d-1}\frac{|\eta_{i}|^{2}}{\alpha_{i}}.$
(5.14)

Thus, we can conclude from (5.12) that the the classical bound of
$I_{2,d,2}(\bm{\alpha})$ is less than or equal to $d$, that is,

$\displaystyle I_{2,d,2}(\bm{\alpha})\leq d\max_{a}\\{|\eta_{a}|^{2}\\}\leq
d.$ (5.15)

We now show that the above inequlity can not be saturated by L.H.S. models. To
this end, let us assume that $I_{2,d,2}(\bm{\alpha})=d$. This implies that the
term inside the square brackets in (5.12) vanishes and thus,

$\displaystyle\left(\sum_{i=0}^{d-1}|\eta_{i}|\right)^{2}=\sum_{i=0}^{d-1}\alpha_{i}\sum_{j=0}^{d-1}\frac{|\eta_{i}|^{2}}{\alpha_{i}}.$
(5.16)

along with the first term

$\max_{a}\\{|\eta_{a}|^{2}\\}=1.$ (5.17)

Recall that equality holds in the Cauchy–Schwarz inequality (5.13) iff
$u_{i}=\kappa v_{i}$ for all $i$ where $\kappa$ is some real coefficient.
Thus, in (5.16) $\alpha_{i}=\kappa|\eta_{i}|$ for each $i$. Using the
condition $\sum_{i}\alpha_{i}^{2}=\sum_{i}|\eta_{i}|^{2}=1$ and that
$\alpha_{i}>0$ for any $i$, imposes that $\kappa=1$, and therefore
$\alpha_{i}=|\eta_{i}|$. But $\alpha_{i}<1$ for any $i$ which contradicts the
second condition (5.17). Thus, our initial assumption is wrong which implies
that the maximal classical value of the steering function
$I_{2,d,2}(\bm{\alpha})$ is strictly less than $d$ for arbitrary collection of
positive numbers $\alpha_{i}$ such that
$\alpha_{0}^{2}+\ldots+\alpha_{d-1}^{2}=1$. Now, we move onto finding the
quantum bound of the steering functional in (2.67).

####  5.2.2 Quantum bound

Here we show that the maximum value of the steering functional
$I_{2,d,2}(\bm{\alpha})$ in Eq. (5.1) obtainable using quantum states and
measurements is given by $\beta_{Q}(\bm{\alpha})=d$. The result is stated
below as a mathematical theorem.

######  Theorem 5.1.

For any collection of positive real numbers
$\bm{\alpha}=\\{\alpha_{0},\alpha_{1},\ldots,\alpha_{d-1}\\}$ such that
$\alpha_{0}^{2}+\ldots+\alpha_{d-1}^{2}=1$, the quantum bound of the steering
functional $I_{2,d,2}(\bm{\alpha})$ is independent of $\bm{\alpha}$ and is
given by $\beta_{Q}(\bm{\alpha})=d$.

###### Proof.

Let us begin by introducing the steering operator corresponding to the
steering functional $I_{2,d,2}(\bm{\alpha})$ in (5.1),

$\hat{I}_{2,d,2}(\bm{\alpha})=\sum_{k=1}^{d-1}\left(A_{0}^{k}\otimes
B_{k|0}+\gamma(\bm{\alpha})A_{1}^{k}\otimes
B_{k|1}+\delta_{k}(\bm{\alpha})A_{0}^{k}\right).$ (5.18)

Recall that $A_{0}=Z_{d}$ and $A_{1}=X_{d}$ and $B_{i}^{\prime}s$ are any
$d$-outcome generalised observables corresponding to the measurements of Bob.
Our aim is to show that

$\beta_{Q}(\bm{\alpha})=\max_{\rho_{AB},B_{i}}\mathrm{Tr}\left[\hat{I}_{2,d,2}(\bm{\alpha})\rho_{AB}\right]=d,$
(5.19)

where $\rho_{AB}$ acting on $\mathbbm{C}^{d}\otimes\mathcal{H}_{B}$ represents
some quantum state shared between Alice and Bob and $\mathcal{H}_{B}$
represents the Hilbert space of Bob of arbitrary but finite dimension. As the
expression (5.19) is linear, we can optimise this over pure states
${|{\psi_{AB}}\rangle}\in\mathbbm{C}^{d}\otimes\mathcal{H}_{B}$, that is,

$\max_{{|{\psi_{AB}}\rangle},B_{i}}\langle\psi_{AB}|\hat{I}_{2,d,2}|\psi_{AB}\rangle=d.$
(5.20)

For simplicity, in the rest of the proof we drop the subscript $AB$ from the
state. We first show that the expectation value of the steering operator for
any ${|{\psi}\rangle}$ is upper bounded by $d$ and then find a quantum
realisation that achieves this bound. For this purpose, let us break the
steering operator $\hat{I}_{2,d,2}(\bm{\alpha})$ into two parts as

$\hat{I}_{2,d,2}(\bm{\alpha})=\sum_{k=1}^{d-1}A_{0}^{k}\otimes
B_{k|0}+S(\bm{\alpha})$ (5.21)

such that

$S(\bm{\alpha})=\sum_{k=1}^{d-1}\left[\gamma(\bm{\alpha})A_{1}^{k}\otimes
B_{k|1}+\delta_{k}(\bm{\alpha})A_{0}^{k}\right].$ (5.22)

Notice that the above operator is hermitian which is due to the fact that
$A_{0}^{d-k}=(A_{0}^{k})^{\dagger}$ and $B_{d-k|0}=B_{k|0}^{\dagger}$ [see Eq.
(2.22)]. It is trivial to see that the absolute values of the expectation
value of each term in the first part of operator in Eq. (5.21) is less than or
equal to $1$ as $A_{0}$ is unitary and
$B_{k|0}^{\dagger}B_{k|0}\leq\mathbb{1}$ for any $k$ [see Eq. (2.23)], that
is,

$|\langle\psi|A^{k}_{0}\otimes B_{k|0}|\psi\rangle|\leq 1,$ (5.23)

for any ${|{\psi}\rangle}$ and $k=1,\ldots,d-1$. This allows us to conclude
from (5.21) that

$\displaystyle\langle\psi|\hat{I}_{2,d,2}(\bm{\alpha})|\psi\rangle$
$\displaystyle\leq$ $\displaystyle
d-1+\langle\psi|S(\bm{\alpha})|\psi\rangle.$ (5.24)

Now, we demonstrate for any ${|{\psi}\rangle}$ that
$\langle\psi|S(\bm{\alpha})|\psi\rangle\leq 1$. For this purpose, let us first
notice that the state ${|{\psi}\rangle}$ belongs to
$\mathbbm{C}^{d}\otimes\mathcal{H}_{B}$. Thus, as discussed before in Chapter
2 any such state can be written using the computational basis in
$\mathbbm{C}^{d}$ as

${|{\psi_{ABE}}\rangle}=\sum_{i=0}^{d-1}\lambda_{i}{|{i}\rangle}_{A}{|{e_{i}}\rangle}_{B},$
(5.25)

where $\lambda_{i}$ are real and non-negative numbers such that
$\lambda_{0}^{2}+\ldots+\lambda_{d-1}^{2}=1$, and ${|{e_{i}}\rangle}$ are
vectors belonging to $\mathcal{H}_{B}$ which are not orthogonal in general.
Plugging in this state, we find the expectation value of $S(\bm{\alpha})$ as

$\displaystyle\langle\psi|S(\bm{\alpha})|\psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{d-1}\sum_{i,j=0}^{d-1}\left[\gamma(\bm{\alpha})\lambda_{i}\lambda_{j}{\langle{i}|}A_{1}^{k}{|{j}\rangle}{\langle{e_{i}}|}B_{k|1}{|{e_{j}}\rangle}+\delta_{k}(\bm{\alpha})\lambda_{i}\lambda_{j}{\langle{i}|}A_{0}^{k}{|{j}\rangle}\\!{\langle{e_{i}}|}e_{j}\rangle\right]$
(5.26) $\displaystyle=$
$\displaystyle\sum_{k=1}^{d-1}\sum_{i=0}^{d-1}\left[\gamma(\bm{\alpha})\lambda_{i}\lambda_{i-k}{\langle{e_{i}}|}B_{k|1}{|{e_{i-k}}\rangle}+\delta_{k}(\bm{\alpha})\lambda_{i}^{2}\omega^{ik}\right],$

where to arrive at the second equality, we plugged in the explicit forms of
$A_{i}^{\prime}s$ and then used the fact that
$Z_{d}^{k}{|{i}\rangle}=\omega^{ki}{|{i}\rangle}$ and
$X_{d}^{k}{|{i}\rangle}={|{i+k}\rangle}$. Focusing on the second term of the
above expression and plugging in $\delta_{0}(\bm{\alpha})=-1$, we obtain that

$\sum_{k=1}^{d-1}\sum_{i=0}^{d-1}\delta_{k}(\bm{\alpha})\lambda_{i}^{2}\omega^{ik}=1+\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\delta_{k}(\bm{\alpha})\lambda_{i}^{2}\omega^{ik}.$
(5.27)

Using then the explicit form of $\delta_{k}(\bm{\alpha})$ given in Eq. (5.2),
we arrive at

$\displaystyle\sum_{k=1}^{d-1}\sum_{i=0}^{d-1}\delta_{k}(\bm{\alpha})\lambda_{i}^{2}\omega^{ik}=1+\gamma(\bm{\alpha})-\gamma(\bm{\alpha})\sum_{i,j=0}^{d-1}\frac{\alpha_{i}}{\alpha_{j}}\lambda_{j}^{2},$
(5.28)

where we also used the identity

$\sum_{k=0}^{d-1}\omega^{k(i-j)}=d\delta_{ij}.$ (5.29)

Notice that since $\gamma(\bm{\alpha})$ is real, the above expression (5.28)
is also real. Since, $S(\bm{\alpha})$ is a Hermitian operator we have that any
expectation value of this operator is real. Plugging the relations (5.28) and
remembering that $\lambda_{i}^{\prime}s$ are also real, we can rewrite Eq.
(5.26) as

$\langle\psi|S(\bm{\alpha})|\psi\rangle=1+\gamma(\bm{\alpha})\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\lambda_{i}\lambda_{i-k}\mathrm{Re}\left({\langle{e_{i}}|}B_{k|1}{|{e_{i-k}}\rangle}\right)-\gamma(\bm{\alpha})\sum_{i,j=0}^{d-1}\frac{\alpha_{i}}{\alpha_{j}}\lambda_{j}^{2}.$
(5.30)

Exploiting the fact that $\mathrm{Re}(z)\leq|z|$ for any $z\in\mathbbm{C}$ and
that $B_{k|1}^{\dagger}B_{k|1}\leq\mathbb{1}$ for any $k$, we get that
$\mathrm{Re}\left({\langle{e_{i}}|}B_{k|1}{|{e_{i-k}}\rangle}\right)\leq 1$,
Thus, we finally arrive at

$\langle\psi|S(\bm{\alpha})|\psi\rangle\leq
1+\gamma(\bm{\alpha})\left[\left(\sum_{i=0}^{d-1}\lambda_{i}\right)^{2}-\sum_{i=0}^{d-1}\alpha_{i}\sum_{j=0}^{d-1}\frac{\lambda_{j}^{2}}{\alpha_{j}}\right].$
(5.31)

Now, using the Cauchy-Schwarz inequality (5.13) in which we substitute
$u_{i}=\lambda_{i}$ and $v_{i}=\alpha_{i}$, we can conclude that the term
inside the square brackets of the above expression is less than or equal to
$0$. Thus, we can finally conclude that the expectation value of
$S(\bm{\alpha})$ for any state ${|{\psi}\rangle}$ is less than or equal to
$1$. that is,

$\langle\psi|S(\bm{\alpha})|\psi\rangle\leq 1$ (5.32)

and hence, putting it back into (5.24), we obtain

$\langle\psi|\hat{I}_{2,d,2}(\bm{\alpha})|\psi\rangle\leq d,$ (5.33)

for any ${|{\psi}\rangle}\in\mathbbm{C}^{d}\otimes\mathcal{H}_{B}$. It is also
interesting to note here that the relation (5.32) is saturated when

$\displaystyle\left(\sum_{i=0}^{d-1}\lambda_{i}\right)^{2}=\sum_{i=0}^{d-1}\alpha_{i}\sum_{j=0}^{d-1}\frac{\lambda_{j}^{2}}{\alpha_{j}}.$
(5.34)

Now, using the fact that Sedrakyan’s inequality (5.13) is saturated if
$u_{i}=\kappa v_{i}$ for some $\kappa\in\mathbbm{C}$. Substituting
$u_{i}=\lambda_{i}$ and $v_{i}=\alpha_{i}$ in the Sedrakyan’s inequality
(5.13), we see that $\lambda_{i}=\kappa\alpha_{i}$ for all $i$. Now, using
normalisation we get that the only solution of Eq. (5.34) is
$\lambda_{i}=e^{i\phi}\alpha_{i}$ for some arbitrary phase $\phi$. As
$\alpha_{i}>0$, we get that $|\lambda_{i}|>0$ for all $i$. Thus, we can simply
conclude that any state saturating the inequality (5.32) is locally full-rank.

Let us now consider a family of states parametrized by the collection of
positive real coefficients $\bm{\alpha}$,

$\displaystyle{|{\psi(\bm{\alpha})}\rangle}_{AB}=\sum_{i=0}^{d-1}\alpha_{i}{|{i}\rangle}_{A}{|{i}\rangle}_{B},$
(5.35)

where the local bases of (5.35) is the computational basis of
$\mathbbm{C}^{d}$. Notice that the above state is a valid normalised quantum
state $\sum_{i=0}^{d-1}\alpha_{i}^{2}=1$. Notice also that every pure
bipartite entangled state of Schmidt rank $d$ can be expressed as these states
(5.35) up to local unitary transformations. Now, consider Bob’s observables to
be projective and to satisfy

$\displaystyle B_{0}=Z_{d}^{*},\qquad B_{1}=X_{d}.$ (5.36)

Plugging this state and the observables in the steering functional in (5.1),
we get that $I_{2,d,2}(\bm{\alpha})=d$. This completes the proof. ∎

Notice that the maximal violation of the steering inequality (5.1) by a state
${|{\psi_{AB}}\rangle}$ and Bob’s observables $B_{i}$ can only be achieved iff
the inequalities (5.40) and (5.43) are saturated. Thus, we arrive at the
following conditions

$\langle\psi|A_{0}^{k}\otimes B_{k|0}|\psi\rangle=1$ (5.37)

for any $k=1,\ldots,d-1$ as well as

$\langle\psi|S(\bm{\alpha})|\psi\rangle=1,$ (5.38)

where $S(\bm{\alpha})$ is defined in Eq. (5.22). Now, consider the Cauchy-
Schwarz inequality for vectors given as

$\displaystyle\mathrm{Re}(\langle u|v\rangle)\leq|\langle
u|v\rangle|\leq\left|\langle{u}{|{u}\rangle}\langle{v}{|{v}\rangle}\right|.$
(5.39)

From the condition (5.37), let us now substitute
${|{u}\rangle}={|{\psi}\rangle}$ and ${|{v}\rangle}=A_{0}^{k}\otimes
B_{k|0}|\psi\rangle$ in the above inequality (5.39). Then using the fact that
$B_{k|0}^{\dagger}B_{k|0}\leq\mathbb{1}$, we get that both the L.H.S. and
R.H.S. of (5.13) are equal to one. This can only happen iff ${|{u}\rangle}$
and ${|{v}\rangle}$ are linearly dependent, that is,
${|{u}\rangle}=\lambda{|{v}\rangle}$ for some $\lambda\in\mathbbm{C}$. As both
${|{u}\rangle}$ and ${|{v}\rangle}$ are normalised, we get that
${|{u}\rangle}=e^{i\phi}{|{v}\rangle}$ where $\phi$ is some arbitrary phase.
Putting it back into Eq. (5.37), we get that $\phi=0$. As a consequence, we
obtain the following relation

$\displaystyle A_{0}^{k}\otimes
B_{k|0}\,{|{\psi_{AB}}\rangle}={|{\psi_{AB}}\rangle}\qquad(k=1,\ldots,d-1).$
(5.40)

Let us now consider the condition (5.38) and express $S(\bm{\alpha})$ as
$S(\bm{\alpha})=S(\bm{\alpha})_{+}-S(\bm{\alpha})_{-}$ where
$S(\bm{\alpha})_{+}$ is a positive matrix spanned by the eigenvectors
corresponding to the positive eigenvalues of $S(\bm{\alpha})$ and
$S(\bm{\alpha})_{-}$ is a positive matrix spanned by the eigenvectors
corresponding to the negative eigenvalues of $S(\bm{\alpha})$. Plugging this
decomposition in (5.38), we get that

$\displaystyle\langle\psi|S(\bm{\alpha})_{+}|\psi\rangle=1+\langle\psi|S(\bm{\alpha})_{-}|\psi\rangle$
(5.41)

Notice now from Eq. (5.32) that the maximum eigenvalue of $S(\bm{\alpha})$ is
one and as a consequence $\langle\psi|S(\bm{\alpha})_{+}|\psi\rangle\leq 1$
and also $\langle\psi|S(\bm{\alpha})_{+}^{2}|\psi\rangle\leq 1$. Thus, from
the above condition (5.41), we can conclude that
$\langle\psi|S(\bm{\alpha})_{-}|\psi\rangle=0$ which implies that
$S(\bm{\alpha})_{-}|\psi\rangle=0$ as $S(\bm{\alpha})_{-}$ is positive. As a
consequence, we also have that $\langle\psi|S(\bm{\alpha})_{+}|\psi\rangle=1$.
Going back to the Cauchy-Schwarz inequality (5.39), we substitute
${|{u}\rangle}={|{\psi}\rangle}$ and
${|{v}\rangle}=S(\bm{\alpha})|\psi\rangle$. Let us now compute
$|\langle{v}{|{v}\rangle}|$ by using the decomposition of $S(\bm{\alpha})$ and
also using the fact that it is hermitian

$\displaystyle\langle\psi|S(\bm{\alpha})|\psi\rangle\leq\langle\psi|S(\bm{\alpha})^{2}|\psi\rangle$
$\displaystyle=$
$\displaystyle\langle\psi|\left(S(\bm{\alpha})_{+}^{2}+S(\bm{\alpha})_{-}^{2}+S(\bm{\alpha})_{+}S(\bm{\alpha})_{-}+S(\bm{\alpha})_{-}S(\bm{\alpha})_{+}\right)|\psi\rangle$
(5.42) $\displaystyle=$
$\displaystyle\langle\psi|S(\bm{\alpha})_{+}^{2}|\psi\rangle\leq 1$

where to get to the second line of the above expression we used that
$S(\bm{\alpha})_{-}|\psi\rangle=0$. Now, as concluded before, we get that
${|{u}\rangle}=e^{i\phi}{|{v}\rangle}$ where $\phi$ is some arbitrary phase.
Now, again using Eq. (5.38) thus we finally arrive at the relation

$\displaystyle\left(\sum_{k=1}^{d-1}\left[\gamma(\bm{\alpha})A_{1}^{k}\otimes
B_{k|1}+\delta_{k}(\bm{\alpha})A_{0}^{k}\right]\right){|{\psi_{AB}}\rangle}={|{\psi_{AB}}\rangle}.$
(5.43)

The relations (5.40) and (5.43) would be particularly useful for certification
of the quantum states and measurements that achieve the maximal violation of
the steering functional $I_{2,d,2}$. Let us now show that observation of
maximal violation of our steering inequalities allows us to certify the state
shared between Alice and Bob and also the measurements performed by Bob.

###  5.3 ISDI certification of all pure bipartite entangled states

Here, we present the 1SDI certification of all pure bipartite entangled states
using the saturation of the quantum bound of the steering functional (5.1)
$I_{2,d,2}=\beta_{Q}$. Let us first recall that we can only characterise Bob’s
observables on the support of his local state $\rho_{B}$. Thus, without loss
of generality we assume it to be full rank. This can also be understood as
that Bob’s observables and local state $\rho_{B}$ act on the same Hilbert
space $\mathcal{H}_{B}$.

######  Theorem 5.2.

Consider that Alice and Bob perform the quantum steering experiment and
observe that the steering functional

$\displaystyle I_{2,d,2}(\bm{\alpha})=\sum_{k=1}^{d-1}\left\langle
A_{0}^{k}\otimes B_{k|0}+\gamma(\bm{\alpha})A_{1}^{k}\otimes
B_{k|1}+\delta_{k}(\bm{\alpha})A_{0}^{k}\right\rangle,$ (5.44)

attains the maximal quantum value $\beta_{Q}=d$ where $d$ denotes the number
of outcomes of each measurement. Alice is trusted and her measurements are
given by $A_{0}=Z_{d}$ and $A_{1}=X_{d}$ [cf. Eq. (2.91)]. Let us say that the
maximal quantum bound is achieved using the state $\rho_{AB}$ acting on
$\mathbbm{C}^{d}\otimes\mathcal{H}_{B}$ and Bob’s generalised observables
$B_{i}\ (i\in\\{1,2\\})$ acting on $\mathcal{H}_{B}$. Then, the following
statements hold true for any integer $d\geq 2$ :

  1. 1.

Bob’s measurements are projective. Equaivalently, the operators $B_{k|i}$ for
all $k,i$ are unitary and $B_{k|i}=B_{1|i}^{k}\equiv B_{i}^{k}$.

  2. 2.

Bob’s Hilbert space $\mathcal{H}_{B}$ admits a decomposition into a
$d-$dimensional Hilbert space $\mathbbm{C}^{d})_{B^{\prime}}$ and some unknown
but finite dimensional auxiliary Hilbert space
$\mathcal{H}_{B^{\prime\prime}}$,

$\displaystyle\mathcal{H}_{B}=(\mathbbm{C}^{d})_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}.$
(5.45)

  3. 3.

There exists a local unitary on Bob’s side
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ such that

$\displaystyle(\mathbb{1}_{A}\otimes U_{B})\rho_{AB}(\mathbb{1}_{A}\otimes
U_{B}^{\dagger})={|{\psi(\bm{\alpha})}\rangle}\\!{\langle{\psi(\bm{\alpha})}|}_{AB^{\prime}}\otimes\rho_{B^{\prime\prime}}^{aux}.$
(5.46)

where ${|{\psi(\bm{\alpha})}\rangle}$ is the state given in (5.35) and

$\displaystyle\forall i,\quad
U_{B}\,B_{i}\,U_{B}^{\dagger}=A_{i}^{*}\otimes\mathbb{1}_{B^{\prime\prime}},$
(5.47)

where $B^{\prime\prime}$ denotes Bob’s auxiliary system.

###### Proof.

The proof is divided into two major steps. In the first step, we exploit the
relations (5.40) and (5.43) to find Bob’s observables that result in the
maximal violation of the steering inequality (5.1). Again, using the relations
(5.40) and (5.43) and the derived Bob’s observables, we find the family of
states shared beween Alice and Bob parametrised by the collection of numbers
$\bm{\alpha}$. For our proof, as was discussed before in Chapter 2, the state
shared between Alice and Bob $\rho_{AB}$ is purified by adding an ancillary
system $E$ possessed by some external agent, named Eve, such that
$\rho_{AB}=\mathrm{Tr}_{E}({|{\psi_{ABE}}\rangle}\\!{\langle{\psi_{ABE}}|})$
where
${|{\psi_{ABE}}\rangle}\in\mathbbm{C}^{d}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{E}$.

##### Bob’s observables

Before finding the explicit forms of Bob’s observables that maximally violate
our steering inequality (5.1), we show that these generalised observables must
correspond to projective measurements. Let us concentrate on Bob’s first
observable and follow the exact same technique as was used in Chapter 4 from
Eqs . (4.38) to (4.42) as the relations (5.40) and (4.38) are identical.
First, we apply $Z_{d}^{d-k}\otimes B_{d-k|0}$ to the relation (5.23) and then
recalling that $Z_{d}$ is unitary as well as that
$B_{d-k|0}=B_{k|0}^{\dagger}$ from (2.22), we obtain that

$\mathbb{1}_{AE}\otimes(B_{k|0}^{\dagger}B_{k|0}){|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}.$
(5.48)

Taking a partial trace over the subsystems $AE$, we arrive at the following
condition

$(B_{k|0}^{\dagger}B_{k|0})\rho_{B}=\rho_{B},$ (5.49)

where
$\rho_{B}=\mathrm{Tr}_{AE}[{|{\psi_{ABE}}\rangle}\\!{\langle{\psi_{ABE}}|}]$.
Recall that $\rho_{B}$ is full-rank and thus it is non-singular and
invertible. This allows us to immediately conclude that
$B_{k|0}^{\dagger}B_{k|0}=\mathbb{1}_{B}$ and consequently
$B_{k|0}B_{k|0}^{\dagger}=\mathbb{1}_{B}$, and thus $B_{k|0}$ is unitary for
any $k=0,\ldots,d-1$. Now using Fact 1, we can conclude that Bob’s
measurements are projective, that is, the positive semi-definite operators
representing the measurement are mutually orthogonal projectors. Further, the
fact that $B_{0}$ is projective imposes that $B_{k|0}$ are powers of $B_{1|0}$
[see Chapter 2]. As a consequence, from now on we can simply denote
$B_{k|0}=B_{0}^{k}$, where $B_{0}\equiv B_{1|0}$.

Let us now focus on Bob’s second observable and show that it corresponds to a
projective measurement too. For this purpose, we refer to the second condition
(5.32) and then consider the general representation of any state
${|{\psi_{ABE}}\rangle}\in\mathbbm{C}^{d}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{E}$
as in Eq. (5.25)

${|{\psi_{ABE}}\rangle}=\sum_{i}\lambda_{i}{|{i}\rangle}_{A}{|{e_{i}}\rangle}_{BE},$
(5.50)

where $\lambda_{i}\geq 0$ and ${|{e_{i}}\rangle}_{BE}$ are vectors that are in
general not orthogonal. Now recall that, we showed in the previous subsection
we showed that any state satisfying the condition (5.32) must be locally full-
rank. Thus, in the state (5.50), $\lambda_{i}>0$ for all $i$. For simplicity,
from here on we drop all the subscripts in the notation of the state. Now, by
plugging this state in the condition
${\langle{\psi_{ABE}}|}S(\bm{\alpha}){|{\psi_{ABE}}\rangle}=1$, we obtain [cf.
Eq. (5.30)]

$\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\lambda_{i}\lambda_{i-k}\mathrm{Re}\left({\langle{e_{i}}|}[B_{k|1}\otimes\mathbb{1}_{E}]{|{e_{i-k}}\rangle}\right)=\sum_{i,j=0}^{d-1}\frac{\alpha_{i}}{\alpha_{j}}\lambda_{j}^{2}.$
(5.51)

Dropping the identity acting on Eve, $\mathbb{1}_{E}$, for the time being and
then using the inequality (5.13) with in $u_{i}=\lambda_{i}$ and
$v_{i}=\alpha_{i}$, we arrive at

$\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\lambda_{i}\lambda_{i-k}\mathrm{Re}\left({\langle{e_{i}}|}B_{k|1}{|{e_{i-k}}\rangle}\right)\geq\left(\sum_{i=0}^{d-1}\lambda_{i}\right)^{2}.$
(5.52)

Notice that the term on the right hand side of the above inequality can be
expanded as
$\left(\sum_{i=0}^{d-1}\lambda_{i}\right)^{2}=\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\lambda_{i}\lambda_{i-k}$.
As a consequence, we immediately obtain that

$\displaystyle\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\lambda_{i}\lambda_{i-k}\mathrm{Re}\left[\left({\langle{e_{i}}|}B_{k|1}{|{e_{i-k}}\rangle}\right)-1\right]\geq
0.$ (5.53)

Again, recalling that $B_{k|1}^{\dagger}B_{k|1}\leq\mathbb{1}_{B}$ for any $k$
allows us to conclude that
$\mathrm{Re}\left({\langle{e_{i}}|}[B_{k|1}{|{e_{i-k}}\rangle}\right)\leq 1$
for any $i$ and $k$. However, $\lambda_{i}$ being non-negative in the
inequality (5.52), forces the term inside the square brackets to be $0$ for
all $i,k$. Thus, we finally obtain

$\mathrm{Re}\left({\langle{e_{i}}|}B_{k|1}{|{e_{i-k}}\rangle}\right)=1.$
(5.54)

As the states ${|{e_{i}}\rangle}$ are normalised, the above condition is
satisfied iff
$B_{k|1}\otimes\mathbb{1}_{E}{|{e_{i-k}}\rangle}={|{e_{i}}\rangle}$. For this
purpose, we again employ the Cauchy-Schwarz inequality (5.39) where
${|{u}\rangle}={|{e_{i}}\rangle}$ and
${|{v}\rangle}=B_{k|1}{|{e_{i-k}}\rangle}$ and then the fact that
$B_{k|1}^{\dagger}B_{k|1}\leq\mathbb{1}$ from which we obtain
$B_{k|1}\otimes\mathbb{1}_{E}{|{e_{i-k}}\rangle}=e^{i\phi}{|{e_{i}}\rangle}$
for some arbitrary phase $\phi$. Again using Eq. (5.54), we get that $\phi=0$.
Now, we multiply this equation with its conjugate transpose, to observe that

${\langle{e_{i-k}}|}[B_{k|1}^{\dagger}B_{k|1}\otimes\mathbb{1}_{E}]{|{e_{i-k}}\rangle}=1.$
(5.55)

for any $i,k$. This implies that for any $k$ the above condition is satisfied
for any $i$. As a consequence, we arrive at a simple relation for any $k$

${\langle{e_{i}}|}[B_{k|1}^{\dagger}B_{k|1}\otimes\mathbb{1}_{E}]{|{e_{i}}\rangle}=1\qquad(i=0,\ldots,d-1).$
(5.56)

Tracing out Eve’s subsystem, further implies that
$\mathrm{Tr}[B_{k|1}^{\dagger}B_{k|1}\rho_{B}^{i}]=1$ for all $i$, where
$\rho_{B}^{i}=\mathrm{Tr}_{E}[{|{e_{i}}\rangle}\\!{\langle{e_{i}}|}_{BE}]$. As
$\rho_{B}^{i}$ is positive and again recalling that
$B_{k|1}^{\dagger}B_{k|1}\leq\mathbb{1}_{B}$ for any $k$ allows us to conclude
that this condition holds true iff $B_{k|1}^{\dagger}B_{k|1}$ is an identity
acting onto the support of $\rho_{B}^{i}$. However, notice that the support of
Bob’s reduced state
$\rho_{B}=\mathrm{Tr}_{AE}[{|{\psi}\rangle}\\!{\langle{\psi}|}_{ABE}]$ is in-
fact composed of the supports of $\rho_{B}^{i}$’s. To see this, we use the
decomposition of the state ${|{\psi}\rangle}_{ABE}$ given in (5.50) to obtain
that

$\displaystyle\rho_{B}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{AE}\left[\sum_{i,j=0}^{d-1}\lambda_{i}\lambda_{j}{|{i}\rangle}\\!{\langle{j}|}\otimes{|{e_{i}}\rangle}\\!{\langle{e_{j}}|}\right]$
(5.57) $\displaystyle=$
$\displaystyle\mathrm{Tr}_{E}\left[\sum_{i=0}^{d-1}\lambda_{i}^{2}{|{e_{i}}\rangle}\\!{\langle{e_{i}}|}\right]=\sum_{i=0}^{d-1}\lambda_{i}^{2}\rho_{B}^{i}.$

As a consequence, $B_{k|1}^{\dagger}B_{k|1}$ is an identity that acts on the
entire support of Bob’s reduced state $\rho_{B}$, and thus we finally have
that $B_{k|1}^{\dagger}B_{k|1}=\mathbb{1}_{B}$ and consequently
$B_{k|1}B_{k|1}^{\dagger}=\mathbb{1}_{B}$ for all $k$. Again, using Fact 1, we
can conclude that the second Bob’s observable corresponds to projective
measurements and hence from here on, we denote $B_{k|1}=B_{1}^{k}$, where
$B_{1}\equiv B_{1|1}$. This completes the part of the proof to show that the
maximal violation of the steering inequality (5.1) can only be achieved when
Bob’s both measurements are projective.

Now, we move onto finding the explicit form of Bob’s both observables. Let us
first consider the relation (5.43) and then apply $\mathbb{1}_{A}\otimes
B_{1}$ to it, which after rearranging some terms gives us

$\displaystyle\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left(X_{d}^{k}\otimes
B_{1}^{k+1}\right){|{\psi_{ABE}}\rangle}=\left[\left(\mathbb{1}_{A}-\sum_{k=1}^{d-1}\delta_{k}(\bm{\alpha})Z_{d}^{k}\right)\otimes
B_{1}\right]{|{\psi_{ABE}}\rangle}.$ (5.58)

To simplify the notation, let us introduce the following operator

$\overline{Z}_{A}:=\mathbb{1}_{A}-\sum_{k=1}^{d-1}\delta_{k}(\bm{\alpha})Z_{d}^{k}.$
(5.59)

Now, an application of $Z_{d}^{-1}\otimes\mathbb{1}_{B}$ to the left hand side
of Eq. (5.58) gives us

$\displaystyle\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left(Z_{d}^{-1}X_{d}^{k}\otimes
B_{1}^{k+1}\right){|{\psi_{ABE}}\rangle}=(\overline{Z}_{A}Z_{d}^{-1}\otimes
B_{1}){|{\psi_{ABE}}\rangle},$ (5.60)

where we can interchange the positioning of $Z_{d}$ and $\overline{Z}_{A}$ as
they commute. Then, by using the commutation relation $Z_{d}X_{d}=\omega
X_{d}Z_{d}$, we can rewrite Eq. (5.60) as

$\displaystyle\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left(\omega^{-k}X_{d}^{k}\otimes
B_{1}^{k+1}\right)\left(Z_{d}^{-1}\otimes\mathbb{1}_{B}\right){|{\psi_{ABE}}\rangle}=(\overline{Z}_{A}\otimes
B_{1})\left(Z_{d}^{-1}\otimes\mathbb{1}_{B}\right){|{\psi_{ABE}}\rangle},$
(5.61)

where we again used the fact that $Z_{d}^{-1}\otimes\mathbb{1}_{B}$ and
$\mathbb{1}_{A}\otimes B_{1}$ commute. Now, using Eq. (5.40) for $k=1$, that
is,
$\left(Z_{d}^{-1}\otimes\mathbb{1}_{B}\right){|{\psi_{ABE}}\rangle}=\mathbb{1}_{A}\otimes
B_{0}{|{\psi_{ABE}}\rangle}$, we finally get

$\displaystyle\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left(\omega^{-k}X_{d}^{k}\otimes
B_{1}^{k+1}B_{0}\right){|{\psi_{ABE}}\rangle}=(\overline{Z}_{A}\otimes
B_{1}B_{0}){|{\psi_{ABE}}\rangle}.$ (5.62)

Applying $B_{0}$ from the left hand side of the expression (5.58), we obtain

$\displaystyle\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left(X_{d}^{k}\otimes
B_{0}B_{1}^{k+1}\right){|{\psi_{ABE}}\rangle}=(\overline{Z}_{A}\otimes
B_{0}B_{1}){|{\psi_{ABE}}\rangle}.$ (5.63)

In the next step, we multiply $\omega^{-1}$ to Eq. (5.62) and then subtract it
from Eq. $\eqref{procond2}$, which immediately gives us

$\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left[X_{d}^{k}\otimes\left(B_{0}B_{1}^{k+1}-\omega^{-(k+1)}B_{1}^{k+1}B_{0}\right)\right]{|{\psi_{ABE}}\rangle}=[\overline{Z}_{A}\otimes(B_{0}B_{1}-\omega^{-1}B_{1}B_{0})]{|{\psi_{ABE}}\rangle}.$
(5.64)

Let us again consider the relation (5.43) and multiply it by
$X_{d}^{-1}\otimes\mathbb{1}_{B}$ from the left hand side, which gives us

$\displaystyle\sum_{k=1}^{d-1}\left(\gamma(\bm{\alpha})X_{d}^{k-1}\otimes
B_{1}^{k}\right){|{\psi_{ABE}}\rangle}=(X_{d}^{-1}\overline{Z}_{A}\otimes\mathbb{1}_{B}){|{\psi_{ABE}}\rangle}.$
(5.65)

Then, after multiplying $\mathbb{1}_{A}\otimes B_{0}$ to the above equation
and then taking into account that it commutes with
$X_{d}\otimes\mathbb{1}_{B}$, it not difficult to see that

$\displaystyle\sum_{k=1}^{d-1}\left(\gamma(\bm{\alpha})X_{d}^{k-1}\otimes
B_{0}B_{1}^{k}\right){|{\psi_{ABE}}\rangle}=(X_{d}^{-1}\overline{Z}_{A}\otimes
B_{0}){|{\psi_{ABE}}\rangle}.$ (5.66)

Now, let us exploit the relation (5.40) for $k=1$, that is,
$\left(Z_{d}^{-1}\otimes\mathbb{1}_{B}\right){|{\psi_{ABE}}\rangle}=\mathbb{1}_{A}\otimes
B_{0}{|{\psi_{ABE}}\rangle}$ and then using the fact that $\overline{Z}_{A}$
and $Z_{d}$ commutes, we finally get,

$\displaystyle\sum_{k=1}^{d-1}\left(\gamma(\bm{\alpha})X_{d}^{k-1}\otimes
B_{0}B_{1}^{k}\right){|{\psi_{ABE}}\rangle}=(X_{d}^{-1}Z_{d}^{-1}\overline{Z}_{A}\otimes\mathbb{1}_{B}){|{\psi_{ABE}}\rangle}.$
(5.67)

Next, we apply $Z_{d}^{-1}\otimes\mathbb{1}_{B}$ to Eq. (5.65) from the left
hand side to obtain,

$\displaystyle\sum_{k=1}^{d-1}\left(\gamma(\bm{\alpha})Z_{d}^{-1}X_{d}^{k-1}\otimes
B_{1}^{k}\right){|{\psi_{ABE}}\rangle}=(Z_{d}^{-1}X_{d}^{-1}\overline{Z}_{A}\otimes\mathbb{1}_{B}){|{\psi_{ABE}}\rangle}.$
(5.68)

Again, by employing the relation $Z_{d}X_{d}=\omega X_{d}Z_{d}$, the above
equation (5.68) can be rewriten as

$\displaystyle\sum_{k=1}^{d-1}\left(\gamma(\bm{\alpha})\omega^{-(k-1)}X_{d}^{k-1}\otimes
B_{1}^{k}B_{0}\right){|{\psi_{ABE}}\rangle}=\left(\omega
X_{d}^{-1}Z_{d}^{-1}\overline{Z}_{A}\otimes\mathbb{1}_{B}\right){|{\psi_{ABE}}\rangle}.$
(5.69)

Notice that in the left hand side of the above equation, we again exploited
the relation (5.40) for $k=1$, that is,
$\left(Z_{d}^{-1}\otimes\mathbb{1}_{B}\right){|{\psi_{ABE}}\rangle}=\mathbb{1}_{A}\otimes
B_{0}{|{\psi_{ABE}}\rangle}$. Now we apply $\omega^{-1}$ to the above equation
(5.69) and then subtract it from Eq. (5.66) to get

$\displaystyle\gamma(\bm{\alpha})\sum_{k=1}^{d-1}\left[X_{d}^{k-1}\otimes\left(B_{0}B_{1}^{k}-\omega^{-k}B_{1}^{k}B_{0}\right)\right]{|{\psi_{ABE}}\rangle}=0,$
(5.70)

which can be divided up into two parts by separating the term corresponding to
$k=1$, as follows

$\displaystyle\gamma(\bm{\alpha})\sum_{k=2}^{d-1}\left[X_{d}^{k-1}\otimes\left(B_{0}B_{1}^{k}-\omega^{-k}B_{1}^{k}B_{0}\right)\right]{|{\psi_{ABE}}\rangle}=-\gamma(\bm{\alpha})\left(B_{0}B_{1}-\omega^{-1}B_{1}B_{0}\right){|{\psi_{ABE}}\rangle}.$
(5.71)

Notice that the left hand side of the expressions (5.64) and (5.71) are
identical, which immediately allows us to get that

$\displaystyle\overline{Z}_{A}\otimes(B_{0}B_{1}-\omega^{-1}B_{1}B_{0}){|{\psi_{ABE}}\rangle}=-\gamma(\bm{\alpha})\left(B_{0}B_{1}-\omega^{-1}B_{1}B_{0}\right){|{\psi_{ABE}}\rangle},$
(5.72)

which after a simple rearrangement of the terms and expanding
$\overline{Z}_{A}$ from (5.59) yields,

$\displaystyle\left[(1+\gamma(\bm{\alpha}))\mathbb{1}_{A}-\sum_{k=1}^{d-1}\delta_{k}(\bm{\alpha})Z_{d}^{k}\right]\otimes\left(B_{0}B_{1}-\omega^{-1}B_{1}B_{0}\right){|{\psi_{ABE}}\rangle}=0.$
(5.73)

As proven in Observation 5.1 stated in Appendix C, the operator
$[1+\gamma(\bm{\alpha})]\mathbb{1}-\sum_{k=1}^{d-1}\delta_{k}(\bm{\alpha})Z_{d}^{k}$
is invertible. Thus, taking trace over the subsystems $A,E$ allows us to
finally conclude that

$(B_{0}B_{1}-\omega^{-1}B_{1}B_{0})\rho_{B}=0,$ (5.74)

where
$\rho_{B}=\mathrm{Tr}_{AE}{|{\psi_{ABE}}\rangle}\\!{\langle{\psi_{ABE}}|}$.
Recalling that $\rho_{B}$ is full-rank and thus invertible, the above
expression (5.74) implies the following commutation relation between Bob’s
both observables

$B_{0}B_{1}=\omega^{-1}B_{1}B_{0}.$ (5.75)

As stated in Fact 2 which was proven in Ref. [46], the above relation along
with the fact that $B_{0}^{d}=B_{1}^{d}=\mathbb{1}_{B}$ imposes that Bob’s
Hilbert space decomposes into a tensor product
$\mathcal{H}_{B}=(\mathbbm{C}^{d})_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$
where $\mathcal{H}_{B^{\prime\prime}}$ is some Hilbert space of unknown but
finite dimension. Along with it, there also exists a unitary transformation
$U_{B}:\mathcal{H}_{B}\to\mathcal{H}_{B}$ such that

$\displaystyle
U_{B}\,B_{0}\,U^{\dagger}_{B}=Z^{*}_{d}\otimes\mathbb{1}_{B^{\prime\prime}},\quad
U_{B}\,B_{1}\,U_{B}^{\dagger}=X_{d}\otimes\mathbb{1}_{B^{\prime\prime}},$
(5.76)

where $\mathbb{1}_{B^{\prime\prime}}$ is the identity acting on
$\mathcal{H}_{B^{\prime\prime}}$. This completes the characterisation of Bob’s
observables that maximally violate the steering inequality (5.1).

##### The state

We finally have all the tools required to find the state that maximally
violates the steering inequality (5.1). As was derived in the previous part,
up to a local unitary Bob’s both observables are the ideal ones (5.76). Thus,
we can rewrite the relation (5.23) and (5.32) by plugging in Bob’s derived
observables as

$(Z_{d}\otimes
Z_{d}^{\dagger}\otimes\mathbb{1}_{B^{\prime\prime}E}){|{\tilde{\psi}_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle},$
(5.77)

and

$\displaystyle\sum_{k=1}^{d-1}\left[\gamma(\bm{\alpha})X_{d}^{k}\otimes
X_{d}^{k}\otimes\mathbb{1}_{B^{\prime\prime}E}+\delta_{k}(\bm{\alpha})Z_{d}^{k}\otimes\mathbb{1}_{BE}\right]{|{\tilde{\psi}_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle},$
(5.78)

where
${|{\tilde{\psi}_{ABE}}\rangle}=U_{B}\otimes\mathbb{1}_{AE}{|{\psi_{ABE}}\rangle}$.
From here on, for convenience we drop the all the identities from the above
relations. As concluded in the previous part of the proof that Bob’s Hilbert
space is of dimension
$(\mathbbm{C}^{d})_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$ due to
which the state ${|{\tilde{\psi}_{ABE}}\rangle}$ belongs to
$(\mathbbm{C}^{d})_{A}\otimes(\mathbbm{C}^{d})_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}\otimes\mathcal{H}_{E}$.
As a consequence, any such state can be written using the computational basis
in $\mathbbm{C}^{d}$ as,

${|{\tilde{\psi}_{ABE}}\rangle}=\sum_{i,j=0}^{d-1}{|{i}\rangle}_{A}{|{j}\rangle}_{B^{\prime}}{|{\psi_{ij}}\rangle}_{B^{\prime\prime}E},$
(5.79)

where ${|{\psi_{ij}}\rangle}_{B^{\prime\prime}E}$ is some unnormalised state
belonging to $\mathcal{H}_{B^{\prime\prime}}\otimes\mathcal{H}_{E}$. After
plugging this state to the condition (5.77) for $k=1$, we arrive at

$\displaystyle\sum_{i,j=0}^{d-1}\omega^{i-j}{|{ij}\rangle}{|{\psi_{ij}}\rangle}=\sum_{i,j=0}^{d-1}{|{ij}\rangle}{|{\psi_{ij}}\rangle},$
(5.80)

which holds true if and only if ${|{\psi_{ij}}\rangle}=0$ for any $i\neq j$.
As a consequence, the only terms in the state (5.79) remains when $i=j$, and
thus we have the simplified form of the state given by

${|{\tilde{\psi}_{ABE}}\rangle}=\sum_{i=0}^{d-1}{|{ii}\rangle}{|{\psi_{ii}}\rangle}.$
(5.81)

Let us now consider the condition (5.78) where we can extend the range of the
sum to $k=0$ by recalling that the $0-th$ power of an observable is identity
and also that $\delta_{0}(\bm{\alpha})=-1$. Thus, after some rearrangement of
the terms we get the following expression

$\displaystyle\sum_{k=0}^{d-1}\left[\gamma(\bm{\alpha})X_{d}^{k}\otimes
X_{d}^{k}+\delta_{k}(\bm{\alpha})Z_{d}^{k}\otimes\mathbb{1}_{B}\right]{|{\tilde{\psi}_{ABE}}\rangle}=\gamma(\bm{\alpha}){|{\tilde{\psi}_{ABE}}\rangle}.$
(5.82)

Plugging in the simplified form of ${|{\tilde{\psi}_{ABE}}\rangle}$ as derived
in (5.81), the above expression turns out to be

$\displaystyle\gamma(\bm{\alpha})\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}{|{i+k}\rangle}{|{i+k}\rangle}{|{\psi_{ii}}\rangle}+\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}\omega^{ki}\delta_{k}(\bm{\alpha}){|{ii}\rangle}{|{\psi_{ii}}\rangle}=\gamma(\bm{\alpha})\sum_{i=0}^{d-1}{|{ii}\rangle}{|{\psi_{ii}}\rangle}.$
(5.83)

Mulitplying the above expression with ${\langle{ss}|}$ from the lest hand
side, we obtain that

$\sum_{k=0}^{d-1}\gamma(\bm{\alpha}){|{\psi_{s\ominus k,s\ominus
k}}\rangle}+\sum_{k=0}^{d-1}\omega^{ks}\delta_{k}(\bm{\alpha}){|{\psi_{ss}}\rangle}=\gamma(\bm{\alpha}){|{\psi_{ss}}\rangle}$
(5.84)

where $s\ominus k$ represents $s-k$ modulo $d$. We can simplify the above
expression by substituting the explicit form of $\delta(\bm{\alpha})$ to
obtain

$\displaystyle\sum_{k=0}^{d-1}{|{\psi_{s\ominus k,s\ominus
k}}\rangle}-\sum_{k=0}^{d-1}\frac{\omega^{k(d-j+s)}}{d}\sum_{\begin{subarray}{c}i,j=0\\\
i\neq
j\end{subarray}}^{d-1}\frac{\alpha_{i}}{\alpha_{j}}{|{\psi_{ss}}\rangle}={|{\psi_{ss}}\rangle}$
(5.85)

Now, using the identity $\sum_{k=0}^{d-1}\omega^{k(j-s)}=d\delta_{j,s}$, we
can simplify the above expression to find the explicit form of the state
${|{\psi_{ss}}\rangle}$ given by

${|{\psi_{ss}}\rangle}=\frac{\alpha_{s}}{\alpha_{0}+\ldots+\alpha_{d-1}}{|{\Psi}\rangle},$
(5.86)

where we denoted ${|{\Psi}\rangle}=\sum_{k=0}^{d-1}{|{\psi_{s\ominus
k,s\ominus k}}\rangle}\equiv\sum_{k=0}^{d-1}{|{\psi_{kk}}\rangle}$ for any
$s$. As a consequence, Eq. (5.81) can be rewritten as

$\displaystyle(U_{B}\otimes\mathbb{1}_{AE}){|{\psi_{ABE}}\rangle}$
$\displaystyle=$
$\displaystyle\left(\sum_{m=0}^{d-1}\alpha_{i}{|{ii}\rangle}_{AB^{\prime}}\right)\otimes{|{\xi}\rangle}_{B^{\prime\prime}E}={|{\psi(\bm{\alpha})}\rangle}_{AB^{\prime}}\otimes{|{\xi}\rangle}_{B^{\prime\prime}E},$
(5.87)

where

${|{\xi}\rangle}_{B^{\prime\prime}E}=\frac{1}{\alpha_{0}+\ldots+\alpha_{d-1}}{|{\Psi}\rangle}.$
(5.88)

This finally completes the proof of certification of all pure bipartite
entangled states along with a pair of arbitrary outcome mutually unbiased
bases in the 1SDI scenario. ∎

Let us now proceed towards another important result of this chapter that
utilises the above theorem (5.2) involving certification of every rank-one
extremal POVM’s.

###  5.4 Certification of all rank-one extremal POVM

As discussed before in Chapter 2, let us consider a rank-one extremal
measurement denoted as $\mathcal{I}=\\{\mathcal{I}_{b}\\}$, such that $b$
represents its outcomes and $\mathcal{I}_{b}$ represents the measurement
operator corresponding to the $b-th$ outcome. These measurement operators are
positive semi-definite and sum up to one. Additionally, it was shown in [80]
that measurement operators of an extremal rank-one POVM are projectors scaled
down by some non-negative real number, that is,
$\mathcal{I}_{b}=\lambda_{b}{|{\mu}\rangle}\\!{\langle{\mu}|}$ with
${|{\mu}\rangle}\in\mathbbm{C}^{d}$ and $0\leq\lambda_{b}\leq 1$.

Figure 5.1: 1SDI scenario to certify any rank-one extremal POVM: Alice and Bob
receive subsystems from the preparation device on which they perform $d+1$ and
$3$ measurements respectively such that Alice is trusted. All the measurements
are $d-$outcome except Bob’s third measurement which is of $d^{2}-$outcome.

It turns our that the observation of the maximal violation of our steering
inequality (5.1) plus an additional set of conditions enables us to design a
simple method that can be used for certification of any extremal rank-one
POVM. For this purpose, we again consider the 1SDI setting such that Alice is
again trusted but now performs $d+1$ measurements corresponding to the
observables $A_{0}=Z_{d}$ and $A_{i+1}=X_{d}Z_{d}^{i}$ for $i=0,1,\ldots,d-1$.
Bob is untrusted and performs three measurements, where the first two
measurements are $d-$outcome and the third one has $d^{2}-$outcomes. The
scenario is depicted in Fig. 5.1.

Notice that the statistics corresponding to both Alice and Bob choosing the
input $0,1$, allow us to employ the steering inequality (5.1) and certify any
pure bipartite entangled state using Theorem 5.2. Without loss of generality,
Bob’s third measurement, which is a $d^{2}-$outcome POVM, is denoted as
$\\{R_{b}\\}$. Further, notice that the operators $X_{d}Z_{d}^{i}$ for
$i=1,2,\ldots,d-1$ are not proper observables based on the definition
introduced in Chapter 2 as $(X_{d}Z_{d}^{i})^{d}=\omega^{id(d-1)}\mathbb{1}$.
Thus, dividing the matrices with the scalar $\omega^{-i(d-1)}$ yields proper
observables, but for simplicity we would drop this factor from further
considerations. It is also worth noting that the statistics obtained from the
$d+1$ observables, $Z_{d}$ and $X_{d}Z_{d}^{i}$, are enough to simulate the
statistics corresponding to the operators $X_{d}^{i}Z_{d}^{j}$ for
$i,j=0,1,\ldots,d-1$ that represent the Heisenberg-Weyl (HW) basis [178]
222The HW basis is a collection of operators that forms a basis for operators
that act on $d-$dimensional Hilbert space with $d$ being any positive
integer.. The reason for this fact is that every element in the HW basis can
be generated by considering the powers of $Z_{d}$ and $X_{d}Z_{d}^{i}$ for all
$i$ and then multiply them with appropriate powers of $\omega$. For instance,
$X_{d}^{2}Z_{d}^{2}$ can be generated by taking $X_{d}Z_{d}$ two times and
then multiplying it with $\omega^{-1}$, that is,
$X_{d}^{2}Z_{d}^{2}=\omega^{-1}(X_{d}Z_{d})^{2}$. The result is stated below
as a simple theorem.

######  Theorem 5.3.

Assume that Alice and Bob perform the quantum steering experiment and are able
to certify that the state shared among them as well as the measurements along
with the Hilbert space of Bob as given in Theorem-5.2. Consider then a POVM
$R=\\{R_{b}\\}$ acting on
$\mathcal{H}_{B}=(\mathbbm{C}^{d})_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$.
If for some extremal POVM $\mathcal{I}=\\{\mathcal{I}_{b}\\}$ acting on
$\mathbbm{C}^{d}$ the following identities

$\displaystyle\langle X^{i}Z^{j}\otimes
R_{b}\otimes\mathbb{1}_{E}\rangle_{{|{\psi_{ABE}}\rangle}}=\langle
X^{i}Z^{j}\otimes\mathcal{I}_{b}\rangle_{{|{\psi(\bm{\alpha})}\rangle}}$
(5.89)

hold true for any $i,j=0,\ldots,d-1$, where
${|{\psi_{ABE}}\rangle}=(\mathbb{1}_{A}\otimes
U_{B}^{\dagger}){|{\psi(\bm{\alpha})}\rangle}_{AB^{\prime}}\otimes{|{\xi_{B^{\prime\prime}E}}\rangle}$
from (5.46) where ${|{\psi(\bm{\alpha})}\rangle}_{AB^{\prime}}$ is the ideal
state defined in Eq. (5.35). Then, there exist a unitary transformation
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ such that the measurement
operators of the POVM $R$ are equivalent to the measurement operators of the
ideal POVM $\mathcal{I}$ as

$\displaystyle U_{B}\ R_{b}\
U_{B}^{\dagger}=\mathcal{I}_{b}\otimes\mathbb{1}_{B^{\prime\prime}}\qquad\forall
b.$ (5.90)

###### Proof.

Our proof takes inspiration from the technique introduced in Ref. [54], where
extremal POVM’s acting on two-dimensional Hilbert space were certified up to
certain equivalences. Here, we generalise that approach to POVM’s that act on
arbitrary dimensional Hilbert space in the scenario where Alice is trusted.
Let us first observe that the statistics one observes from the actual
experiment must be equivalent to one observed in the ideal experiment, that
is,

$\displaystyle\forall\ b\quad\forall
i,j\qquad{\langle{\psi_{ABE}}|}X^{i}Z^{j}\otimes
R_{b}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={\langle{\psi(\bm{\alpha})}|}X^{i}Z^{j}\otimes\mathcal{I}_{b}{|{\psi(\bm{\alpha})}\rangle}.$
(5.91)

Solving the above condition is enough to certify the POVM $R$. For this
purpose, as was done in Chapter 4, we first rewrite the state
${|{\psi(\bm{\alpha})}\rangle}$ in terms of the maximally entangled state of
two qudits ${|{\phi_{+}^{d}}\rangle}$ [see Eq. (4.16)] as

${|{\psi(\bm{\alpha})}\rangle}=[\mathbb{1}_{A}\otimes
P(\bm{\alpha})]{|{\phi_{+}^{d}}\rangle},$ (5.92)

where

$P(\bm{\alpha})=\sum_{i=0}^{d-1}\alpha_{i}{|{i}\rangle}\\!{\langle{i}|}.$
(5.93)

Recall that $\alpha_{i}>0$ for all $i$ and $\sum_{i}\alpha_{i}^{2}=1$. Let us
also introduce another set of $d^{2}$ number of operators, derived from HW
basis

$W_{i,j}:=P(\bm{\alpha})^{-1}\left(X^{i}Z^{j}\right)^{*}P(\bm{\alpha})^{-1}.$
(5.94)

Let us observe that the above operators are linearly independent as
$P(\bm{\alpha})$ is invertible and $X^{i}Z^{j}$ are orthogonal in the Hilbert-
Schmidt scalar product, that is,
$\mathrm{Tr}[X^{i}Z^{j}(X^{i^{\prime}}Z^{j^{\prime}})^{\dagger}]=d\delta_{i,i^{\prime}}\delta_{j,j^{\prime}}$.
As a consequence, the set $\\{W_{i,j}\\}$ forms a complete basis for operators
acting on $d-$dimensional Hilbert space. Recalling that the measurement
operators $\mathcal{I}_{b}$ of the ideal POVM act on $d-$dimensional Hilbert
space and thus, using the newly defined operator basis (5.94), we can express
them as

$\mathcal{I}_{b}=\sum_{i,j=0}^{d-1}l^{b}_{i,j}W_{i,j}\qquad\forall b,$ (5.95)

where $l_{i,j}^{b}$ are in general complex coefficients. Let us now compute
the right hand side of the expression (5.91) by plugging in the above
representation of the POVM $\mathcal{I}$,

$\displaystyle{\langle{\psi(\bm{\alpha})}|}X^{i}Z^{j}\otimes\mathcal{I}_{b}{|{\psi(\bm{\alpha})}\rangle}$
$\displaystyle=$
$\displaystyle\sum_{m,n}l^{b}_{m,n}\\!{\langle{\psi(\bm{\alpha})}|}X^{i}Z^{j}\otimes
P(\bm{\alpha})^{-1}\left(X^{m}Z^{n}\right)^{*}P(\bm{\alpha})^{-1}{|{\psi(\bm{\alpha})}\rangle}$
(5.96) $\displaystyle=$
$\displaystyle\sum_{m,n}l^{b}_{m,n}\\!{\langle{\phi_{+}^{d}}|}X^{i}Z^{j}\otimes\left(X^{m}Z^{n}\right)^{*}{|{\phi_{+}^{d}}\rangle},$

where we exploited the form of the state ${|{\psi(\bm{\alpha})}\rangle}$ given
in (5.92). Now, exploiting the identity $(R\otimes
Q)\,{|{\phi_{+}^{d}}\rangle}=(RQ^{T}\otimes\mathbb{1}){|{\phi_{+}^{d}}\rangle}$
that is satisfied for any two matrices $Q$ and $R$ acting on $d-$dimensional
Hilbert space [see Fact 5 in Appendix A] and also the fact that $X^{i}Z^{j}$
form an orthogonal basis as mentioned above, we finally obtain that

${\langle{\psi(\bm{\alpha})}|}X^{i}Z^{j}\otimes\mathcal{I}_{b}{|{\psi(\bm{\alpha})}\rangle}=l^{b}_{i,j}\qquad\forall
b.$ (5.97)

Next, our aim is to compute the left hand side of the expression (5.91). We
use the fact that according to Theorem 5.2, Bob’s Hilbert space decomposes as
$\mathcal{H}_{B}=\mathbbm{C}^{d}\otimes\mathcal{H}_{B^{\prime\prime}}$ and
there exist a unitary $U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ that
transforms that transforms the state ${|{\psi_{ABE}}\rangle}$ as

$\displaystyle(\mathbb{1}_{AE}\otimes
U_{B}){|{\psi_{ABE}}\rangle}={|{\psi(\bm{\alpha})}\rangle}_{AB^{\prime}}\otimes{|{\xi_{B^{\prime\prime}E}}\rangle}.$
(5.98)

Now, any measurement operator of the POVM $R$ acting on $\mathcal{H}_{B}$ can
be expressed using the basis (5.94) as,

$\displaystyle
U_{B}R_{b}U_{B}^{\dagger}=\sum_{i,j=0}^{d-1}W_{i,j}\otimes\widetilde{R}_{i,j}^{b},$
(5.99)

where $\widetilde{R}_{i,j}^{b}$ are general operators acting on
$\mathcal{H}_{B^{\prime\prime}}$. Now, computing the left hand side of (5.91)
by plugging in it the above mentioned form of $R_{b}$ (5.99) and the state
${|{\psi_{ABE}}\rangle}$, we have

${\langle{\psi_{ABE}}|}X^{i}Z^{j}\otimes
R_{b}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}=\left[\sum_{m,n}\\!{\langle{\psi(\bm{\alpha})}|}X^{i}Z^{j}\otimes
W_{m,n}{|{\psi(\bm{\alpha})}\rangle}\right]{\langle{\xi_{B^{\prime\prime}E}}|}\widetilde{R}_{i,j}^{b}\otimes\mathbb{1}_{E}{|{\xi_{B^{\prime\prime}E}}\rangle}.$
(5.100)

As was computed above to get (5.97) from (5.96), the term inside the square
bracket in the above expression is just $1$. Thus, we finally arrive at

${\langle{\psi_{ABE}}|}X^{i}Z^{j}\otimes
R_{b}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={\langle{\xi_{B^{\prime\prime}E}}|}\widetilde{R}_{i,j}^{b}\otimes\mathbb{1}_{E}{|{\xi_{B^{\prime\prime}E}}\rangle}=\mathrm{Tr}\left(\widetilde{R}_{i,j}^{b}\sigma_{B^{\prime\prime}}\right)\qquad\forall
b,$ (5.101)

where
$\sigma_{B^{\prime\prime}}=\mathrm{Tr}_{E}({|{\xi_{B^{\prime\prime}E}}\rangle}\\!{\langle{\xi_{B^{\prime\prime}E}}|})$.
Let us now decompose $\sigma_{B^{\prime\prime}}$ using its eigenvectors
denoted by ${|{k}\rangle}$ as
$\sigma_{B^{\prime\prime}}=\sum_{k}p_{k}{|{k}\rangle}\\!{\langle{k}|}$.
Plugging this into the above expression, we get that

$\displaystyle\mathrm{Tr}\left(\widetilde{R}_{i,j}^{b}\sigma_{B^{\prime\prime}}\right)=\sum_{k}p_{k}{\langle{k}|}\widetilde{R}_{i,j}^{b}{|{k}\rangle}.$
(5.102)

Recalling again the identity (5.91) and then using Eq. (5.97), we finally
arrive at

$\displaystyle\sum_{k}p_{k}{\langle{k}|}\widetilde{R}_{i,j}^{b}{|{k}\rangle}=l_{i,j}^{b}.$
(5.103)

Next, we introduce a family of POVM’s $k$ as,
$\mathcal{I}_{k}=\\{\mathcal{I}_{b,k}\\}$, whose measurement operators are
given by

$\displaystyle\mathcal{I}_{b,k}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{B^{\prime\prime}}\left[\left(\mathbb{1}_{B^{\prime}}\otimes{|{k}\rangle}\\!{\langle{k}|}_{B^{\prime\prime}}\right)R_{b}\right]$
(5.104) $\displaystyle=$
$\displaystyle\sum_{i,j=0}^{d-1}\\!{\langle{k}|}\widetilde{R}_{i,j}^{b}{|{k}\rangle}W_{i,j}.$

As $R$ is a valid POVM, as discussed in Chapter 2 all its measurement
operators are hermitian and positive semi-definite, that is, $R_{b}\geq 0$ for
all $b$. Consequently, from the first line of Eq. (5.104), we can see that
$\mathcal{I}_{b,k}$ is also positive semi-definite, that is,
$\mathcal{I}_{b,k}\geq 0$ for any $k$ and $b$, as product of two positive
semi-definite matrices is also positive semi-definite. Further,
$\sum_{b}R_{b}=\mathbb{1}_{B}$ is identity and then using the first line of of
Eq. (5.104), we can directly see that
$\sum_{b}\mathcal{I}_{b,k}=\mathbb{1}_{B^{\prime}}$ for any $k$. As a
consequence, the family of POVM’s $\\{\mathcal{I}_{b}^{k}\\}_{b}$ are valid
quantum measurements. Let us now go back to Eq. (5.103), and then rewrite it
using the family of POVM’s (5.104) as

$\displaystyle\mathcal{I}_{b}=\sum_{i,j=0}^{d-1}l_{i,j}^{b}W_{i,j}=\sum_{i,j=0}^{d-1}\sum_{k}p_{k}{\langle{k}|}\widetilde{R}_{i,j}^{b}{|{k}\rangle}W_{i,j}=\sum_{k}p_{k}\mathcal{I}_{b,k}.$
(5.105)

However, the POVM $\mathcal{I}$ is extremal and can not be decomposed as a
convex mixture of other POVM’s. Thus, we can immediately conclude that

$\displaystyle\forall k\qquad\mathcal{I}_{b,k}=\mathcal{I}_{b},$ (5.106)

which is equivalent to the condition,

$\displaystyle\forall
k\qquad{\langle{k}|}\widetilde{R}_{i,j}^{b}{|{k}\rangle}=l_{i,j}^{b}.$ (5.107)

Next, we consider the following vectors belonging to $\mathbbm{C}^{d}$:

$\displaystyle{|{\varphi_{a,s,t}}\rangle}=\frac{1}{\sqrt{2}}\left({|{s}\rangle}\pm\mathbbm{i}^{a}{|{t}\rangle}\right),$
(5.108)

where $a=0,1$ and ${|{s}\rangle}$ and ${|{t}\rangle}$ are two distinct vectors
belonging to the eigenbasis $\\{{|{k}\rangle}\\}$ of
$\sigma_{B^{\prime\prime}}$. We now compute the following quantity

$\displaystyle\mathrm{Tr}_{B^{\prime\prime}}\left[\left(\mathbb{1}_{B^{\prime}}\otimes|\varphi_{a,s,t}\rangle\\!\langle\varphi_{a,s,t}|_{B^{\prime\prime}}\right)R_{b}\right]=\sum_{i,j}\mathrm{Tr}(|\varphi_{a,s,t}\rangle\\!\langle\varphi_{a,s,t}|_{B^{\prime\prime}}\widetilde{R}_{i,j}^{b})W_{i,j}.$
(5.109)

Expanding the above quantity using the explicit form of the vectors given in
(5.108), we obtain

$\displaystyle\mathrm{Tr}_{B^{\prime\prime}}\left[\left(\mathbb{1}_{B^{\prime}}\otimes|\varphi_{a,s,t}\rangle\\!\langle\varphi_{a,s,t}|_{B^{\prime\prime}}\right)R_{b}\right]=\mathcal{I}_{b}\pm\mathrm{Tr}_{B^{\prime\prime}}\left[(\mathbb{1}_{B^{\prime}}\otimes
L^{a}_{B^{\prime\prime}})R_{b}\right],$ (5.110)

where

$\displaystyle
L_{B^{\prime\prime}}^{a}=(\mathbbm{i}^{a}/2)\left(|t\rangle\\!\langle
s|+(-1)^{a}|s\rangle\\!\langle t|\right).$ (5.111)

The fact that $R_{b}\geq 0$, imposes that left-hand side of the above
expression is non-negative as it is a product of two matrices that are
positive semi-definite matrices. This allows us to conclude that

$\displaystyle\mathcal{I}_{b}\geq\pm\mathrm{Tr}_{B^{\prime\prime}}\left[(\mathbb{1}_{B^{\prime}}\otimes
L^{a}_{B^{\prime\prime}})R_{b}\right].$ (5.112)

As discussed above, the measurement operators of any rank-one extremal POVM
acting on $d-$dimensional Hilbert space can be expressed as
$\mathcal{I}_{b}=\lambda_{b}{|{\mu_{b}}\rangle}\\!{\langle{\mu_{b}}|}$, where
the vectors ${|{\mu_{b}}\rangle}$ are normalized and belong to
$\mathbbm{C}^{d}$. Using this fact, we show in Observation 5.2 stated in
Appendix C that the operator appearing on the right-hand side of the above
expression (5.112) must be rank one as well and must admit the following form

$\displaystyle\mathrm{Tr}_{B^{\prime\prime}}\left[(\mathbb{1}_{B^{\prime}}\otimes
L^{a}_{B^{\prime\prime}})R_{b}\right]=\lambda^{\prime}_{b}{|{\mu_{b}}\rangle}\\!{\langle{\mu_{b}}|},$
(5.113)

such that $\lambda_{b}\geq\pm\lambda^{\prime}_{b}$. Recalling that
$\sum_{b}R_{b}=\mathbb{1}_{B}$ and that
$\mathrm{Tr}L_{B^{\prime\prime}}^{a}=0$ for any $a$, we can finally conclude
that

$\displaystyle\sum_{b}\mathrm{Tr}_{B^{\prime\prime}}\left[(\mathbb{1}_{B^{\prime}}\otimes
L^{a}_{B^{\prime\prime}})R_{b}\right]=0=\sum_{b}\lambda_{b}^{\prime}{|{\mu_{b}}\rangle}\\!{\langle{\mu_{b}}|}.$
(5.114)

Since $\mathcal{I}_{b}$ are linearly independent, we learn from the above
condition that $\lambda^{\prime}_{b}=0$ for all $b$ which in turn implies from
(5.113) that
$\mathrm{Tr}_{B^{\prime\prime}}\left[(\mathbb{1}_{B^{\prime}}\otimes
L^{a}_{B^{\prime\prime}})R_{b}\right]=0$ for any $b$ and $a$. Finally
expanding the left hand side of (5.113) by plugging in the explicit form of
$L^{a}_{B^{\prime\prime}}$ and also taking into account that $X^{i}Z^{j}$ are
linearly independent for any $i,j$ gives us two simple conditions:

$\displaystyle\left(X^{i}Z^{j}\right)^{*}\left({\langle{s}|}\widetilde{R}_{i,j}^{b}{|{t}\rangle}+{\langle{t}|}\widetilde{R}_{i,j}^{b}{|{s}\rangle}\right)=0,$
(5.115)

for $a=0$ and

$\displaystyle\left(X^{i}Z^{j}\right)^{*}\left({\langle{t}|}\widetilde{R}_{i,j}^{b}{|{s}\rangle}-{\langle{s}|}\widetilde{R}_{i,j}^{b}{|{t}\rangle}\right)=0,$
(5.116)

for $a=1$. One can immediately see that the only possible solution of the
above conditions (5.115) and (5.116) is
${\langle{s}|}\widetilde{R}_{i,j}^{b}{|{t}\rangle}=0$ for $s\neq t$. Thus,
from Eq. (5.107) we can conclude that the POVM acting on the support of Bob’s
local state is given by
$R_{b}=\mathcal{I}_{b}\otimes\mathbb{1}_{B^{\prime\prime}}$ for all
$b^{\prime}s$. This completes the proof. ∎

It is worth noting that the above certification scheme works for any rank-one
extremal POVM with arbitrary number of outcomes. However, it was shown in [80]
that any extremal POVM with $d^{2}$ outcomes have to be rank-one. As a
consequence, we certify every $d^{2}-$outcome extremal POVM. We now show that
the certified state and the certified POVM can be used for optimal randomness
certification.

###  5.5 Optimal randomness certification

Let us again go back to the previous scenario depicted in Fig. 5.1, but now
let us assume that there is another party Eve who wants to guess Bob’s
outcome. As discussed in Chapter 2, Eve has full control on Bob’s lab and also
has access to the state sent by the preparation device. However, unlike
randomness certification in the Bell scenario, in the 1SDI scenario Eve has no
access to Alice’s lab as it is trusted. This is depicted in Fig. 5.2.

Figure 5.2: Optimal randomness certification in the 1SDI scenario: Alice
(trusted) and Bob receive subsystems from the preparation device on which they
perform $d+1$ and $3$ measurements respectively. All the measurements are
$d-$outcome except Bob’s third measurement which has $d^{2}$ outcomes. Using
this measurement Bob wishes to generate $2\log_{2}d$ bits of randomness. Eve
has knowledge about Bob’s measurement choices. She also might receive an
additional system from the preparation device. Using her measuring device and
the received subsystem from the preparation she wants to guess Bob’s outcome.

Let us now say that Alice and Bob observe the maximal violation of the
steering inequality (5.1) using the observables corresponding to the inputs
$x,y=0,1$ where $A_{0}=Z_{d}$ and $A_{1}=X_{d}$. Now, using Theorem 5.2, Alice
and Bob can certify the quantum state shared between Alice and Bob up to local
unitaries and additional degrees of freedom [see Eq. (5.46)]. Notice that in
the proof we considered an external system $E$ that was used to purify the
state shared between Alice and Bob. Without loss of generality, this external
system in fact denotes the subsystem possessed by Eve and her Hilbert space is
denoted by $\mathcal{H}_{E}$. Again, in the previous section, using this
result we certified Bob’s third measurement to be an ideal extremal
measurement up to some local unitary [see Eq. (5.90)]. Notice that in the
proof, the only condition we used apart from Theorem 5.2 was that the
statistics one obtains in the actual experiment is the same as in the ideal
experiment, even with the presence of the external subsystem $E$ which is held
with Eve [cf. Eq. (5.91)]. Given the certified state and the measurements, let
us now compute the probability of Eve to guess Bob’s outcomes corresponding to
the POVM $R$ [cf. Chapter 2],

$\displaystyle G(y=2,\vec{p})$ $\displaystyle=$
$\displaystyle\sup_{S_{p}}\sum_{b}{\langle{\psi_{ABE}}|}\mathbb{1}_{A}\otimes
R_{b}\otimes E^{(b)}{|{\psi_{ABE}}\rangle}$ (5.117) $\displaystyle=$
$\displaystyle\sup_{S_{p}}\\!{\langle{\psi(\bm{\alpha})}|}\mathbb{1}_{A}\otimes\mathcal{I}_{b}{|{\psi(\bm{\alpha})}\rangle}\\!{\langle{\xi_{B^{\prime\prime}E}}|}\mathbb{1}_{B^{\prime\prime}}\otimes
E^{(b)}{|{\xi_{B^{\prime\prime}E}}\rangle}.$

Eve’s strategy is composed of the states
$\sigma_{E}=\mathrm{Tr}_{B^{\prime\prime}}{|{\xi_{B^{\prime\prime}E}}\rangle}\\!{\langle{\xi_{B^{\prime\prime}E}}|}$
and a measurement $Z=\\{E^{(b)}\\}$. Thus, simplifying the above expression we
obtain

$\displaystyle
G(y=2,\vec{p})=\sup_{\sigma_{E},Z}\sum_{b}\mathrm{Tr}[\mathcal{I}_{b}\rho_{B^{\prime}}(\bm{\alpha})]\mathrm{Tr}[E^{(b)}\sigma_{E}]$
(5.118)

where
$\rho_{B^{\prime}}(\bm{\alpha})=\mathrm{Tr}_{A}{|{{\psi(\bm{\alpha})}}\rangle}\\!{\langle{{\psi(\bm{\alpha})}}|}_{AB^{\prime}}$.
Using the fact $\sum_{b}E^{(b)}=\mathbb{1}_{E}$, we can immediately observe
from the above expression that for any extremal POVM $\\{\mathcal{I}_{b}\\}$
if

$\mathrm{Tr}[\mathcal{I}_{b}\rho_{B}(\bm{\alpha})]=\frac{1}{d^{2}}\qquad\forall
b,$ (5.119)

then the guessing probability (5.117) is $G(y=2,\vec{p})=1/d^{2}$. As a
consequence, the maximal violation of the steering inequality (5.1) along with
with conditions (5.91), can be used to certify $2\log_{2}d$ bits of randomness
from Bob’s POVM using any pure bipartite entangled state provided there exists
an extremal POVM $\\{\mathcal{I}_{b}\\}$ that satisfies the condition (5.119)
for any $\rho_{B}(\bm{\alpha})$.

Here we show an example of extremal qudit POVM that can be used to generate
the optimal amount of randomness by Bob when he and Alice and share the two-
qudit maximally entangled state ${|{\phi^{d}_{+}}\rangle}$. We consider a
simple construction of family of extremal $d^{2}-$outcome POVM’s acting on
arbitrary dimensional Hilbert space introduced in Ref. [80]. It turns out that
such POVM’s serve as a perfect example to obtain the desired result. Consider
the following $d^{2}$ unitary operators defined as

$U_{k,l}=X_{d}^{k}Z_{d}^{l},$ (5.120)

where $k,l=0\dots d-1$ and a vector ${|{\nu}\rangle}\in\mathbbm{C}^{d}$ such
that $\mathrm{Tr}[U^{\dagger}_{k,l}{|{\nu}\rangle}]\neq 0$ for any $k,l$. As
proven in [80], the following $d^{2}-$outcome POVM given by

$\mathcal{I}_{k,l}:=\frac{1}{d}U_{k,l}{|{\nu}\rangle}\\!{\langle{\nu}|}U^{\dagger}_{k,l},$
(5.121)

is extremal. Notice that when
${|{\psi_{AB}}\rangle}={|{\phi_{+}^{d}}\rangle}$, then the local states of
Alice and Bob are $\rho_{A}=\rho_{B}=\mathbb{1}_{d}/d$. Now, plugging this
reduced state and the above POVM (5.121) into the left hand side of the
condition (5.119), we see that

$\displaystyle\mathrm{Tr}[\mathcal{I}_{b}\rho_{B}(\bm{\alpha})]=\frac{1}{d^{2}}\mathrm{Tr}(U_{k,l}{|{\nu}\rangle}\\!{\langle{\nu}|}U^{\dagger}_{k,l})=\frac{1}{d^{2}}\qquad\forall
k,l.$ (5.122)

Thus, the example provided above can be used to obtain optimal randomness.

Now, we also demonstrate that the optimal randomness can be certified when
Alice and Bob share partially entangled states. For this purpose, we find
examples of extremal qudit POVMs with $d^{2}$ outcomes when $d=3,4,5,6$ that
can be used to securely generate $2\log_{2}d$ amount of random bits using a
particular class of partially entangled pure states
${|{\psi(\bm{\alpha})}\rangle}_{AB}$ such that $\alpha_{i}\geq 1/d$ for
$i=0,1,\ldots,d-2$. These extremal POVM’s are given by,

$\mathcal{I}_{b}:=\lambda_{b}{|{\delta_{b}}\rangle}\\!{\langle{\delta_{b}}|}.$
(5.123)

For $b=0,\ldots,d-2$, the vectors ${|{\delta_{b}}\rangle}$ are given by

$\displaystyle{|{\delta_{b}}\rangle}={|{b}\rangle},$ (5.124)

whereas, for $b=d-1,\ldots,d^{2}-1$ they are defined as

$\displaystyle{|{\delta_{b}}\rangle}=\sum_{i=0}^{d-1}\mu_{i}\exp{\left(\frac{2\pi\mathbbm{i}\xi_{i}(b-d+1)}{d^{2}-d+1}\right)}{|{i}\rangle}$
(5.125)

where

$\displaystyle\mu_{i}=\sqrt{\frac{1-\lambda_{i}}{(d^{2}-d+1)\lambda_{d}}}\qquad(i=0,1,\ldots,d-2)$
(5.126)

and

$\quad\mu_{d-1}=\sqrt{\frac{1}{(d^{2}-d+1)\lambda_{d}}}.$ (5.127)

The $\lambda_{b}^{\prime}s$ are given by

$\displaystyle\lambda_{i}=\frac{1}{d^{2}\alpha_{i}^{2}}\qquad(i=0,1,\ldots,d-2)$
(5.128)

and

$\lambda_{d-1}=\lambda_{d}=\ldots=\lambda_{d^{2}-1}=\frac{1}{d^{2}-d+1}\left(d-\sum_{i=0}^{d-2}\lambda_{i}\right).$
(5.129)

Finally, below we provide the $\xi_{i}$ coefficients.
For $d=3$:

$\displaystyle\xi_{0}=0,\quad\xi_{1}=1\quad\text{and}\quad\xi_{2}=3.$ (5.130)

For $d=4$:

$\displaystyle\xi_{0}=0,\quad\xi_{1}=1,\quad\xi_{2}=3\quad\text{and}\quad\xi_{3}=9.$
(5.131)

For $d=5$:

$\displaystyle\xi_{0}=0,\quad\xi_{1}=1,\quad\xi_{2}=4,\quad\xi_{3}=14\quad\text{and}\quad\xi_{4}=16.$
(5.132)

For $d=6$:

$\displaystyle\xi_{0}=0,\quad\xi_{1}=1,\quad\xi_{2}=3,\quad\xi_{3}=8,\quad\xi_{4}=12\quad\text{and}\quad\xi_{4}=18.$
(5.133)

All these coefficients were found numerically such that the above constructed
POVM is extremal and satisfies the condition (5.119).

###  5.6 Conclusions and Discussions

In this chapter, we first constructed a family of steering inequalities that
are maximally violated by every pure entangled bipartite state. The only other
work that provides a steering inequality that is maximally violated by any
pure entangled state is Ref. [176]. However, the inequality proposed in Ref.
[176] requires the trusted party to perform the full tomography on her
subsystem. On the other hand, our scheme is the most efficient in terms of the
number of measurements, as we require only two measurements to be performed by
both the parties. This is the minimal number required to observe any form of
quantum non-locality. We then showed that the maximal violation of our
inequality allows one to certify any pure bipartite entangled state in the
1SDI scenario. A method for certification of any pure entangled bipartite
state in the 1SDI scenario was also proposed in [139], but their approach is a
direct translation of the method of Ref. [179] to the 1SDI scenario that
relies on certification of two-qubit states as discussed before in Chapter 3.
The scheme also requires both the parties to perform three and four
measurements respectively. Contrary to this, our certification scheme relies
only on two genuinely $d$-outcome measurements per party making it extremely
useful for experimental implementation. Moreover, our scheme does not need to
assume that the state shared among Alice and Bob is pure and the measurements
performed by them are projective.

Improving on the results presented in the previous Chapter 4, we went on
further with our self-testing result and used it to certify any rank-one
extremal POVM in 1SDI scenario. Apart from it, we also showed that this task
can be accomplished with quantum states that are close to separable states,
that is, quantum states with a low level of entanglement. This makes our
scheme resource friendly. Finally, we utilised both these results to devise a
simple scheme for certification of the optimal amount of $2\log_{2}d$ bits of
randomness using quantum systems of local dimension $d$ in the 1SDI scenario.
Apart from its importance towards application in quantum cryptography, our
result answers a long-standing question in the quantum foundations community
of whether one can securely generate this optimal amount of randomness.
Moreover, for a few finite dimensions, we showed that we can also accomplish
this task using low levels of entanglement.

Some interesting follow-up questions arise from our work. First, and the most
important one would be to explore whether our construction of the steering
functionals can be used to design Bell functionals whose maximal quantum value
is achieved by any pure entangled bipartite state and two measurements per
site which can be later used for self-testing. Another interesting problem
would be to devise a scheme for certification of extremal measurements of
arbitary rank, and thus certify any quantum measurement in the 1SDI scenario.
A direct follow-up problem from our work is to find extremal POVM’s satisfying
the condition (5.119) for any $\rho_{B}(\bm{\alpha})$. A more challenging
problem would be then to find a fully device-independent scheme for
certification of optimal randomness using quantum systems of arbitrary local
dimension. For the particular case of $d=2$ and $d=3$, schemes have been
devised that can certify $2\log_{2}2$ and $2\log_{2}3$ bits of local
randomness in [54] and [7] respectively (see also Ref. [180]).

##  Chapter 6 Concluding remarks

###  6.1 Summary of the thesis

Let us finally summarise the major points presented throughout this thesis and
their relevance to the current literature in quantum information and
foundations.

##### From a theoretical perspective

  1. 1.

All the results presented in this thesis are very general, in the sense that,
they are applicable to composite quantum systems of arbitrary local
dimensions. The results presented in Chapter 3 are even applicable to
arbitrary number of parties. As a matter of fact, the results presented in
Chapter 4 and Chapter 5 can also be generalised to arbitrary number of parties
using similar mathematical techniques.

  2. 2.

Most of the known certification schemes in the device-independent regime deal
with states that are locally qubits and thus can utilise the Jordan’s lemma
[74] which simplifies the mathematical considerations significantly. Our work
is thus interesting from a mathematical point of view, as we develop new
mathematical techniques that are applicable to arbitrary finite dimensional
quantum systems.

  3. 3.

The result presented in Chapter 3 provide an interesting insight into the
structure of quantum sets. For instance, it was exploited in Ref. [164] to
show that the set of quantum correlations in a certain Bell scenario is not
closed.

  4. 4.

An open question in quantum information has been whether one can securely
generate the optimal randomness using quantum systems of arbitrary local
dimension. In Chapter 5, we show that it is possible in the 1SDI scenario.

##### From a practical perspective

  1. 1.

We showed in Chapter 3 that one can certify the generalised GHZ state using
just two genuinely $d-$outcome measurements per party in a fully device-
independent way. This is the minimum number of measurements required to
observe quantum non-locality and thus more efficient as compared to the
existing schemes with regards to implementing it in experiments as one needs
to observe minimum number of correlations to certify the state.

  2. 2.

Mutually unbiased bases are essential for quantum cryptographic tasks. In
Chapter 4, we presented robust certification of mutually unbiased bases in the
1SDI scenario. This is the first instance where a method for certification of
a general family of measurements, termed genuinely incompatible, have been
introduced, which is based on quantum steering.

  3. 3.

In Chapter 5, we demonstrated a method to certify any pure bipartite entangled
state using just two genuinely $d-$outcome measurements per party in the 1SDI
scenario. This is again the most efficient protocol till date that can be used
to certify such states. We then showed that any rank-one extremal measurement
can be certified in the 1SDI scenario.

  4. 4.

Any cryptographic task requires access to sources generating random bits. In
Chapter 3, we demonstrated a protocol to generate randomness of amount
$\log_{2}d$ bits using projective measurements in a fully device-independent
way. Then, in Chapter 5, we showed that optimal randomness of amount
$2\log_{2}d$ bits can be certified using a quantum system and a generalised
measurement in the 1SDI scenario. Both of these protocols are secure against
any eavesdropper who has access to quantum resources.

Let us list some of the interesting open questions that stem from this work.

###  6.2 Open questions for further exploration

  1. 1.

The first open question that naturally stems from this work is to find Bell
inequalities inspired from the construction of our steering inequalities in
Chapter 5 that are maximally violated by any pure bipartite entangled state
and utilises only two measurements per party. Then, building on our techniques
described in Chapter 3, it would be extremely interesting to prove self-
testing statements of such states in the fully DI scenario using minimal
number of measurements.

  2. 2.

Generalising the steering inequalities in Chapter 5 to the multipartite
scenario in order for 1SDI certification of any pure multipartite entangled
state. The author of this thesis is currently investigating the possibility to
certify certain class of multipartite states in the 1SDI scenario such as
graph states and Schmidt states by extending the approach presented in Chapter
4 and Chapter 5.

  3. 3.

Generalising the certification method of rank-one extremal measurements in the
1SDI scenario to measurements of arbitrary rank. Further, it would be
interesting to explore whether one can reduce the number of measurements
performed by each party, as in our scheme, the trusted party needs to perform
$d+1$ measurements.

  4. 4.

Finding ideal POVM’s that can be used to locally generate the optimal amount
of randomness using any pure partially entangled bipartite state, or putting
it simply, finding extremal POVM’s $\\{\mathcal{I}_{b}\\}$ that satisfies the
condition (5.91) for any non-singular local state $\rho_{B}$.

  5. 5.

Extending all the 1SDI schemes presented in this thesis to the fully device-
independent scenario. In particular, it would be extremely interesting to
provide a way to certify the optimal amount of randomness in the fully device-
independent scenario. Further, certifying the mutually unbiased bases in a
fully device-independent way has been a long sought after question in quantum
information community. For instance, if one can fully characterise the trusted
side in the scheme presented in Chapter 4, then this scheme becomes fully
device-independent.

## Bibliography

  * [1] Shubhayan Sarkar and Remigiusz Augusiak  “Self-testing of multipartite Greenberger-Horne-Zeilinger states of arbitrary local dimension with arbitrary number of measurements per party”  In Phys. Rev. A 105 American Physical Society, 2022, pp. 032416  DOI: 10.1103/PhysRevA.105.032416
  * [2] Shubhayan Sarkar et al.  “Self-testing of any pure entangled state with minimal number of measurements and optimal randomness certification in one-sided device-independent scenario”, 2021  arXiv:2110.15176 [quant-ph]
  * [3] Shubhayan Sarkar, Debashis Saha and Remigiusz Augusiak  “Certification of incompatible measurements using quantum steering”  In Phys. Rev. A 106 American Physical Society, 2022, pp. L040402  DOI: 10.1103/PhysRevA.106.L040402
  * [4] Shubhayan Sarkar, Debashis Saha, Jędrzej Kaniewski and Remigiusz Augusiak  In npj Quantum Information 7, 2021, pp. 151  DOI: 10.1038/s41534-021-00490-3
  * [5] Shubhayan Sarkar, Chandan Datta, Saronath Halder and Remigiusz Augusiak  “Self-testing composite measurements and bound entangled state in a unified framework”  arXiv, 2023  DOI: 10.48550/ARXIV.2301.11409
  * [6] Shubhayan Sarkar  “Certification of the maximally entangled state using non-projective measurements”  arXiv, 2022  DOI: 10.48550/ARXIV.2210.14099
  * [7] Jakub J. Borkała, Chellasamy Jebarathinam, Shubhayan Sarkar and Remigiusz Augusiak  “Device-Independent Certification of Maximal Randomness from Pure Entangled Two-Qutrit States Using Non-Projective Measurements”  In Entropy 24.3, 2022  DOI: 10.3390/e24030350
  * [8] Shubhayan Sarkar and Debashis Saha  “Probing measurement problem of quantum theory with an operational approach”  arXiv, 2021  DOI: 10.48550/ARXIV.2107.08447
  * [9] Shubhayan Sarkar  “Universal notion of classicality based on ontological framework”  arXiv, 2021  DOI: 10.48550/ARXIV.2104.14355
  * [10] Colin Benjamin and Shubhayan Sarkar  “The emergence of cooperation in the thermodynamic limit”  In Chaos, Solitons & Fractals 135, 2020, pp. 109762  DOI: https://doi.org/10.1016/j.chaos.2020.109762
  * [11] Shubhayan Sarkar and Colin Benjamin  “Triggers for cooperative behavior in the thermodynamic limit: A case study in Public goods game”  In Chaos: An Interdisciplinary Journal of Nonlinear Science 29.5, 2019, pp. 053131  DOI: 10.1063/1.5085076
  * [12] Shubhayan Sarkar and Colin Benjamin  “Quantum Nash equilibrium in the thermodynamic limit”  In Quantum Inf Process 18, 2019, pp. 122  DOI: https://doi.org/10.1007/s11128-019-2237-2
  * [13] Colin Benjamin and Shubhayan Sarkar  “Entanglement renders free riding redundant in the thermodynamic limit”  In Physica A: Statistical Mechanics and its Applications 521, 2019, pp. 607–613  DOI: https://doi.org/10.1016/j.physa.2019.01.085
  * [14] Shubhayan Sarkar  “Entropy as a bound for expectation values and variances of a general quantum mechanical observable”  In International Journal of Quantum Information 16.04, 2018, pp. 1850036  DOI: 10.1142/S0219749918500363
  * [15] Shubhayan Sarkar and Chandan Datta  “Can quantum discord increase in a quantum communication task?”  In Quantum Inf Process 17, 2018, pp. 248  DOI: https://doi.org/10.1007/s11128-018-2019-2
  * [16] John Preskill  “Course Information for Physics 219/Computer Science 219 Quantum Computation”, 2015  URL: http://theory.caltech.edu/~preskill/ph229/
  * [17] Michael A. Nielsen and Isaac L. Chuang  “Quantum Computation and Quantum Information: 10th Anniversary Edition”  Cambridge University Press, 2010  URL: https://doi.org/10.1017/CBO9780511976667
  * [18] A. Einstein, B. Podolsky and N. Rosen  “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”  In Phys. Rev. 47 American Physical Society, 1935, pp. 777–780  DOI: 10.1103/PhysRev.47.777
  * [19] John S Bell  “On the Einstein-Podolsky-Rosen paradox”  In Physics Physique Fizika 1.3 APS, 1964, pp. 195  URL: https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
  * [20] Alain Aspect, Philippe Grangier and Gérard Roger  “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedanken experiment: A New Violation of Bell’s Inequalities”  In Phys. Rev. Lett. 49 American Physical Society, 1982, pp. 91–94  DOI: 10.1103/PhysRevLett.49.91
  * [21] A. Aspect, P. Grangier and G. Roger  “Experimental Tests of Realistic Local Theories via Bell’s Theorem”  In Phys. Rev. Lett. 47 American Physical Society, 1981, pp. 460–463  DOI: 10.1103/PhysRevLett.47.460
  * [22] M. Giustina, A. Mech, S.Ramelow and al.  In Nature 497, 2013, pp. 227–230  URL: https://doi.org/10.1038/nature12012
  * [23] B. Hensen, H. Bernien, A. Dréau and al.  “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres”  In Nature 526, 2015, pp. 682–686  URL: https://doi.org/10.1038/nature15759
  * [24] Antonio Acín et al.  “Device-Independent Security of Quantum Cryptography against Collective Attacks”  In Phys. Rev. Lett. 98 American Physical Society, 2007, pp. 230501  DOI: 10.1103/PhysRevLett.98.230501
  * [25] Dominic Mayers and Andrew Chi-Chih Yao  “Quantum cryptography with imperfect apparatus”  In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), 1998, pp. 503–509  URL: https://arxiv.org/abs/quant-ph/9809039
  * [26] D. Mayers and A. Yao  “Self Testing Quantum Apparatus”  In Quantum Inf. Comput. 4.4 Paramus, NJ: Rinton Press, Incorporated, 2004, pp. 273–286  DOI: doi.org/10.26421/QIC4.4
  * [27] M McKague, T H Yang and V Scarani  “Robust self-testing of the singlet”  In Journal of Physics A: Mathematical and Theoretical 45.45 IOP Publishing, 2012, pp. 455304  DOI: 10.1088/1751-8113/45/45/455304
  * [28] Tzyh Haur Yang and Miguel Navascués  “Robust self-testing of unknown quantum systems into any entangled two-qubit states”  In Phys. Rev. A 87 American Physical Society, 2013, pp. 050102  DOI: 10.1103/PhysRevA.87.050102
  * [29] Cédric Bamps and Stefano Pironio  “Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing”  In Phys. Rev. A 91 American Physical Society, 2015, pp. 052111  DOI: 10.1103/PhysRevA.91.052111
  * [30] Yukun Wang, Xingyao Wu and Valerio Scarani  “All the self-testings of the singlet for two binary measurements”  In New Journal of Physics 18.2 IOP Publishing, 2016, pp. 025021  DOI: 10.1088/1367-2630/18/2/025021
  * [31] I Šupić, R Augusiak, A Salavrakos and A Acín  “Self-testing protocols based on the chained Bell inequalities”  In New Journal of Physics 18.3 IOP Publishing, 2016, pp. 035013  DOI: 10.1088/1367-2630/18/3/035013
  * [32] Tzyh Haur Yang et al.  “Robust and Versatile Black-Box Certification of Quantum Devices”  In Phys. Rev. Lett. 113 American Physical Society, 2014, pp. 040401  DOI: 10.1103/PhysRevLett.113.040401
  * [33] Tim Coopmans, Jedrzej Kaniewski and Christian Schaffner  “Robust self-testing of two-qubit states”  In Phys. Rev. A 99 American Physical Society, 2019, pp. 052123  DOI: 10.1103/PhysRevA.99.052123
  * [34] Jedrzej Kaniewski  “Self-testing of binary observables based on commutation”  In Phys. Rev. A 95 American Physical Society, 2017, pp. 062323  DOI: 10.1103/PhysRevA.95.062323
  * [35] Xinhui Li et al.  “Analytic robustness bound for self-testing of the singlet with two binary measurements”  In J. Opt. Soc. Am. B 36.2 OSA, 2019, pp. 457–463  DOI: 10.1364/JOSAB.36.000457
  * [36] C.-E. Bardyn et al.  “Device-independent state estimation based on Bell’s inequalities”  In Phys. Rev. A 80 American Physical Society, 2009, pp. 062327  DOI: 10.1103/PhysRevA.80.062327
  * [37] Matthew McKague  “Self-Testing Graph States”  In Theory of Quantum Computation, Communication, and Cryptography Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 104–120
  * [38] Jedrzej Kaniewski  “Analytic and Nearly Optimal Self-Testing Bounds for the Clauser-Horne-Shimony-Holt and Mermin Inequalities”  In Phys. Rev. Lett. 117 American Physical Society, 2016, pp. 070402  DOI: 10.1103/PhysRevLett.117.070402
  * [39] F. Baccari et al.  “Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing”  In Phys. Rev. Lett. 124 American Physical Society, 2020, pp. 020402  DOI: 10.1103/PhysRevLett.124.020402
  * [40] I Šupić, A Coladangelo, R Augusiak and A Acín  “Self-testing multipartite entangled states through projections onto two systems”  In New Journal of Physics 20.8 IOP Publishing, 2018, pp. 083041  DOI: 10.1088/1367-2630/aad89b
  * [41] Matteo Fadel  “Self-testing Dicke states”  arXiv, 2017  DOI: 10.48550/ARXIV.1707.01215
  * [42] Xinhui Li et al.  “Self-testing using only marginal information”  In Phys. Rev. A 98 American Physical Society, 2018, pp. 052331  DOI: 10.1103/PhysRevA.98.052331
  * [43] Xingyao Wu et al.  “Robust self-testing of the three-qubit $W$ state”  In Phys. Rev. A 90 American Physical Society, 2014, pp. 042339  DOI: 10.1103/PhysRevA.90.042339
  * [44] Károly F. Pál, Tamás Vértesi and Miguel Navascués  “Device-independent tomography of multipartite quantum states”  In Phys. Rev. A 90 American Physical Society, 2014, pp. 042340  DOI: 10.1103/PhysRevA.90.042340
  * [45] Pavel Sekatski, Jean-Daniel Bancal, Sebastian Wagner and Nicolas Sangouard  “Certifying the Building Blocks of Quantum Computers from $B$ell’s Theorem”  In Phys. Rev. Lett. 121 American Physical Society, 2018, pp. 180505  DOI: 10.1103/PhysRevLett.121.180505
  * [46] Jędrzej Kaniewski et al.  “Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems”  In Quantum 3 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2019, pp. 198  DOI: 10.22331/q-2019-10-24-198
  * [47] A. Coladangelo, K. Goh and V. Scarani  “All pure bipartite entangled states can be self-tested”  In Nat Commun 8, 2017, pp. 15485  URL: https://doi.org/10.1038/ncomms15485
  * [48] Andrea Coladangelo  “Generalization of the Clauser-Horne-Shimony-Holt inequality self-testing maximally entangled states of any local dimension”  In Phys. Rev. A 98 American Physical Society, 2018, pp. 052115  DOI: 10.1103/PhysRevA.98.052115
  * [49] Tzyh Haur Yang et al.  “Robust and Versatile Black-Box Certification of Quantum Devices”  In Phys. Rev. Lett. 113 American Physical Society, 2014, pp. 040401  DOI: 10.1103/PhysRevLett.113.040401
  * [50] Jean-Daniel Bancal et al.  “Physical characterization of quantum devices from nonlocal correlations”  In Phys. Rev. A 91 American Physical Society, 2015, pp. 022115  DOI: 10.1103/PhysRevA.91.022115
  * [51] D. M. Greenberger, M. A. Horne and A. Zeilinger  “Going Beyond Bell’s Theorem”, 2021  arXiv: https://doi.org/10.48550/arXiv.0712.0921
  * [52] Alexia Salavrakos et al.  “Bell Inequalities Tailored to Maximally Entangled States”  In Phys. Rev. Lett. 119 American Physical Society, 2017, pp. 040402  DOI: 10.1103/PhysRevLett.119.040402
  * [53] R Augusiak, A Salavrakos, J Tura and A Acín  “Bell inequalities tailored to the Greenberger–Horne–Zeilinger states of arbitrary local dimension”  In New Journal of Physics 21.11 IOP Publishing, 2019, pp. 113001  DOI: 10.1088/1367-2630/ab4d9f
  * [54] Erik Woodhead et al.  “Maximal randomness from partially entangled states”  In Phys. Rev. Research 2 American Physical Society, 2020, pp. 042028  DOI: https://doi.org/10.1103/PhysRevResearch.2.042028
  * [55] Ivan Šupić, Daniel Cavalcanti and Joseph Bowles  “Device-independent certification of tensor products of quantum states using single-copy self-testing protocols”  In Quantum 5 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2021, pp. 418  DOI: 10.22331/q-2021-03-23-418
  * [56] Joseph Bowles, Ivan Šupić, Daniel Cavalcanti and Antonio Acín  “Self-testing of Pauli observables for device-independent entanglement certification”  In Phys. Rev. A 98 American Physical Society, 2018, pp. 042336  DOI: 10.1103/PhysRevA.98.042336
  * [57] Matthew McKague  “Self-testing in parallel with CHSH”  In Quantum 1 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2017, pp. 1  DOI: 10.22331/q-2017-04-25-1
  * [58] Xingyao Wu, Jean-Daniel Bancal, Matthew McKague and Valerio Scarani  “Device-independent parallel self-testing of two singlets”  In Phys. Rev. A 93 American Physical Society, 2016, pp. 062121  DOI: 10.1103/PhysRevA.93.062121
  * [59] Jean-Daniel Bancal, Nicolas Sangouard and Pavel Sekatski  “Noise-Resistant Device-Independent Certification of Bell State Measurements”  In Phys. Rev. Lett. 121 American Physical Society, 2018, pp. 250506  DOI: 10.1103/PhysRevLett.121.250506
  * [60] Marc Olivier Renou, Jedrzej Kaniewski and Nicolas Brunner  “Self-Testing Entangled Measurements in Quantum Networks”  In Phys. Rev. Lett. 121 American Physical Society, 2018, pp. 250507  DOI: 10.1103/PhysRevLett.121.250507
  * [61] E. Schrödinger  “Discussion of Probability Relations between Separated Systems”  In Math. Proc. Camb. Phil. Soc. 31.4 Cambridge University Press (CUP), 1935, pp. 555–563  DOI: 10.1017/s0305004100013554
  * [62] H. M. Wiseman, S. J. Jones and A. C. Doherty  “Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox”  In Phys. Rev. Lett. 98 American Physical Society, 2007, pp. 140402  DOI: 10.1103/PhysRevLett.98.140402
  * [63] Ivan Šupic and Matty J Hoban  “Self-testing through EPR-steering”  In New J. Phys. 18.7 IOP Publishing, 2016, pp. 075006  DOI: 10.1088/1367-2630/18/7/075006
  * [64] Alexandru Gheorghiu, Petros Wallden and Elham Kashefi  “Rigidity of quantum steering and one-sided device-independent verifiable quantum computation”  In New J. Phys. 19.2 IOP Publishing, 2017, pp. 023043  DOI: 10.1088/1367-2630/aa5cff
  * [65] Dagmar Bruß  “Optimal Eavesdropping in Quantum Cryptography with Six States”  In Phys. Rev. Lett. 81 American Physical Society, 1998, pp. 3018–3021  DOI: 10.1103/PhysRevLett.81.3018
  * [66] D. Bruß and C. Macchiavello  “Optimal Eavesdropping in Cryptography with Three-Dimensional Quantum States”  In Phys. Rev. Lett. 88 American Physical Society, 2002, pp. 127901  DOI: 10.1103/PhysRevLett.88.127901
  * [67] P. Oscar Boykin and Vwani Roychowdhury  “Optimal encryption of quantum bits”  In Phys. Rev. A 67 American Physical Society, 2003, pp. 042317  DOI: 10.1103/PhysRevA.67.042317
  * [68] H. Bechmann-Pasquinucci and W. Tittel  “Quantum cryptography using larger alphabets”  In Phys. Rev. A 61 American Physical Society, 2000, pp. 062308  DOI: 10.1103/PhysRevA.61.062308
  * [69] Charles H. Bennett and Gilles Brassard  “Quantum cryptography: Public key distribution and coin tossing” Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84  In Theoretical Computer Science 560, 2014, pp. 7–11  DOI: https://doi.org/10.1016/j.tcs.2014.05.025
  * [70] Thomas Durt and Bob Nagler  “Covariant cloning machines for four-level systems”  In Phys. Rev. A 68 American Physical Society, 2003, pp. 042323  DOI: 10.1103/PhysRevA.68.042323
  * [71] Nicolas J. Cerf  “Pauli Cloning of a Quantum Bit”  In Phys. Rev. Lett. 84 American Physical Society, 2000, pp. 4497–4500  DOI: 10.1103/PhysRevLett.84.4497
  * [72] Nicolas Cerf, Thomas Durt and Nicolas Gisin  “Cloning a qutrit”  In Journal of Modern Optics 49.8 Taylor & Francis, 2002, pp. 1355–1373  DOI: 10.1080/09500340110109043
  * [73] THOMAS Durt, BERTHOLD-GEORG Englert, INGEMAR Bengtsson and KAROL Życzkowski  “ON MUTUALLY UNBIASED BASES”  In International Journal of Quantum Information 08.04, 2010, pp. 535–640  DOI: 10.1142/S0219749910006502
  * [74] Stefano Pironio et al.  “Device-independent quantum key distribution secure against collective attacks”  In New Journal of Physics 11.4 IOP Publishing, 2009, pp. 045021  DOI: 10.1088/1367-2630/11/4/045021
  * [75] Roger Colbeck and Renato Renner  “Hidden Variable Models for Quantum Theory Cannot Have Any Local Part”  In Phys. Rev. Lett. 101 American Physical Society, 2008, pp. 050403  DOI: 10.1103/PhysRevLett.101.050403
  * [76] S. Pironio et al.  “Random numbers certified by Bell’s theorem”  In Nature 464.7291, 2010, pp. 1021–1024  DOI: 10.1038/nature09008
  * [77] R. F. Werner and M. M. Wolf  “Bound Entangled Gaussian States”  In Phys. Rev. Lett. 86 American Physical Society, 2001, pp. 3658–3661  DOI: 10.1103/PhysRevLett.86.3658
  * [78] W. Forrest Stinespring  “Positive functions on $C*$-algebras”  In Proc. Amer. Math. Soc. 6, 1955, pp. 211–216  URL: https://doi.org/10.1090/S0002-9939-1955-0069403-4
  * [79] F. S. Beckman and D. A. Quarles  “On isometries of Euclidean spaces”  In Proc. Amer. Math. Soc. 4, 1953, pp. 810–815  URL: https://doi.org/10.1090/S0002-9939-1953-0058193-5
  * [80] Giacomo Mauro D’Ariano, Paoloplacido Lo Presti and Paolo Perinotti  “Classical randomness in quantum measurements”  In J. Phys. A: Math. Gen. 38.26 IOP Publishing, 2005, pp. 5979–5991  DOI: 10.1088/0305-4470/38/26/010
  * [81] Paul Busch, Pekka Lahti and Reinhard F. Werner  “Colloquium: Quantum root-mean-square error and measurement uncertainty relations”  In Rev. Mod. Phys. 86 American Physical Society, 2014, pp. 1261–1281  DOI: 10.1103/RevModPhys.86.1261
  * [82] Otfried Gühne et al.  “Incompatible measurements in quantum information science”, 2021  arXiv:2112.06784 [quant-ph]
  * [83] Vern Paulsen  “Completely Bounded Maps and Operator Algebras”, Cambridge Studies in Advanced Mathematics  Cambridge University Press, 2003  URL: https://doi.org/10.1017/CBO9780511546631
  * [84] John Watrous  “The Theory of Quantum Information”  Cambridge University Press, 2018  URL: https://doi.org/10.1017/9781316848142
  * [85] John S. Bell  “On the Problem of Hidden Variables in Quantum Mechanics”  In Rev. Mod. Phys. 38 American Physical Society, 1966, pp. 447–452  DOI: 10.1103/RevModPhys.38.447
  * [86] Cédric Bamps and Stefano Pironio  “Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing”  In Phys. Rev. A 91 American Physical Society, 2015, pp. 052111  DOI: 10.1103/PhysRevA.91.052111
  * [87] B. S. Tsirel’son  “Quantum analogues of the Bell inequalities. The case of two spatially separated domains”  In J Math Sci 36, 1987, pp. 557–570  URL: https://doi.org/10.1007/BF01663472
  * [88] S. Popescu and D. Rohrlich  “Quantum nonlocality as an axiom”  In Found Phys 24, 1994, pp. 379–385  URL: https://doi.org/10.1007/BF02058098
  * [89] Nicolas Brunner et al.  “Bell nonlocality”  In Rev. Mod. Phys. 86 American Physical Society, 2014, pp. 419–478  DOI: 10.1103/RevModPhys.86.419
  * [90] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman and M. D. Reid  “Experimental criteria for steering and the $\mathrm{E}$instein-$\mathrm{P}$odolsky-$\mathrm{R}$osen paradox”  In Phys. Rev. A 80 American Physical Society, 2009, pp. 032112  DOI: 10.1103/PhysRevA.80.032112
  * [91] Roope Uola, Ana C. S. Costa, H. Chau Nguyen and Otfried Gühne  “Quantum steering”  In Rev. Mod. Phys. 92 American Physical Society, 2020, pp. 015001  DOI: 10.1103/RevModPhys.92.015001
  * [92] Richard Cleve and Harry Buhrman  “Substituting quantum entanglement for communication”  In Phys. Rev. A 56 American Physical Society, 1997, pp. 1201–1204  DOI: 10.1103/PhysRevA.56.1201
  * [93] Nick Herbert  “Cryptographic approach to hidden variables”  In American Journal of Physics 43.4, 1975, pp. 315–316  URL: https://doi.org/10.1119/1.9861
  * [94] Artur K. Ekert  “Quantum cryptography based on Bell’s theorem”  In Phys. Rev. Lett. 67 American Physical Society, 1991, pp. 661–663  DOI: 10.1103/PhysRevLett.67.661
  * [95] Antonio Acín, Nicolas Gisin and Lluis Masanes  “From Bell’s Theorem to Secure Quantum Key Distribution”  In Phys. Rev. Lett. 97 American Physical Society, 2006, pp. 120405  DOI: 10.1103/PhysRevLett.97.120405
  * [96] Antonio Acín, Serge Massar and Stefano Pironio  “Efficient quantum key distribution secure against no-signalling eavesdroppers”  In New Journal of Physics 8.8 IOP Publishing, 2006, pp. 126–126  DOI: 10.1088/1367-2630/8/8/126
  * [97] Lluís Masanes  “Universally Composable Privacy Amplification from Causality Constraints”  In Phys. Rev. Lett. 102 American Physical Society, 2009, pp. 140501  DOI: 10.1103/PhysRevLett.102.140501
  * [98] Lluís Masanes et al.  “Full Security of Quantum Key Distribution From No-Signaling Constraints”  In IEEE Transactions on Information Theory 60.8, 2014, pp. 4973–4986  URL: https://doi.org/10.1109/TIT.2014.2329417
  * [99] L. Masanes, S. Pironio and A. Acín  “Secure device-independent quantum key distribution with causally independent measurement devices.”  In Nat Commun 2.238, 2011  URL: https://doi.org/10.1038/ncomms1244
  * [100] Jonathan Barrett, Lucien Hardy and Adrian Kent  “No Signaling and Quantum Key Distribution”  In Phys. Rev. Lett. 95 American Physical Society, 2005, pp. 010503  DOI: 10.1103/PhysRevLett.95.010503
  * [101] Jonathan Barrett, Roger Colbeck and Adrian Kent  “Unconditionally secure device-independent quantum key distribution with only two devices”  In Phys. Rev. A 86 American Physical Society, 2012, pp. 062326  DOI: 10.1103/PhysRevA.86.062326
  * [102] Valerio Scarani et al.  “Secrecy extraction from no-signaling correlations”  In Phys. Rev. A 74 American Physical Society, 2006, pp. 042339  DOI: 10.1103/PhysRevA.74.042339
  * [103] Esther Hänggi and Renato Renner  “Device-Independent Quantum Key Distribution with Commuting Measurements”  arXiv, 2010  DOI: 10.48550/ARXIV.1009.1833
  * [104] Nicolas Brunner et al.  “Testing the Dimension of Hilbert Spaces”  In Phys. Rev. Lett. 100 American Physical Society, 2008, pp. 210503  DOI: 10.1103/PhysRevLett.100.210503
  * [105] Jean-Daniel Bancal, Nicolas Gisin, Yeong-Cherng Liang and Stefano Pironio  “Device-Independent Witnesses of Genuine Multipartite Entanglement”  In Phys. Rev. Lett. 106 American Physical Society, 2011, pp. 250404  DOI: 10.1103/PhysRevLett.106.250404
  * [106] Tobias Moroder et al.  “Device-Independent Entanglement Quantification and Related Applications”  In Phys. Rev. Lett. 111 American Physical Society, 2013, pp. 030501  DOI: 10.1103/PhysRevLett.111.030501
  * [107] Koon Tong Goh et al.  “Experimental comparison of tomography and self-testing in certifying entanglement”  In Phys. Rev. A 100 American Physical Society, 2019, pp. 022305  DOI: 10.1103/PhysRevA.100.022305
  * [108] Rotem Arnon-Friedman and Jean-Daniel Bancal  “Device-independent certification of one-shot distillable entanglement”  In New Journal of Physics 21.3 IOP Publishing, 2019, pp. 033010  DOI: 10.1088/1367-2630/aafef6
  * [109] Antonio Acín and Lluis Masanes  “Certified randomness in quantum physics”  In Nature 540.7632, 2016, pp. 213–219  DOI: 10.1038/nature20119
  * [110] Stefano Pironio and Serge Massar  “Security of practical private randomness generation”  In Phys. Rev. A 87 American Physical Society, 2013, pp. 012336  DOI: 10.1103/PhysRevA.87.012336
  * [111] R. Colbeck and R. Renner  “No extension of quantum theory can have improved predictive power”  In Nat Commun 2, 2011, pp. 411  URL: https://doi.org/10.1038/ncomms1416
  * [112] Roger Colbeck and Renato Renner  “Free randomness can be amplified”  In Nature Phys 8, 2012, pp. 450–453  URL: https://doi.org/10.1038/nphys2300
  * [113] Michael J. W. Hall  “Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement Independence”  In Phys. Rev. Lett. 105 American Physical Society, 2010, pp. 250404  DOI: 10.1103/PhysRevLett.105.250404
  * [114] R. Gallego et al.  “Full randomness from arbitrarily deterministic events”  In Nat Commun 4, 2013, pp. 2654  URL: https://doi.org/10.1038/ncomms3654
  * [115] Serge Fehr, Ran Gelles and Christian Schaffner  “Security and composability of randomness expansion from Bell inequalities”  In Phys. Rev. A 87 American Physical Society, 2013, pp. 012335  DOI: 10.1103/PhysRevA.87.012335
  * [116] Dax Enshan Koh et al.  “Effects of Reduced Measurement Independence on Bell-Based Randomness Expansion”  In Phys. Rev. Lett. 109 American Physical Society, 2012, pp. 160404  DOI: 10.1103/PhysRevLett.109.160404
  * [117] S. Gómez et al.  “Experimental investigation of partially entangled states for device-independent randomness generation and self-testing protocols”  In Phys. Rev. A 99 American Physical Society, 2019, pp. 032108  DOI: 10.1103/PhysRevA.99.032108
  * [118] Ole Andersson, Piotr Badziąg, Irina Dumitru and Adán Cabello  “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality”  In Phys. Rev. A 97 American Physical Society, 2018, pp. 012314  DOI: 10.1103/PhysRevA.97.012314
  * [119] F. J. Curchod et al.  “Unbounded randomness certification using sequences of measurements”  In Phys. Rev. A 95 American Physical Society, 2017, pp. 020102  DOI: 10.1103/PhysRevA.95.020102
  * [120] Antonio Acín, Stefano Pironio, Tamás Vértesi and Peter Wittek  “Optimal randomness certification from one entangled bit”  In Phys. Rev. A 93 American Physical Society, 2016, pp. 040102  DOI: 10.1103/PhysRevA.93.040102
  * [121] Cyril Branciard et al.  “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering”  In Phys. Rev. A 85 American Physical Society, 2012, pp. 010301  DOI: 10.1103/PhysRevA.85.010301
  * [122] Eneet Kaur, Mark M Wilde and Andreas Winter  “Fundamental limits on key rates in device-independent quantum key distribution”  In New Journal of Physics 22.2 IOP Publishing, 2020, pp. 023039  DOI: 10.1088/1367-2630/ab6eaa
  * [123] Karol Bartkiewicz et al.  “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks”  In Phys. Rev. A 93 American Physical Society, 2016, pp. 062345  DOI: 10.1103/PhysRevA.93.062345
  * [124] Yang Wang et al.  “Finite-key analysis for one-sided device-independent quantum key distribution”  In Phys. Rev. A 88 American Physical Society, 2013, pp. 052322  DOI: 10.1103/PhysRevA.88.052322
  * [125] Chun Zhou et al.  “Finite-key bound for semi-device-independent quantum key distribution”  In Opt. Express 25.15 OSA, 2017, pp. 16971–16980  DOI: 10.1364/OE.25.016971
  * [126] T. Gehring et al.  “Implementation of continuous-variable quantum key distribution with composable and one-sided-device-independent security against coherent attacks”  In Nat Commun 6 OSA, 2015, pp. 8795  URL: https://doi.org/10.1038/ncomms9795
  * [127] Nathan Walk et al.  “Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution”  In Optica 3.6 OSA, 2016, pp. 634–642  DOI: 10.1364/OPTICA.3.000634
  * [128] Q. Y. He and M. D. Reid  “Genuine Multipartite Einstein-Podolsky-Rosen Steering”  In Phys. Rev. Lett. 111 American Physical Society, 2013, pp. 250403  DOI: 10.1103/PhysRevLett.111.250403
  * [129] Yu Xiang, Ioannis Kogias, Gerardo Adesso and Qiongyi He  “Multipartite Gaussian steering: Monogamy constraints and quantum cryptography applications”  In Phys. Rev. A 95 American Physical Society, 2017, pp. 010101  DOI: 10.1103/PhysRevA.95.010101
  * [130] Ioannis Kogias et al.  “Hierarchy of Steering Criteria Based on Moments for All Bipartite Quantum Systems”  In Phys. Rev. Lett. 115 American Physical Society, 2015, pp. 210401  DOI: 10.1103/PhysRevLett.115.210401
  * [131] Ioannis Kogias, Yu Xiang, Qiongyi He and Gerardo Adesso  “Unconditional security of entanglement-based continuous-variable quantum secret sharing”  In Phys. Rev. A 95 American Physical Society, 2017, pp. 012315  DOI: 10.1103/PhysRevA.95.012315
  * [132] Elsa Passaro, Daniel Cavalcanti, Paul Skrzypczyk and Antonio Acín  “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios”  In New Journal of Physics 17.11 IOP Publishing, 2015, pp. 113010  DOI: 10.1088/1367-2630/17/11/113010
  * [133] Brian Coyle, Matty J. Hoban and Elham Kashefi  “One-Sided Device-Independent Certification of Unbounded Random Numbers”  In Electronic Proceedings in Theoretical Computer Science 273 Open Publishing Association, 2018, pp. 14–26  DOI: 10.4204/eptcs.273.2
  * [134] Paul Skrzypczyk and Daniel Cavalcanti  “Maximal Randomness Generation from Steering Inequality Violations Using Qudits”  In Phys. Rev. Lett. 120 American Physical Society, 2018, pp. 260401  DOI: 10.1103/PhysRevLett.120.260401
  * [135] Yun Zhi Law, Le Phuc Thinh, Jean-Daniel Bancal and Valerio Scarani  “Quantum randomness extraction for various levels of characterization of the devices”  In Journal of Physics A: Mathematical and Theoretical 47.42 IOP Publishing, 2014, pp. 424028  DOI: 10.1088/1751-8113/47/42/424028
  * [136] Marco Piani and John Watrous  “Necessary and Sufficient Quantum Information Characterization of Einstein-Podolsky-Rosen Steering”  In Phys. Rev. Lett. 114 American Physical Society, 2015, pp. 060404  DOI: 10.1103/PhysRevLett.114.060404
  * [137] Roope Uola et al.  “Quantifying Quantum Resources with Conic Programming”  In Phys. Rev. Lett. 122 American Physical Society, 2019, pp. 130404  DOI: 10.1103/PhysRevLett.122.130404
  * [138] Suchetana Goswami et al.  “One-sided device-independent self-testing of any pure two-qubit entangled state”  In Phys. Rev. A 98 American Physical Society, 2018, pp. 022311  DOI: 10.1103/PhysRevA.98.022311
  * [139] Harshank Shrotriya, Kishor Bharti and Leong-Chuan Kwek  “Robust Self Testing of All Pure Bipartite Maximally Entangled States via Quantum Steering”  In Phys. Rev. Research 3 American Physical Society, 2021, pp. 033093  DOI: 10.1103/PhysRevResearch.3.033093
  * [140] Matthew McKague and Michele Mosca  “Generalized Self-testing and the Security of the 6-State Protocol”  In Theory of Quantum Computation, Communication, and Cryptography Berlin, Heidelberg: Springer Berlin Heidelberg, 2011, pp. 113–130
  * [141] Ivan Šupić and Joseph Bowles  “Self-testing of quantum systems: a review”  In Quantum 4 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2020, pp. 337  DOI: 10.22331/q-2020-09-30-337
  * [142] Koon Tong Goh et al.  “Geometry of the set of quantum correlations”  In Phys. Rev. A 97 American Physical Society, 2018, pp. 022104  DOI: 10.1103/PhysRevA.97.022104
  * [143] Flavio Baccari, Remigiusz Augusiak, Ivan Šupić and Antonio Acín  “Device-Independent Certification of Genuinely Entangled Subspaces”  In Phys. Rev. Lett. 125 American Physical Society, 2020, pp. 260507  DOI: 10.1103/PhysRevLett.125.260507
  * [144] Jędrzej Kaniewski et al.  “Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems”  In Quantum 3 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2019, pp. 198  DOI: 10.22331/q-2019-10-24-198
  * [145] J. Kaniewski  “Weak form of self-testing”  In Phys. Rev. Research 2 American Physical Society, 2020, pp. 033420  DOI: 10.1103/PhysRevResearch.2.033420
  * [146] Lluís Masanes  “Asymptotic Violation of Bell Inequalities and Distillability”  In Phys. Rev. Lett. 97 American Physical Society, 2006, pp. 050503  DOI: 10.1103/PhysRevLett.97.050503
  * [147] Miguel Navascués, Stefano Pironio and Antonio Acín  “Bounding the Set of Quantum Correlations”  In Phys. Rev. Lett. 98 American Physical Society, 2007, pp. 010401  DOI: 10.1103/PhysRevLett.98.010401
  * [148] Miguel Navascués, Stefano Pironio and Antonio Acín  “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations”  In New Journal of Physics 10.7 IOP Publishing, 2008, pp. 073013  DOI: 10.1088/1367-2630/10/7/073013
  * [149] S. Pironio, M. Navascués and A. Acín  “Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables”  In SIAM Journal on Optimization 20.5, 2010, pp. 2157–2180  DOI: 10.1137/090760155
  * [150] Shin-Liang Chen et al.  “Robust self-testing of steerable quantum assemblages and its applications on device-independent quantum certification”  In Quantum 5, 2021, pp. 552  DOI: 10.22331/q-2021-09-28-552
  * [151] R. L. Rivest, A. Shamir and L. Adleman  “A Method for Obtaining Digital Signatures and Public-Key Cryptosystems”  In Communications of the ACM 21.2 New York, NY, USA: Association for Computing Machinery, 1978  DOI: 10.1145/359340.359342
  * [152] P.W. Shor  “Algorithms for quantum computation: discrete logarithms and factoring”  In Proceedings 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124–134  URL: https://doi.org/10.1109/SFCS.1994.365700
  * [153] Robert Konig, Renato Renner and Christian Schaffner  “The Operational Meaning of Min- and Max-Entropy”  In IEEE Transactions on Information Theory 55.9, 2009, pp. 4337–4347  DOI: 10.1109/TIT.2009.2025545
  * [154] Antonio Acín, Serge Massar and Stefano Pironio  “Randomness versus Nonlocality and Entanglement”  In Phys. Rev. Lett. 108 American Physical Society, 2012, pp. 100402  DOI: 10.1103/PhysRevLett.108.100402
  * [155] O Nieto-Silleras, S Pironio and J Silman  “Using complete measurement statistics for optimal device-independent randomness evaluation”  In New Journal of Physics 16.1 IOP Publishing, 2014, pp. 013035  DOI: 10.1088/1367-2630/16/1/013035
  * [156] Jean-Daniel Bancal, Lana Sheridan and Valerio Scarani  “More randomness from the same data”  In New Journal of Physics 16.3 IOP Publishing, 2014, pp. 033011  DOI: 10.1088/1367-2630/16/3/033011
  * [157] Ivan Šupić et al.  “Quantum networks self-test all entangled states”  arXiv, 2022  DOI: 10.48550/ARXIV.2201.05032
  * [158] Daniel Collins et al.  “Bell Inequalities for Arbitrarily High-Dimensional Systems”  In Phys. Rev. Lett. 88 American Physical Society, 2002, pp. 040404  DOI: 10.1103/PhysRevLett.88.040404
  * [159] Jonathan Barrett, Adrian Kent and Stefano Pironio  “Maximally Nonlocal and Monogamous Quantum Correlations”  In Phys. Rev. Lett. 97 American Physical Society, 2006, pp. 170409  DOI: 10.1103/PhysRevLett.97.170409
  * [160] Marek Zukowski, Anton Zeilinger and Michael A. Horne  “Realizable higher-dimensional two-particle entanglements via multiport beam splitters”  In Phys. Rev. A 55 American Physical Society, 1997, pp. 2564–2579  DOI: 10.1103/PhysRevA.55.2564
  * [161] Leandro Aolita, Rodrigo Gallego, Adán Cabello and Antonio Acín  “Fully Nonlocal, Monogamous, and Random Genuinely Multipartite Quantum Correlations”  In Phys. Rev. Lett. 108 American Physical Society, 2012, pp. 100401  DOI: 10.1103/PhysRevLett.108.100401
  * [162] George Svetlichny  “Distinguishing three-body from two-body nonseparability by a Bell-type inequality”  In Phys. Rev. D 35 American Physical Society, 1987, pp. 3066–3069  DOI: 10.1103/PhysRevD.35.3066
  * [163] Jianwei Wang et al.  “Multidimensional quantum entanglement with large-scale integrated optics”  In Science 360.6386, 2018, pp. 285–291  DOI: 10.1126/science.aar7053
  * [164] Salman Beigi  “Separation of quantum, spatial quantum, and approximate quantum correlations”  In Quantum 5 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2021, pp. 389  DOI: 10.22331/q-2021-01-28-389