# Certifying measurement incompatibility in prepare-and-measure and Bell
scenarios

Sophie Egelhaaf  Department of Applied Physics, University of Geneva, 1211
Geneva, Switzerland  Jef Pauwels  Department of Applied Physics, University of
Geneva, 1211 Geneva, Switzerland  Constructor University, Geneva, Switzerland
Marco Túlio Quintino  Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Roope Uola  Department of Applied Physics, University of Geneva, 1211 Geneva,
Switzerland

###### Abstract

We consider the problem of certifying measurement incompatibility in a
prepare-and-measure (PM) scenario. We present different families of sets of
qubit measurements which are incompatible, but cannot lead to any quantum over
classical advantage in PM scenarios. Our examples are obtained via a general
theorem which proves a set of qubit dichotomic measurements can have their
incompatibility certified in a PM scenario if and only if their
incompatibility can be certified in a bipartite Bell scenario where the
parties share a maximally entangled state. Our framework naturally suggests a
hierarchy of increasingly stronger notions of incompatibility, in which more
power is given to the classical simulation by increasing its dimensionality.
For qubits, we give an example of measurements whose incompatibility can be
certified against trit simulations, which we show is the strongest possible
notion for qubits in this framework.

##  I Introduction

Measurement incompatibility, the existence of measurements that cannot be
carried out simultaneously on a single system, is a defining feature of
quantum theory. It represents a radical departure from classical theory and
underpins central aspects in quantum foundations, such as the resolution of
the Heisenberg microscope [1], fundamental bounds in interferometry [2], and
Fine’s theorem [3]. More recently, measurement incompatibility has found
various uses in quantum information theory [4] as a key ingredient for quantum
information protocols, such as quantum key distribution [5, 6], channel
verification [7], device-independent entanglement certification [8], EPR
steering [9, 10, 11, 12, 13, 14, 15], Bell nonlocality [16, 17], temporal
correlations [18, 19, 20], Random Access Coding [21], state discrimination
tasks [22, 23, 24, 25, 26], distributed sampling [27], contextuality [28, 29],
and programmability [30].

Incompatibility is defined at the abstract level of Hilbert spaces, as a
relational property between operator measures. In practice, however, neither
these abstract objects nor the relations between them, are directly
observable. We only have access to the outcome statistics. While
incompatibility can be formulated operationally, its direct detection hinges
on the assumption that appropriately chosen sets of trusted test-states can be
prepared.

Another feature of quantum theory is the ability to establish correlations
between distant parties that defy classical explanations either by measuring
quantum messages, as in the prepare-and-measure scenario [31, 32, 33, 34], or
by performing local measurements on pre-shared entangled states, as in the
Bell scenario [35, 16]. In the Bell scenario, reaching such non-classical
correlations is known to require incompatible measurements and, hence, can be
used to certify incompatibility in a device-independent way, a fact studied
extensively [36, 37, 38, 39].

Here, we analyse the problem of certifying measurement incompatibility in a PM
scenario. This problem has appeared in some previous works, either implicitly
or explicitly [40, 41, 42, 43, 44]. Here, we provide a rigorous definition for
the phenomenon, consistent with the previous works, and discuss some important
properties. Focusing in particular on the simplest case of qubit measurements,
we show that various fundamental results in the literature [45, 46, 40, 42]
can be linked to our scenario, providing a better structural understanding of
the concept of PM certifiable incompatibility, together with classical models
and witnessing techniques for the phenomenon.

The paper is structured as follows. We present basic definitions and discuss
their operational role in the second section. In the third section, we
summarize the central correlation scenarios: the prepare-and-measure and the
Bell scenario. In the fourth section, we illustrate incompatibility in the PM
scenario against classical simulation protocols. More precisely, we show that
all sets of qubit measurements have a four-dimensional classical model and
show that the sets of qubit measurements which have a two-dimensional
classical model also have a local model when used in a Bell scenario with a
maximally entangled state. We further provide examples of measurement sets
illustrating this. Our notion of incompatibility naturally suggests a
hierarchy of increasingly stronger notions of incompatibility corresponding to
the dimensionality of the underlying classical simulation. For qubits, we show
that all measurements can be simulated with quart models, and give an example
of incompatible measurements that can be certified incompatible assuming trit
models. We conclude the paper in the fifth section with discussion and open
questions.

##  II Joint measurability

Figure 1: Operational definition of joint measurability. Joint measurability
asks whether there exists a single POVM $\\{E_{\lambda}\\}_{\lambda}$ from
which the original set of POVMs $\\{B_{b|y}\\}_{b,y}$ can be classically post-
processed.

A quantum measurement is modelled as a positive operator-valued measure
(POVM), i.e. a collection
$\\{B_{b|y}\\}_{b}\subset\mathcal{L}(\mathbb{C}_{d})$ for which $B_{b|y}\geq
0$ and $\sum_{b}B_{b|y}=\mathds{1}$. Here the index $y$ labels the choice of
measurement and $b$ is the corresponding outcome. A central concept for sets
of measurements $\\{B_{b|y}\\}_{b,y}$ is _joint measurability_. This asks
whether there exists some further POVM from which the set can be post-
processed, cf. Fig. 1. More formally, we have the following definition:

######  Definition 1 (Joint measurability (JM)).

A set of quantum measurements $\\{B_{b|y}\\}_{y}$ is jointly measurable (JM)
if there exists a POVM $\\{E_{\lambda}\\}_{\lambda}$ and probability
distributions $\\{p(\cdot|y,\lambda)\\}_{y\lambda}$ such that

$\displaystyle
B_{b|y}=\sum_{\lambda}p(b|y,\lambda)E_{\lambda},\quad\quad\forall b,y$ (1)

A set of measurements which is not jointly measurable is called
_incompatible_. The POVM $\\{E_{\lambda}\\}_{\lambda}$ is called a joint or
mother POVM.

This definition of joint measurability, formulated at the level of relations
between operators in Hilbert space, is entirely formal. Nonetheless, in the
context of quantum theory, our observations are limited to the _outcome
statistics of measurements_. To make the definition more operational
practical, consider a bipartite correlation experiment involving two parties,
Alice and Bob. Bob possesses a device capable of implementing the various
measurements $\\{B_{b|y}\\}_{y}$ based on some independently chosen input $y$.
In each round of the experiment, Alice sends a trusted quantum state
$\rho_{x}$ based on some input $x$. Such prepare-and-measure experiments are
characterized by a set of conditional probabilities $\\{p(b|x,y)\\}_{bxy}$
referred to as the correlations in the experiment. Quantum theory predicts
that given these states and measurements, $p(b|x,y)=\tr[\rho_{x}B_{b|y}]$.

If the preparation device of Alice is completely trusted, we can give the
practical, but equivalent definition of joint measurability:

Definition 1∗ (Joint measurability (JM), practical) _A set of quantum
measurements $\\{B_{b|y}\\}_{b,y}$ is jointly measurable (JM) if for _every_
set of trusted preparations $\\{\rho_{x}\\}_{x}$ we can write:_

$p(b|\rho_{x},y)=\sum_{a}p_{A}(a|\rho_{x})p_{B}(b|y,a),$ (2)

_where $p_{B}(b|y,a)$ are classical post-processings and one requires
$p_{A}(a|\rho_{x})$ to be linear in the second argument. From Fréchet-Riesz
representation theorem, it then follows that
$p_{A}(a|\rho_{x})=\tr(\rho_{x}E_{a})$ for some POVM $\\{E_{a}\\}_{a}$._

This equivalent definition is advantageous as it directly connects to what we
can observe, namely outcomes of experiments. Specifically, it asserts that the
joint measurability of the set of quantum measurements is confirmed if, for
any conceivable set of trusted preparations $\\{\rho_{x}\\}_{x}$, the observed
probabilities $p(b|\rho_{x},y)$ can be expressed as a combination of the
probabilities $p_{A}(a|\rho_{x})$ related to a mother POVM $\\{E_{a}\\}_{a}$
and classical post-processing $p_{B}(b|y,a)$. However, it is important to note
that this definition assumes perfect trust in Alice’s device, implying that
the device precisely prepares the states $\\{\rho_{x}\\}_{x}$.

Nonetheless, it is easy to see that the requirement of perfectly trusted
preparations is not needed to check the incompatibility of various
incompatible measurements. For example, consider that the preparations of
Alice are noisy, i.e. instead of sending the states $\rho_{x}$, she sends the
states $\eta\rho_{x}+(1-\eta)\frac{\mathds{1}}{d}$. This is equivalent to
assuming that the measurements performed by Bob are noisy. For any set of
$d$-dimensional measurements $\\{B_{b|y}\\}_{y}$ and $\eta\in[0,1]$, we define
the set of white noisy measurements $\\{B_{b|y}^{\eta}\\}_{y}$ as

$\displaystyle B_{b|y}^{\eta}:=\eta
B_{b|y}+(1-\eta)\frac{\mathds{1}}{d}\tr(B_{b|y}),\;\forall b,y.$ (3)

For any set of incompatible measurements $\\{B_{b|y}^{\eta}\\}_{y}$, the noisy
set $\\{B_{b|y}^{\eta}\\}_{y}$ is still incompatible up to some level of noise
$\eta$. This can be seen by noting that the set of jointly measurable sets of
measurements is closed [47]. As a standard example, one can consider the pair
of noisy spin measurements in the directions $x$ and $z$. The spin
measurements are given by
$B_{+|1}=|+\rangle\langle+|,B_{-|1}=|-\rangle\langle-|$,
$B_{+|2}=|0\rangle\langle 0|$ and $B_{-|2}=|1\rangle\langle 1|$. The resulting
noisy measurements $B_{b|y}^{\eta}$ are jointly measurable for $\eta\leq
1/\sqrt{2}$ with the mother POVM
$E_{i,j}=\frac{1}{4}(\openone+\eta(i\sigma_{x}+j\sigma_{z}))$, where
$i,j\in\\{-1,1\\}$, and otherwise incompatible [48]. The post-processing is
simply marginalisation, i.e. summing over the index $j$ gives the noisy $x$
measurement and summing over $i$ gives the noisy $z$ measurement. The critical
value of $\eta$ at which a set of measurements $\\{B_{b|y}^{\eta}\\}_{y}$
becomes jointly measurable serves as a simple metric for noise robustness.

This raises the question of whether we can get a stronger certificate of joint
measurability, by lifting the level of trust on Alice’s preparations.

##  III Device-independent certification of joint measurability

Figure 2: Left: The prepare-and-measure scenario for a qubit message and a
dichotomic measurement. Right: The Bell scenario where Alice and Bob share a
$\phi^{+}$ state and perform dichotomic measurements. The set of measurements
that violate the 2-dimensional classical model and the local hidden variable
model, respectively, are equivalent.

If Alice’s preparations $\rho_{x}$ are completely untrusted, can the parties
certify the incompatibility of a set of measurements $\\{B_{b|y}\\}_{y}$? The
usual approach to analysing the prepare-and-measure scenario (semi-)device-
independently, assumes some limitation on the kind of states Alice can send,
or, equivalently, the systems supported by the channel connecting the parties
[49]. Indeed, if the message $\rho_{x}$ can encode perfect information about
the input $x$, the parties can reproduce any correlations $p$. To have a non-
trivial scenario, we can limit the information that $\rho_{x}$ can contain
about $x$. A natural and common approach is to limit the dimension $d$ of the
Hilbert space of $\rho_{x}$, and to allow for shared randomness between the
sender and the receiver. Note that for our purposes, this dimension is
naturally equal to the dimension of the space that the measurements
$\\{B_{b|y}\\}_{y}$ act on.

By definition, to certify that a set of measurements $\\{B_{b|y}\\}_{y}$ is
incompatible entails proving that the correlations cannot be created by
performing a single mother POVM plus classical post-processing. Since all
correlations in a prepare-and-measure scenario with a single measurement
setting can be simulated with classical systems of dimension equaling that of
the measurement [40], showing incompatibility with untrusted preparations is
thus equivalent to demonstrating that, for _some set of states_ , the
resulting correlations $p$ cannot be classically simulated. We refer to the
correlations in a prepare-and-measure experiment $p$ as _classical_ or
$\text{PM}_{d}$ if $p$ can be explained by Alice communicating $d$-dimensional
classical systems:

######  Definition 2 ($\text{PM}_{d}$).

A set of probabilities $\\{p(b|x,y)\\}_{b,x,y}$ can be obtained in a prepare-
and-measure scenario with $d$-dimensional classical communication if there
exists probability measures $\mu$, $p_{A}(\cdot|x,\lambda)$,
$p_{B}(\cdot|a,y,\lambda)$, where $a\in{1,\ldots,d}$ such that

$\displaystyle
p(b|x,y)=\int\pi(\lambda)\sum_{a=1}^{d}p_{A}(a|x,\lambda)p_{B}(b|a,y,\lambda)d\mu(\lambda)$
(4)

for all $b,x,y$.

The smallest $d$ for which given PM correlations do have PMd model is
sometimes called the signalling dimension [50]. This leads to the following
definition:

######  Definition 3 ($\text{PM}_{d}$-JM).

A set of measurements $\\{B_{b|y}\\}_{y}$, $B_{b|y}\in L(\mathbb{C}_{d})$ is
$\text{PM}_{d}$-JM if, for every set of quantum states $\\{\rho_{x}\\}_{x}$,
where $\rho_{x}\in L(\mathbb{C}_{d})$, the set with probabilities given by

$\displaystyle p(b|x,y)=\tr(\rho_{x}B_{b|y})$ (5)

is $\text{PM}_{d}$.

Note that in the above definitions, we can allow the dimension $d_{C}$ of the
classical simulation to be different from the dimension $d_{Q}$ of the
measurements. This enables us to formulate stronger statements about the
incompatibility of a set of measurements when $d_{C}>d_{Q}$.

In Ref. [42], the authors employ a computational polytope approximation method
to demonstrate the existence of qubit measurements that are not jointly
measurable (JM) but are $\text{PM}_{2}$-JM. In the next section, we will
analytically prove this result by presenting several explicit examples of
qubit measurements that are incompatible (not JM) but $\text{PM}_{2}$-JM.

Another natural correlation experiment which can be used to certify the
incompatiblity of quantum measurements is the Bell scenario. Here, Alice and
Bob, who both receive an input, respectively $x$ and $y$, produce outcomes $a$
and $b$, see Fig. 2. The parties can exchange physical systems before the
experiment, but are not allowed to communicate after receiving their inputs. A
Bell experiment is described by the conditional probabilities
$\\{p(a,b|x,y)\\}_{a,b,x,y}$, referred to as correlations. The most general
correlations allowed by quantum theory, are obtained by Alice and Bob
performing quantum measurements, respectively $\\{A_{a|x}\\}_{x}$ and
$\\{B_{b|y}\\}_{y}$ on a shared quantum state $\rho_{AB}$. The correlations
are given by the Born rule $p(a,b|x,y)=\tr[\rho_{AB}A_{a|x}\otimes B_{b|y}]$.

If the correlations $p$ can be described by Alice and Bob exchanging classical
systems before the experiment, we call $p$ _Bell local_. More precisely, we
have the following definition:

######  Definition 4 (Bell locality).

A set of probabilities $\\{p(a,b|x,y)\\}_{a,b,x,y}$ is Bell local if there
exists sets of probability measures $\mu$, $p_{A}(\cdot|x,\lambda)$, and
$p(\cdot|y,\lambda)$ such that

$\displaystyle
p(a,b|x,y)=\int\pi(\lambda)p_{A}(a|x,\lambda)p_{B}(b|y,\lambda)d\mu(\lambda)\quad$
(6)

for all $a,b,x,y$.

This naturally leads to the following definition:

######  Definition 5 (Bell-JM and Bell${}^{\phi^{+}}$-JM).

A set of measurements $\\{B_{b|y}\\}_{y}$, $B_{b|y}\in L(\mathbb{C}_{d})$ is
Bell-JM if for every quantum state $\rho_{AB}\in
L(\mathbb{C}_{d^{\prime}}\otimes\mathbb{C}_{d})$ and every set of measurements
$\\{A_{a|x}\\}_{a,x}$ the set with probabilities given by

$\displaystyle p(a,b|x,y)=\tr(\rho_{AB}A_{a|x}\otimes B_{b|y})$ (7)

is Bell local.

The set of measurements $B_{b|y}\in L(\mathbb{C}_{d})$ is
Bell${}^{\phi^{+}}$-JM if

$\displaystyle
p(a,b|x,y)=\tr(\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}A_{a|x}\otimes
B_{b|y})$ (8)

is Bell local, where
$\ket{\phi^{+}_{d}}:=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii}$ is the
maximally entangled state.

The relationship of between JM and Bell-JM has been analysed in Refs. [12,
13], which present examples of measurements which are not JM but are
Bell${}^{\phi^{+}}$-JM in a scenario where Alice’s measurements are
dichotomic. Later, Ref. [51] shows an example of measurements which are not JM
but are Bell-JM in a scenario where Alice’s measurements are dichotomic.
Finally, Ref. [52] and [53] showed that there exists sets of qubit
measurements which are not JM, but they are Bell-JM, regardless of what Alice
measures.

##  IV Main results

In this section, we establish and demonstrate results pertaining to the
certification of joint measurability in both prepare-and-measure and Bell
correlations, with a specific focus on qubit scenarios. To commence, we have
the following theorem [46]:

######  Theorem 1 (PM4-JM for qubits).

Any set of qubit measurements $\\{B_{b|y}\\}_{y}$ is PM4-JM.

###### Proof.

Ref. [46] shows that all qubit quantum PM strategies can be perfectly
simulated classically with two bits of communication. In other words, for
qubits, all sets with quantum probabilities $p(b|x,y)=\tr(\rho_{x}B_{b|y})$
are PM4. ∎

Consequently, the following theorem represents the strongest form of
incompatibility in prepare-and-measure qubit systems concerning classical
simulability:

######  Theorem 2 (PM3-JM for qubits).

There exists sets of qubit measurements $\\{B_{b|y}\\}_{y}$ which are not
PM3-JM.

###### Proof.

Ref. [46] presents an explicit example of a qubit strategy in a PM scenario
which cannot be obtained via trit classical strategies. Hence, the qubit
measurements used in this work are not PM3-JM. In the example of Ref. [46],
Alice’s states are the 6 eigenvectors of Pauli matrices and Bob has 24
different projective measurements, which are chosen such that their
representation on the Bloch sphere is the vertices of a snub cube. ∎

Our subsequent theorem strongly builds upon the results and methodologies
outlined in Ref. [45], which establishes a one-to-one connection between
specific full-correlation dichotomic Bell scenarios and dichotomic prepare-
and-measure scenarios.

######  Theorem 3.

[PM2-JM and Bell${}^{\phi^{+}}$-JM for qubits] Let
$\\{B_{b|y}^{\eta_{y}}\\}_{y}$ be a set of qubit noisy projective
measurements, that is, $B_{b|y}\in L(\mathbb{C}_{2})$ and we can write
$B_{0|y}^{\eta_{y}}=\eta_{y}\outerproduct{\phi_{y}}{\phi_{y}}+(1-\eta_{y})\frac{\mathds{1}}{2}$,
for some qubit pure state $\ket{\phi_{y}}$ and $\eta_{y}\in[0,1]$. The set of
noisy measurements $\\{B_{b|y}^{\eta_{y}}\\}_{y}$ is Bell${}^{\phi^{+}}$-JM if
and only if it is $\text{PM}_{2}$-JM.

The proof of Thm. 3 is presented in Appendix A. We note that the measurements
used in the above Theorem are exactly the ones, whose POVM elements
$\\{B_{b|y}^{\eta_{y}}\\}_{b,y}$ admit a Bloch vector representation similar
to quantum states, that is, we can write
$B_{0|y}^{\eta_{y}}:=\frac{1}{2}(\openone+\vec{b}_{y}\cdot\vec{\sigma})$,
where $\vec{b}_{y}\in\mathbb{R}^{3}$ is a vector with Euclidean norm
$\norm{\vec{b}_{y}}=\eta_{y}\leq 1$, and
$\vec{b}_{y}\cdot\vec{\sigma}:=b_{y}^{1}\sigma_{x}+b_{y}^{2}\sigma_{y}+b_{y}^{3}\sigma_{z}.$

We notice that the proof is constructive in the follwoing sense: let
$\\{B_{b|y}\\}$ be a set of measurements which are not PM2, i.e. there exist a
set of states $\\{\rho_{x}\\}_{x}$ such that
$\tr(\rho_{x}B_{b|y})\notin\text{PM}_{2}$. Then,the behaviour
$p(ab|xy)=\tr(\phi_{2}^{+}A_{a|x}\otimes B_{b|y})\notin\text{Bell}$, where
$A_{0|x}:=\rho_{x}$ and $A_{1|x}:=\mathds{1}-\rho_{x}$. Conversely, if for the
set of measurements $\\{A_{a|x}\\}_{x}$, we have
$p(ab|xy)=\tr(\phi_{2}^{+}A_{a|x}\otimes B_{b|y})\notin\text{Bell}$ for some
$\\{B_{b|y}\\}_{y}$, it holds that $\tr(\rho_{x}B_{b|y})\notin\text{PM}_{2}$,
with the states $\rho_{x}=A_{0|x}$. We now discuss some consequences of Thm.
3.

######  Corollary 1.

Let $\left\\{X^{\eta},Y^{\eta},Z^{\eta}\right\\}$ be the three noisy Pauli
measurements. This set is JM iff $\eta\leq\frac{1}{\sqrt{3}}$ and PM2-JM iff
$\eta\leq\frac{1}{\sqrt{2}}$. Hence, this yields a simple example that
$PM_{2}-JM$ does not imply $JM$ even for qubits.

###### Proof.

We start by noticing that, for the CHSH Bell inequality, if Alice and Bob
share a maximally entangled state and Alice has $\eta$ white noisy Pauli
measurements, it is possible to obtain a CHSH violation if and only if
$\eta>1/\sqrt{2}$. To finish the proof, we simply notice that in dichotomic
full correlations scenarios where one party has three measurements, the only
non-trivial Bell inequality is the CHSH [54]. ∎

######  Corollary 2.

Let $\left\\{\Pi_{b|y}^{\eta}\right\\}$ be the set of all noisy projective
qubit measurements. This set is $PM_{2}-JM$ iff $\eta\leq\frac{1}{K_{3}}$,
where $K_{3}$ is the real Grothendieck constant of order three, which is known
to respect [55] $0.6875\leq\frac{1}{K_{3}}<0.6961$.

Let $\left\\{\Pi_{b|y}^{2,\eta}\right\\}$ be the set of all qubit noisy
projective planar measurements, i.e. measrurements whose Bloch vectors lie in
a plane. This set is $PM_{2}-JM$ iff $\eta\leq\frac{1}{K_{2}}$, where
$K_{2}=\sqrt{2}$ is the real Grothendieck constant of order two.

###### Proof.

The two-qubit isotropic state
$\phi^{+}_{\eta}:=\eta\outerproduct{\phi^{+}_{2}}{\phi^{+}_{2}}+(1-\eta)\frac{\mathds{1}_{4}}{4}$
violates a Bell inequality with projective measurements if and only if
$\eta>\frac{1}{K_{3}}$ [56, 57]. That is, if $\eta>\frac{1}{K_{3}}$, there
exists measurements such that
$\tr(\phi^{+}_{\eta}\,\Pi^{A}_{a|x}\otimes\Pi_{b|y})$ violates a Bell
inequality. To finish the first part of the proof, we just notice that
$\tr(\phi^{+}_{\eta}\,\Pi^{A}_{a|x}\otimes\Pi^{B}_{b|y})=\tr(\phi^{+}\,\Pi^{A}_{a|x}\otimes\Pi^{\eta}_{b|y})$.

To prove the result for planar measurement, we notice that the two-qubit
maximally entangled state violates a Bell inequality with projective planar
measurements if and only if $\eta>\frac{1}{K_{2}}$ [57]. ∎

We notice that, although stated differently, corollary 2 is equivalent to
Lemma 2 and Lemma 3 from Ref. [45], and that the relationship between the
Grothendieck constant with Bell and PM scenarios has been extensively
discussed in Refs. [56, 57, 58, 59, 60, 55, 45].

We notice that corollary 2 provides various examples of qubit measurements
which are not JM but are $\text{PM}_{2}$-JM. For example, any set of
$\eta$-noisy planar measurements with $\eta=1/\sqrt{2}$ is PM2-JM. However, if
a set of $\eta=1/\sqrt{2}$ noisy planar measurements includes two Pauli
measurements and a non-Pauli one, i.e. one measurement has a Bloch vector not
parallel to the other two, it is not JM. This provides several examples of
incompatible measurements which cannot be certified in a PM scenario. Also,
since $0.6875\leq\frac{1}{K_{3}}$, any set of $\eta$-noisy qubit measurements
with $\eta\leq 0.6875$ cannot be certified in PM scenario.

By using the connection between PM2-JM and Bell${}^{\phi^{+}}$-JM together
with a known characterization of joint measurability [48], we get the
following operator-inequality based connection between the three concepts.

######  Theorem 4.

Let $\\{B_{b|y}^{\eta}\\}_{y}$ be a pair of qubit white noisy projective
measurements, that is $y\in\\{1,2\\}$. The set $\\{B_{b|y}^{\eta}\\}_{y}$ is
JM iff it is Bell-JM, iff it is PM2-JM, and iff
$\norm{B_{0}^{\eta}+B_{1}^{\eta}}+\norm{B_{0}^{\eta}-B_{1}^{\eta}}\leq 2$,
where $B_{y}:=B_{0|y}-B_{1|y}$.

###### Proof.

The claim on JM follows from the known criterion of Busch [48] together with a
suitable argument using the CHSH operator, cf. Appendix B, and Theorem 3.∎

##  V Discussions and open questions

A crucial question in quantum communications is under which trust assumptions
we can certify quantum properties. A common and natural approach is to limit
the dimension of the Hilbert space, but various other alternative assumptions
have been studied [61]. This work presents such analysis for measurement
resources in a prepare-and-measure setting, concentrating on a dimension-
bounded generalisation of joint measurability. Here, we perform a case study
for qubit measurements. These examples showcase the impact of trust
assumptions on incompatibility witnesses (cf. Corollary 2).

The present work suggests several interesting directions for future research.
For example, a major open question is to establish a hierarchy between the set
of PMd and Bell-local measurements beyond the simple case of qubits and the
$\phi^{+}$ state. More precisely, given a set of arbitrary $d$-dimensional
measurements, does $\\{B_{b|y}\\}_{y}\in$ PMd imply $\\{B_{b|y}\\}_{y}\in$
Bell-Local? Or, does $\\{B_{b|y}\\}_{y}\in$ Bell-Local imply
$\\{B_{b|y}\\}_{y}\in$ PMd? The intimate connection between the Bell and
prepare-and-measure scenarios [62, 63] might naively suggest the existence of
such a hierarchy. Notably, an invertible map exists between correlations in a
Bell scenario and a subset of prepare-and-measure scenarios with preparations
satisfying appropriate equivalences [64], see also Ref. [15]. However, the
role of classical simulation dimensions in this relationship remains unclear.

In the context of qubits, the set of all possible POVMs is PM4-JM, implying
that qubit measurement incompatibility cannot be certified in a scenario where
Alice can transmit two classical bits to Bob. The extension of this concept to
qudits prompts intriguing questions. For instance, consider the set of all
qutrit measurements $\\{B_{b|y}\\}_{b,y}$. Is there a classical dimension
$d\in\mathbb{N}$ such that $\\{B_{b|y}\\}_{b,y}$ is PMd-JM?

For qubits, does Bell${}^{\phi^{+}_{2}}$-JM imply Bell-JM in general? This
question was already addressed in various previous references, e.g. [12, 13,
51, 52, 53], but it seems very hard to solve.

Another interesting direction is to generalise the definition of PMd-JM, by
asking for the existence of a simulation with dimension $d$ only _on average_
(cf. [65] Appendix C), i.e. can the correlations be simulated by
probabilistically choosing a simulation $\lambda$ of dimension $d_{\lambda}$
with probability $p_{\lambda}$ such that
$d=\sum_{\lambda}p_{\lambda}d_{\lambda}$? This promotes $d$, as a measure of
joint measurability, to a continuous quantity 111While it was proven in [46]
that bit simulation strategies for the set of all qubit correlations have
measure zero, for trit simulations this is still an open problem.. A further
interesting question is to investigate how PMd-JM relates to signaling
dimension of channels [66], classical simulation of temporal correlations
[67], and how it generalizes to GPTs [68, 69].

More broadly, one could study the impact of different trust assumptions on
incompatibility and other measurements resources. For example, can one
formulate an analogue of energy bounds for measurements and how does a
distrust assumption affect measurement simulability [70, 71, 72].

##  VI Acknowledgments

S.E. and R.U. are grateful for the financial support from the Swiss National
Science Foundation (Ambizione PZ00P2- 202179). J.P. acknowledges support from
NCCR-SwissMAP.

##  Appendix A Proof of Thm. 3

The idea of the proof is to show an explicit bijective map between qubit PM
strategies and Bell strategies with a pair of maximally entangled qubit pairs.
Before presenting this construction, we establish some notation. We define the
(full) correlator in a Bell scenario

$\displaystyle C_{xy}:=p(a=b|x,y)-p(a\neq b|x,y)\,.$ (9)

In a dichotomic Bell scenario with measurements $\\{A_{a|x}\\}_{a,x}$ and
$\\{B_{b|y}\\}_{b,y}$, we define the observables

$\displaystyle A_{x}:=A_{0|x}-A_{1|x}\,,$ (10) $\displaystyle
B_{y}:=B_{0|y}-B_{1|y}\,.$ (11)

With these definitions, if the parties share the state $\rho$, the correlator
is given by

$\displaystyle C^{\rho}_{xy}=\tr(\rho A_{x}\otimes B_{y})\,.$ (12)

Furthermore, for $\rho=\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}$ we have the
identity

$\displaystyle C^{\phi^{+}}_{xy}=\frac{1}{d}\tr(A_{x}^{T}B_{y})\,.$ (13)

Also, if the observable $A_{x}$ and $B_{y}$ respect $\tr(A_{x})=\tr(B_{y})=0$,
if we know the value of $P_{xy}:=\tr(\rho_{x}B_{y})$, we can always recover
the probabilities via the identity $p(a,b|x,y)=\frac{1+(-1)^{a\oplus
b}C^{\phi^{+}}_{xy}}{4}$, where $a\oplus b$ is the addition modulo two. A
proof of this identity follows from direct calculation.

Similarly, in the dichotomic PM-scenario we define the (single) correlator as

$\displaystyle P_{xy}:=p(b=0|x,y)-p(b=1|x,y)\,.$ (14)

If Alice sends the states $\rho_{x}$ and Bob performs measurements $B_{b|y}$
the correlator is given by

$\displaystyle P_{xy}:=\tr(\rho_{x}B_{y})\,.$ (15)

Notice that since $p(b=0|x,y)+p(b=1|x,y)=1$, if we know the value of
$P_{xy}:=\tr(\rho_{x}B_{y})$, we can always recover the probabilities via the
identity $p(b=0|x,y)=\frac{P_{xy}+1}{2}$.

We are now ready to state Lemma 1, which employs ideas from Lemma II.1, Lemma
II.2, and Lemma II.3 from Ref. [60].

######  Lemma 1.

Let $P_{xy}:=\tr(\rho_{x}B_{y})$ be qubit correlations in a dichotomic PM
scenario where Alice has $N_{A}$ inputs and Bob has $N_{B}$ inputs
with222Assuming $\tr(B_{y})=0$ is equivalent to saying that Bob’s POVMs are
given by
$B_{0|y}=\eta_{y}\outerproduct{\phi_{y}}{\phi_{y}}+(1-\eta_{y})\frac{\mathds{1}}{2}$
for some state $\ket{\phi_{y}}$ and $\eta_{y}\in[0,1]$. $\tr(B_{y})=0$. Let us
define the new correlator $P^{\prime}_{xy}$ in a scenario where Alice includes
the states given by $\mathds{1}-\rho_{x}$. That is, Alice and Bob have
$2N_{A}$ and $N_{B}$ inputs respectively, and

$\displaystyle P^{\prime}_{xy}:=$
$\displaystyle\tr(\rho_{x}B_{y})=P_{xy},\quad\quad$ $\displaystyle\forall
x\in\\{1,\ldots,N_{A}\\},\forall y\in\\{1,\ldots,N_{B}\\}$ (16) $\displaystyle
P^{\prime}_{xy}:=$
$\displaystyle\tr((\mathds{1}-\rho_{(x-N_{A})})B_{y})=-P_{(x-N_{A})y},$
$\displaystyle\forall x\in\\{N_{A}+1,\ldots,2N_{A}\\},\forall
y\in\\{1,\ldots,N_{B}\\}.$ (17)

We now define the POVMs

$\displaystyle A_{0|x}:=\rho_{x}^{T}\,,$ (18) $\displaystyle
A_{1|x}:=\mathds{1}-\rho_{x}^{T}\,.$ (19)

The probabilities
$p(a,b|x,y)=\tr(\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}A_{a|x}\otimes
B_{b|y})$ are Bell local if and only if the probabilities associated to
$P^{\prime}_{xy}:=\tr(\rho_{x}B_{y})$ are in PM2.

###### Proof.

We start the proof by defining the Bell correlator,

$\displaystyle C^{\prime}_{xy}:=$
$\displaystyle\tr(\outerproduct{\phi^{+}_{2}}{\phi^{+}_{2}}A_{x}\otimes
B_{y})\quad\quad$ $\displaystyle\forall x\in\\{1,\ldots,N_{A}\\},\forall
y\in\\{1,\ldots,N_{B}\\}$ (20) $\displaystyle C^{\prime}_{xy}:=$
$\displaystyle-\tr(\outerproduct{\phi^{+}_{2}}{\phi^{+}_{2}}A_{x}\otimes
B_{y})\quad\quad$ $\displaystyle\forall
x\in\\{N_{A}+1,\ldots,2N_{A}\\},\forall y\in\\{1,\ldots,N_{B}\\}\,,$ (21)

where $A_{x}=2\rho_{x}^{T}-\mathds{1}$ is the observable associated to the
measurements $A_{a|x}$ defined in Eq. (18).

Our next step is to show that $C^{\prime}_{xy}=P^{\prime}_{xy}$. This will
show that, if the probabilities of $P^{\prime}_{xy}$ in the PM scenario can be
obtained with qubits, we can construct a behaviour
$p(a,b|x,y)=\tr(\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}A_{a|x}\otimes
B_{b|y})$ with correlator $C^{\prime}_{xy}$. Also, given the behaviour
$p(a,b|x,y)=\tr(\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}A_{a|x}\otimes
B_{b|y})$, we can construct a qubit PM strategy respecting
$C^{\prime}_{xy}=P^{\prime}_{xy}$ for every $x$ and $y$. For this, it is
enough to notice that

$\displaystyle C^{\prime}_{xy}:=$
$\displaystyle\tr(\outerproduct{\phi^{+}_{2}}{\phi^{+}_{2}}A_{x}\otimes
B_{y})\quad\quad$ $\displaystyle\forall x\in\\{1,\ldots,N_{A}\\},\forall
y\in\\{1,\ldots,N_{B}\\}$ (22) $\displaystyle=$
$\displaystyle\frac{1}{2}\tr(A_{x}^{T}\otimes B_{y})$ (23) $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}2\tr(\rho_{x}B_{y})-\tr(B_{y})\Big{)}$ (24)
$\displaystyle=$ $\displaystyle\tr(\rho_{x}B_{y})$ (25) $\displaystyle=$
$\displaystyle P_{xy}$ (26) $\displaystyle=$ $\displaystyle
P_{xy}^{\prime}\,,$ (27)

where the last inequality holds because $x\in\\{1,\ldots,N_{A}\\},\forall
y\in\\{1,\ldots,N_{B}\\}$. Analogous calculation with
$x\in\\{N_{A}+1,\ldots,2N_{A}\\}$ shows that $C^{\prime}_{xy}=P^{\prime}_{xy}$
for all admissible values of $x$ and $y$.

We now prove that if $P^{\prime}_{xy}\in\text{PM}_{2}$, then $C^{\prime}_{xy}$
is Bell local. The intuition of the proof is that since
$P^{\prime}_{xy}\in\text{PM}_{2}$, the behaviour of $P^{\prime}_{xy}$ may be
obtained in a quantum scenario (potentially with shared randomness) where the
measurements performed by Bob are jointly measurable. Hence, the corresponding
quantum behaviour $C^{\prime}_{xy}=P^{\prime}_{xy}$ is necessarily Bell local.
Below, we describe this proof in more details.
If $P^{\prime}_{xy}\in\text{PM}_{2}$, then for any fixed $\lambda$, by
definition
$p(b|x,y,\lambda)=\sum_{a=0}^{1}p_{A}(a|x,\lambda)p_{B}(b|y,a,\lambda)$. Every
such strategy can be viewed as a quantum strategy where Alice sends a qubit
which is diagonal in the computational basis, that is,
$\rho_{x,\lambda}^{c}=\sum_{a}p_{A}(a|x,\lambda)\outerproduct{a}{a}$ and Bob
measures in the computational basis followed by classical post-processing,
i.e., $M_{b|y,\lambda}^{c}:=\sum_{a}p_{B}(b|y,a,\lambda)\outerproduct{a}{a}$.
Hence, $P^{\prime}_{xy,\lambda}\in\text{PM}_{2}$ can be written as

$\displaystyle
P^{\prime}_{xy,\lambda}=\tr(\rho_{x,\lambda}^{c}M_{y,\lambda}^{c})\,,$ (28)

where the operators $\rho_{x,\lambda}^{c}$ and $M_{b|y,\lambda}^{c}$ are
diagonal in the same basis. From the construction we presented in Eq. (18), we
see that for each $\lambda$ the correlator $C^{\prime}_{xy,\lambda}$ is
obtained in a scenario where Alice’s measurements are given by

$\displaystyle
A_{0|x,\lambda}^{c}:={\rho_{x,\lambda}^{c}}^{T}\leavevmode\nobreak\ ,$ (29)
$\displaystyle A_{0|x,\lambda}^{c}:=\openone-{\rho_{x,\lambda}^{c}}^{T}$ (30)

Since all POVM elements of Alice’s measurements are diagonal in the same
basis, her set of measurements is jointly measurable and cannot lead to Bell
nonlocal correlations, regardless of the shared state or the measurements
performed by Bob [3, 17, 12, 13], hence for each fixed $\lambda$, the
correlator $C^{\prime}_{xy,\lambda}$ is Bell local. Since the set of Bell
local correlations is convex, convex combination of local correlators are also
Bell local, that is, $\sum_{\lambda}\pi(\lambda)C^{\prime}_{xy,\lambda}$ is
necessarily Bell local.

We now prove that if the behaviour with probabilities
$p(a,b|x,y)=\tr(\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}A_{a|x}\otimes
B_{b|y})$ is Bell local, then $P^{\prime}_{xy}\in\text{PM}_{2}$. For that, we
will show that, if $P^{\prime}_{xy}\notin\text{PM}_{2}$, then the correlator
$C_{xy}^{\prime}$ violates a correlator Bell inequality.

If $P^{\prime}_{xy}\notin\text{PM}_{2}$, there exists a hyperplane that
separates it from the classical set, that is, there exists real numbers
$M_{xy}$ such that,

$\displaystyle\sum_{xy}M_{xy}P^{\prime}_{xy}=Q\,,$ (31)

but, for all $P^{\text{PM}_{2}}_{xy}\in\text{PM}_{2}$ ,

$\displaystyle\max_{P^{\text{PM}_{2}}_{xy}\in\text{PM}_{2}}\Big{[}\sum_{xy}M_{xy}P^{\text{PM}_{2}}_{xy}\Big{]}=L<Q\,.$
(32)

Since, $P^{\prime}_{xy}=C^{\prime}_{xy}$, it follows

$\displaystyle\sum_{xy}M_{xy}C^{\prime}_{xy}=Q\,.$ (33)

Finally, we have to show that all local Bell correlations
$C^{\text{LHV}}_{xy}$ respect

$\displaystyle\max_{C^{\text{LHV}}_{xy}\in\text{LHV}}\Big{[}\sum_{xy}M_{xy}C^{\text{LHV}}_{xy}\Big{]}=L\,,$
(34)

that is, we have to show that $\sum_{xy}M_{xy}C_{xy}\leq L$ is a Bell
inequality. This is ensured by Lemma II.1 of Ref. [60], which shows that if
$\max_{P^{\text{PM}_{2}}_{xy}\in\text{PM}_{2}}\Big{[}\sum_{xy}M^{\prime}_{xy}P^{\text{PM}_{2}}_{xy}\Big{]}=L$,
and the coefficients’ $M^{\prime}_{xy}$ can be written
$M^{\prime}_{xy}=M_{xy}$ for $x\in\\{1,\ldots N_{A}\\}$ and
$M^{\prime}_{xy}=-M_{xy}$ for $x\in\\{N+A+1,\ldots 2N_{A}\\}$, it holds that
$\max_{C^{\text{LHV}}_{xy}\in\text{LHV}}\Big{[}\sum_{xy}M_{xy}C^{\text{LHV}}_{xy}\Big{]}=L$.
And, since our correlator $P^{\prime}_{xy}$ respects $P^{\prime}_{xy}=P_{xy}$
for $x\in\\{1,\ldots N_{A}\\}$ and $P^{\prime}_{xy}=-P_{xy}$ for
$x\in\\{N+A+1,\ldots 2N_{A}\\}$, we can always find coefficients
$M^{\prime}_{xy}$ meeting the hypothesis of Lemma II.1 of Ref. [60].

∎

Using Lemma 1, the proof of Thm. 3 follows immediately. Let $B_{b|y}$ be a set
of qubit measurements which can be written as
$B_{0|y}=\eta_{y}\outerproduct{\phi_{y}}{\phi_{y}}+(1-\eta_{y})\frac{\mathds{1}}{2}$,
for some qubit pure state $\ket{\phi_{y}}$ and $\eta_{y}\in[0,1]$.

If there exists qubit states $\rho_{x}$ such that
$p(b|x,y)=\tr(\rho_{x}B_{b|y})\notin PM_{2}$, then there exist measurements
$A_{a|x}$ such that $p(a,b|x,y)=\Tr(A_{a|x}\otimes B_{b|y}\phi^{+}_{2})$ is
not Bell local.

##  Appendix B Proof of Theorem 4

######  Definition 6.

Let $A_{0},A_{1},B_{0},B_{1}\in L(\mathbb{C}_{d})$ be self-adjoint operators
with eigenvalues contained in $[-1,1]$. The CHSH operator is defined as

$\displaystyle\text{CHSH}:=A_{0}\otimes B_{0}+A_{1}\otimes B_{0}+A_{0}\otimes
B_{1}-A_{1}\otimes B_{1}\,.$ (35)

######  Lemma 2.

The CHSH operator satisfies the norm inequality
$\norm{CHSH}\leq\norm{B_{0}+B_{1}}+\norm{B_{0}-B_{1}}$.

###### Proof.

$\displaystyle\norm{CHSH}=$
$\displaystyle\norm{A_{0}\otimes(B_{0}+B_{1})+A_{1}\otimes(B_{0}-B_{1})}$ (36)
$\displaystyle\leq$
$\displaystyle\norm{A_{0}\otimes(B_{0}+B_{1})}+\norm{A_{1}\otimes(B_{0}-B_{1})}$
(37) $\displaystyle=$
$\displaystyle\norm{A_{0}}\norm{B_{0}+B_{1}}+\norm{A_{1}}\norm{B_{0}-B_{1}}$
(38) $\displaystyle\leq$
$\displaystyle\norm{B_{0}+B_{1}}+\norm{B_{0}-B_{1}}\,.$ (39)

∎

This shows that if Bob performs measurements which satisfy
$\norm{B_{0}+B_{1}}+\norm{B_{0}-B_{1}}\leq 2$, Alice and Bob cannot violate
the CHSH inequality, regardless of the shared state, i.e. the measurements are
Bell-JM. The following Theorem shows that the above upper bound can be reached
with $\ket{\phi^{+}}$.

######  Theorem 5.

Let $B_{0},B_{1}\in L(\mathbb{C}_{2})$ be qubit self-adjoint operators with
eigenvalues in $[-1,1]$ and $\tr(B_{y})=0$. There exist self-adjoint operators
$A_{0},A_{1}\in L(\mathbb{C}_{2})$ with eigenvalues contained in $\\{-1,1\\}$
and a quantum state $\ket{\psi}\in\mathbb{C}_{2}\otimes\mathbb{C}_{2}$ such
that

$\displaystyle\bra{\psi}CHSH\ket{\psi}=\norm{B_{0}+B_{1}}+\norm{B_{0}-B_{1}}\,.$
(40)

Moreover, the state $\ket{\psi}\in\mathbb{C}_{2}\otimes\mathbb{C}_{2}$ may be
taken as the maximally entangled state
$\ket{\phi^{+}}:=\frac{1}{\sqrt{2}}\sum_{i=1}^{2}\ket{ii}$.

###### Proof.

First, notice that, for any operators $A_{0},A_{1},B_{0},B_{1}\in
L(\mathbb{C}_{2})$, we have that

$\displaystyle\bra{\phi^{+}}CHSH\ket{\phi^{+}}=\frac{\tr\Big(A_{0}^{T}\otimes(B_{0}+B_{1})\Big{missing})+\tr\Big(A_{1}^{T}\otimes(B_{0}-B_{1})\Big{missing})}{2}\,.$
(41)

By definition of operator norm, there exists normalised vectors
$\ket{\psi_{0}^{\prime}},\ket{\psi_{1}^{\prime}}\in L(\mathbb{C}_{2})$ such
that

$\displaystyle\absolutevalue{\bra{\psi_{0}^{\prime}}B_{0}+B_{1}\ket{\psi_{0}^{\prime}}}=$
$\displaystyle\norm{B_{0}+B_{1}}\,,$ (42)
$\displaystyle\absolutevalue{\bra{\psi_{1}^{\prime}}B_{0}-B_{1}\ket{\psi_{1}^{\prime}}}=$
$\displaystyle\norm{B_{0}-B_{1}}\,.$ (43)

Since $\tr(B_{y})=0$, there exists $\ket{\psi_{0}},\ket{\psi_{1}}\in
L(\mathbb{C}_{2})$ such that

$\displaystyle\bra{\psi_{0}}B_{0}+B_{1}\ket{\psi_{0}}=$
$\displaystyle\norm{B_{0}+B_{1}}\,,$ (44)
$\displaystyle\bra{\psi_{1}}B_{0}-B_{1}\ket{\psi_{1}}=$
$\displaystyle\norm{B_{0}-B_{1}}\,.$ (45)

Indeed, if $\bra{\psi_{0}^{\prime}}B_{0}+B_{1}\ket{\psi_{0}^{\prime}}\geq 0$,
take $\ket{\psi_{0}}:=\ket{\psi_{0}^{\prime}}$. If
$\bra{\psi_{0}^{\prime}}B_{0}+B_{1}\ket{\psi_{0}^{\prime}}<0$, take
$\ket{\psi_{0}}$ as a vector which satisfies
$\outerproduct{\psi_{0}}{\psi_{0}}=\mathds{1}-\outerproduct{\psi_{0}^{\prime}}{\psi_{0}^{\prime}}$,
that is, $\ket{\psi_{0}}$ is the universal NOT of the state
$\ket{\psi_{0}^{\prime}}$. It follows that

$\displaystyle\bra{\psi_{0}}B_{0}+B_{1}\ket{\psi_{0}}$
$\displaystyle=-\tr(\outerproduct{\psi_{0}^{\prime}}{\psi_{0}^{\prime}}(B_{0}+B_{1}))$
$\displaystyle=-\bra{\psi_{0}^{\prime}}B_{0}+B_{1}\ket{\psi_{0}^{\prime}}$
(46)

is a positive number. An analogous argument works for $\ket{\psi_{1}}$.

We now define $A_{x}=:2\outerproduct{\psi_{x}}{\psi_{x}}^{T}-\mathds{1}$. We
have

$\displaystyle\bra{\phi^{+}}CHSH\ket{\phi^{+}}$
$\displaystyle=\frac{\tr\Big(A_{0}^{T}\otimes(B_{0}+B_{1})\Big{missing})+\tr\Big(A_{1}^{T}\otimes(B_{0}-B_{1})\Big{missing})}{d}$
(47)
$\displaystyle=\frac{2}{d}\left(\bra{\psi_{0}}B_{0}+B_{1}\ket{\psi_{0}}+\bra{\psi_{1}}B_{0}-B_{1}\ket{\psi_{1}}\right)-\Big{(}\tr(B_{0})+\tr(B_{1})+\tr(B_{0})-\tr(B_{1})\Big{)}$
(48)
$\displaystyle=\frac{2}{d}\left(\bra{\psi_{0}}B_{0}+B_{1}\ket{\psi_{0}}+\bra{\psi_{1}}B_{0}-B_{1}\ket{\psi_{1}}\right)-2\tr(B_{0})\,.$
(49)

Since $\tr(B_{0})=0$, we have

$\displaystyle\bra{\phi^{+}}CHSH\ket{\phi^{+}}$
$\displaystyle=\frac{2}{2}\left(\bra{\psi_{0}}B_{0}+B_{1}\ket{\psi_{0}}+\bra{\psi_{1}}B_{0}-B_{1}\ket{\psi_{1}}\right)\,,$
(50) $\displaystyle=\norm{B_{0}+B_{1}}+\norm{B_{0}-B_{1}}\,.$ (51)

∎

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