††thanks: H. A. and N. W. contributed equally to this work.††thanks: H. A. and
N. W. contributed equally to this work.

# Noise-robust preparation contextuality shared between any number of
observers via unsharp measurements

Hammad Anwer  Department of Physics, Stockholm University, S-10691 Stockholm,
Sweden  Natalie Wilson  Department of Physics, Stockholm University, S-10691
Stockholm, Sweden  Ralph Silva  Institute for Theoretical Physics, ETH Zurich,
Switzerland  Sadiq Muhammad  Department of Physics, Stockholm University,
S-10691 Stockholm, Sweden  Armin Tavakoli  Département de Physique Appliquée,
Université de Genève, CH-1211 Genève, Switzerland  Mohamed Bourennane
Department of Physics, Stockholm University, S-10691 Stockholm, Sweden

###### Abstract

Multiple observers who independently harvest nonclassical correlations from a
single physical system share the system’s ability to enable quantum
correlations. We show that any number of independent observers can share the
preparation contextual outcome statistics enabled by state ensembles in
quantum theory. Furthermore, we show that even in the presence of any amount
of white noise, there exists quantum ensembles that enable such shared
preparation contextuality. The findings are experimentally realised by
applying sequential unsharp measurements to an optical qubit ensemble which
reveals three shared demonstrations of preparation contextuality.

Introduction.— Quantum correlations can surpass the limitations of
corresponding classical models. In their most common form, quantum
correlations are obtained from the outcomes of single (albeit randomly chosen)
measurements performed on a physical system. After the measurement, the
physical system can be discarded, or even demolished by the measurement
apparatus. Therefore, since one does not need to consider the measurement-
induced decoherence in the state of the physical system, optimal quantum
correlations are often obtained from sharp (projective) measurements that
extract a maximal amount of information from the physical system while also
inducing a maximal disturbance in its state Fuchs .

Arguably, the fact that measurements disturb physical states should have
interesting consequences for more general quantum correlations. To reveal the
influence of measurement-induced disturbances on observed outcome statistics,
one requires systems to undergo more than a single measurement. A simple
scenario for studying the trade-off between the strength of quantum
correlations and the disturbance induced by extracting them is one in which
quantum correlations are shared between many observers. Sharing quantum
correlations means that a physical system is measured by a sequence of
independent observers, each of whom are tasked with falsifying the existence
of a classical model for their observed correlations. Hence, the stronger the
correlations extracted by the first observer, the larger the disturbance
induced in the state of the system, and thus the weaker the correlations that
can possibly be extracted by a second observer. Sharing quantum correlations
requires the first observer to measure in such a way that the outcome
correlations are strong enough to elude all classical models while the induced
disturbance is small enough to enable a second observer independently repeat
the same feat. Understanding and characterising quantum correlations obtained
via sequential measurements is a conceptually interesting problem SG16 ; BM13
; GW14 ; TC18  which has promising applications in quantum information
protocols CJ17 ; CH18 .

Sharing quantum correlations was first studied in the context of Bell
inequality tests SG16  where it was found that a pair of qubits in a singlet
state can enable two sequential Bell inequality violations. This has also been
experimentally demonstrated Exp1 ; Exp2 . In addition, the number of
sequential Bell inequality violations can be indefinitely extended at the
price of all observers strongly biasing their choice of measurement and
therefore rendering the quantum correlations super-exponentially fragile to
noise SG16 . Moreover, the shared quantum correlations have recently also been
studied in other entanglement-based tasks such as entanglement witnessing BM18
and quantum steering SD18 ; SDH18 .

Here, we theoretically and experimentally study the sharing of quantum
correlations that demonstrate preparation contextuality. These are
correlations that cannot be reproduced in a hidden variable theory that
ascribes equivalent representations to indistinguishable preparations, i.e. it
disregards the context (specific procedure) underlying a state preparation
Sp05 . Such quantum contextuality does not require entanglement but only
single quantum systems, and is well-studied both in theory (see e.g. Refs.Sp05
; Negativity ; SpekkensPOM ; HT17 ; LM13 ; BB15 ; SC18 ) and experiment (see
e.g. Refs. SpekkensPOM ; HT17 ; MP16 ). In our scenario, states are sampled
from an ensemble and communicated sequentially between independent observers,
each of whom performs a measurement with the aim of obtaining preparation
contextual outcome statistics. We show that preparation contextuality can be
shared between any number of sequential observers. Furthermore, we show that
the sharing is robust to noise, in the sense that for any given number of
independent observers and exposure to any nontrivial amount of white noise,
one can find an ensemble whose contextuality can be shared between all the
observers. We proceed to experimentally demonstrate the sharing of preparation
contextuality. We realise a four-observer scenario in which the first observer
prepares an optical qubit ensemble and the remaining three observers perform
sequential unsharp (non-maximally disturbing) measurements. Thus, we obtain
three shared demonstrations of preparation contextuality.

Nonclassicality via preparation contextuality.— The impossibility of
describing the set of observables in quantum theory by underlying classical
(noncontextual) quantities originates in the arguments of Bell, Kochen and
Specker KochenSpecker . More recently, the notion of contextuality has seen a
generalisation formulated in operational terms (i.e., in terms of
probabilities) applying to measurements, transformations and preparations Sp05
. Here, we are interested in contextuality in terms of preparations.

The predictions of an operational theory (e.g. quantum theory) may be
explained by an ontological model HS10  that ascribes each physical system S
to a set $\Lambda$ of ontic (objective) states $\lambda$. A particular
preparation $P$ of the system is associated to a distribution
$\mu_{P}(\lambda)$ over the ontic state space. Similarly, the probability of
outcome $b$ of a measurement $M$ is described by a response function
$\xi_{b,M}(\lambda)$. The ontological model thus seeks a $\mu$ and a $\xi$ to
explain the observed statistics by
$p(b|P,M)=\int_{\Lambda}\mu_{P}(\lambda)\xi_{b,M}(\lambda)d\lambda$. The model
is said to be preparation noncontextual if two different preparations $P$ and
$P^{\prime}$ that cannot be distinguished by the statistics generated by any
measurement (that is; $\forall M:p(b|P,M)=p(b|P^{\prime},M)$) are associated
to the same distribution over ontic states, i.e., $\mu_{P}=\mu_{P^{\prime}}$.
If observed statistics falsify this assumption, then it is said to be
preparation contextual. Quantum state ensembles are known to enable
preparation contextuality.

A family of preparation noncontextuality inequalities.— In order to prove
preparation contextuality, it is sufficient to violate an inequality bounding
the correlations attainable in a preparation noncontextual model. We focus on
a family of such inequalities introduced in Ref. SpekkensPOM  related to a
variant of Random Access Coding Ambainis ; RAC . Consider a party Alice
receiving a random input string $x=x_{1}\ldots x_{n}\in\\{0,1\\}^{n}$. Her
input is associated to a preparation $P_{x}$ (one of $2^{n}$ possible) which
is sent to a receiver Bob. Her preparations are constrained to satisfy certain
indistinguishability relations: there must exist no measurement that can
reveal any information about the parity of the string $r\cdot x$ for every
$r\in\\{0,1\\}^{n}$ with $|r|\geq 2$. Bob receives a random input
$y\in\\{1,\ldots,n\\}$, and performs a measurement $\\{M_{y}^{b}\\}$ with
outcome $b\in\\{0,1\\}$. The partnership is awarded a point if the outcome of
Bob coincides with the $y$th entry in Alice’s string. In any preparation
noncontextual theory, the probability of winning obeys the following bound
SpekkensPOM :

$\mathcal{A}^{(n)}\equiv\frac{1}{n2^{n}}\sum_{x,y}p(b=x_{y}\lvert
x,y)\leq\frac{n+1}{2n}.$ (1)

Due to the contextual nature of quantum theory, these inequalities can be
violated. Maximal quantum violations for any $n\geq 2$ are known IndexGame .
Bob performs dichotomic measurements characterised by an observable
$G_{n,y}^{T}$. These are recursively defined from $G_{2,1}=\sigma_{x}$,
$G_{2,2}=\sigma_{y}$, and $G_{3,1}=\sigma_{x}$, $G_{3,2}=\sigma_{y}$ and
$G_{3,3}=\sigma_{z}$, and

$n$ even: $\displaystyle G_{n,k}=G_{n-1,k}\otimes\sigma_{x}\hskip
14.22636pt\forall i\in\\{1,\ldots,n-1\\},$ $n$ odd: $\displaystyle
G_{n,k}=G_{n-2,k}\otimes\sigma_{x}\hskip 14.22636pt\forall
i\in\\{1,\ldots,n-2\\}$ (2)

with $G_{n,n}=\openone\otimes\sigma_{y}$ if $n$ is even, and
$G_{n,n}=\openone\otimes\sigma_{z}$ and $G_{n,n-1}=\openone\otimes\sigma_{y}$
of $n$ is odd. The optimal preparations are states of $\lfloor n/2\rfloor$
qubits specified by

$\rho_{x}=\operatorname{tr}_{\text{A}}\left[\left(\openone+A_{x}\right)\otimes\openone\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right],$ (3)

where $A_{x}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(-1)^{x_{i}}G_{n,i}$,
$\phi_{\text{max}}$ corresponds to the maximally entangled state
$\left(|0,0\rangle+|1,1\rangle\right)/\sqrt{2}$, and the trace is taken over
the first system in every entangled pair. Note that Alice’s preparations are
single quantum systems, and only for simplicity written in terms of post-
measurement states of a collection of entangled states. The presented strategy
leads to the maximal quantum value $\mathcal{A}^{(n)}=1/2(1+1/\sqrt{n})$ for
every $n$ IndexGame .

Sequential scenario.— We consider a scenario in which the ability to violate
the inequality (1) is shared between many independent observers, named Bob1,…,
Bobm, each of whom receive an independent random input
$y_{k}\in\\{1,\ldots,n\\}$ and output $b_{k}\in\\{0,1\\}$. Alice’s randomly
chosen preparation is sent to Bob1 who performs a measurement and passes the
post-measurement state to Bob2 who performs a measurement and passes the post-
measurement state to Bob3 etc. The scenario is illustrated in Fig. 1. The pair
Alice-Bobk uses the marginal distribution $p(b_{k}|x,y_{k})$ to compute the
witness (1) (here labelled $\mathcal{A}_{k}^{(n)}$) to check for preparation
contextuality.

Figure 1: Alice’s preparations are sent from one observer to the next, each
performing a measurement aiming to independently reveal preparation contextual
statistics. To this end, only the average post-measurement state
$\tilde{\rho}_{x}^{(k)}$ is relevant.

In a quantum approach, we may denote Alice’s preparations by $\rho_{x}$ which
must satisfy the indistinguishability relation $\sum_{r\cdot
x=0}\rho_{x}=\sum_{r\cdot x=1}\rho_{x}$ for every string $r$ with $|r|\geq 2$.
Since one has to keep track of both the statistics and the post-measurement
states of each Bob, we require the detailed set of Kraus operators for each
measurement. By $K_{y_{k}}^{b_{k}}$ we denote the Kraus operators of Bobk
associated to the $y_{k}$th measurement and $b_{k}$th outcome. The state
received by Bobk is specified by Alice’s input $x$, and the strings of inputs
$(y_{1},\ldots,y_{k-1})$ and outputs $(b_{1},\ldots,b_{k-1})$ of all previous
Bobs. However, we treat each Bob in the sequence as independent from the rest,
meaning that they do not know the specific inputs or outputs of the other Bobs
in each run of the experiment. Thus, in order to calculate the relevant
marginal distributions $p(b_{k}|x,y_{k})$, only the average state
$\tilde{\rho}_{x}^{(k)}$ received by Bobk is required, i.e., the state
obtained from averaging a preparation $\rho_{x}$ of Alice over all the inputs
and outputs of all previous Bobs:

$\tilde{\rho}_{x}^{(k)}=\frac{1}{n}\sum_{y_{k-1},b_{k-1}}K_{y_{k-1}}^{b_{k-1}}\tilde{\rho}_{x}^{(k-1)}(K_{y_{k-1}}^{b_{k-1}})^{\dagger},$
(4)

with $\tilde{\rho}_{x}^{(1)}=\rho_{x}$. Consequently, the desired marginal
statistics for Bobk are
$p(b_{k}|x,y_{k})=\operatorname{tr}\left(\tilde{\rho}_{x}^{(k)}(K_{y_{k}}^{b_{k}})^{\dagger}K_{y_{k}}^{b_{k}}\right)$.
This constitutes a description of general quantum strategies in the sequential
scenario.

Sharing preparation contextuality.— We apply the above general description to
construct a specific family of quantum strategies for sharing preparation
contextuality, that is inspired by the previously described optimal quantum
strategy for the maximal violation of the inequalities (1). Alice prepares the
states (3) while each Bob performs an unsharp variant of the measurements
optimal for violating (1). In that strategy the measurements of Bob are the
dichotomic observables $G_{n,y_{k}}^{T}$ defined in (Noise-robust preparation
contextuality shared between any number of observers via unsharp
measurements), corresponding to the projectors
$\Pi_{n,y}^{b}=(\openone+(-1)^{b}G_{n,y}^{T})/2$ that are both the Kraus
operators and POVM elements. For a weaker measurement, one modifies the POVM
element to $(\openone+(-1)^{b}\;\eta_{k}\;G_{n,y}^{T})/2$, for some
$\eta_{k}\in[0,1]$. If $\eta_{k}=1$ ($\eta_{k}=0$), the measurement is sharp
(non-interacting). Choosing $0<\eta_{k}<1$ corresponds to an unsharp
measurement. The corresponding Kraus operator is given by

$K_{y_{k}}^{b_{k}}=\sqrt{\frac{1+\eta_{k}}{2}}\Pi_{n,y_{k}}^{b_{k}}+\sqrt{\frac{1-\eta_{k}}{2}}\Pi_{n,y_{k}}^{\bar{b}_{k}},$
(5)

where the bar-sign denotes a bit-flip. This class of strategies has the
following convenient property.

######  Lemma 1.

If Alice prepares the states in Eq. (3) and the Bobs each measure
$G^{T}_{n,y_{k}}$ with sharpness $\eta_{k}$, the average state received by
Bobk is

$\tilde{\rho}_{x}^{(k)}=v_{k}\rho_{x}+\left(1-v_{k}\right)\rho_{\text{mix}},$
(6)

where $\rho_{\text{mix}}$ is the maximally mixed state and the visibility
$v_{k}\in[0,1]$ is given recursively by

$v_{k}=v_{k-1}f_{k-1}=\prod_{j=1}^{k-1}f_{j},$ (7)

where $v_{1}=1$ by definition, and the “quality factor" $f_{k}$ of the
measurement of Bobk is defined from the sharpness $\eta_{k}$ as
$f_{k}=(1+(n-1)\sqrt{1-\eta_{k}^{2}})/n$.

###### Proof.

The proof is technical in character and is given in Appendix (section A).

Using Eq. (6), the figure of merit (1) for the pair Alice and Bobk reads

$\mathcal{A}_{k}^{(n)}=\frac{1}{2}\left(1+\frac{v_{k}\eta_{k}}{\sqrt{n}}\right).$
(8)

This leads to preparation contextuality whenever $\eta_{k}>1/(v_{k}\sqrt{n})$.
This can be used to recursively calculate the critical pairs
$(\eta_{k},v_{k})$. Thusly, we arrive at the following result.

######  Result 1.

The number of observers who can independently share the preparation
contextuality enabled by Alice’s ensemble is at least $n$.

###### Proof.

Consider that each Bob tunes the sharpness of his measurement so as to just
violate the inequality (1), but not more. Expressing the measurement sharpness
$\eta_{k}=\sin\theta_{k}$, where $\theta_{k}\in[0,\pi/2]$, we thus require
$\sin\theta_{k}=1/(v_{k}\sqrt{n})$. On the other hand, a trivial lower bound
on the quality factor of Bobk’s measurement is
$f_{k}=\left(1+(n-1)\cos\theta_{k}\right)/n\geq\cos\theta_{k}$. Squaring, and
using the expression for the critical value of $\sin\theta_{k}$ above, we find
that $f_{k}^{2}\geq 1-1/(v_{k}^{2}n)$. Since the visibility of the next Bob is
$v_{k+1}=v_{k}f_{k}$, we have $v_{k+1}^{2}=v_{k}^{2}f_{k}^{2}\geq
v_{k}^{2}\left(1-1/(v_{k}^{2}n)\right)$. Hence, the decrease in visibility
from each Bob to the next is bounded by $v_{k}^{2}-v_{k+1}^{2}\leq 1/n$ which
together with $v_{1}=1$ gives $v_{k+1}^{2}\geq 1-k/n$. This implies that the
visibility of the $n$th Bob is at least $v_{n}\geq 1/\sqrt{n}$, which is
precisely the condition for violating the preparation noncontextuality
inequality.

Thus by suitably choosing $n$, an arbitrary long sequence of observers can
share the preparation contextual correlations enabled by Alice’s ensemble.
Moreover, we show in Appendix (section B) that for the considered class of
quantum strategies, the number of observers who share preparation
contextuality can be no more than $n$. Also, as shown in Appendix (section C),
one can share preparation contextuality between any number of observers also
in a scenario in which none of the Bob’s knows his position in the sequence.

Noise-robustness.— The scenario we have considered so far is an idealisation
in which no noise appears. In addition to this not being realistic in any
experiment, it is interesting to consider whether the noiseless scenario is
distinctive, or also significantly noisy ensembles footnote  enable shared
preparation contextuality. To address this matter, we let Alice’s preparations
be mixtures of the intended state $\rho_{x}$ with the maximally mixed state:
$\rho_{x}(q)=q\rho_{x}+(1-q)\rho_{\text{mix}}$ for some visibility
$q\in[0,1]$. For a given number of observers, what is the smallest $q$ such
that preparation contextuality can be shared between all observers?

######  Result 2.

For any given number of independent observers $m$, there exists an ensemble
whose contextuality can be shared between all observers for any $q>0$.

###### Proof.

We substitute $\rho_{x}$ for $\rho_{x}(q)$ in the proof of Result 1. This
means $v_{1}=q$, and leads to $v_{k+1}^{2}\geq q-k/n$. Thus, in order to
observe $m$ violations, one must choose $n\geq\lceil\frac{m}{q}\rceil$.

Hence, preparation contextuality can be shared between any number of observers
using ensembles with an arbitrarily large noise-component by choosing a
sufficiently large $n$. The price to pay for this property is that when
$q\rightarrow 0$, both the Hilbert space dimension of Alice’s ensemble and the
number of preparations and measurements diverge.

Experiment.— We demonstrate the theoretical findings in an experiment with
three ($n=3$) sequential tests of preparation contextuality. Alice prepares
the eight qubit states (3) with Bloch vectors
$\vec{a}_{x}=\left[(-1)^{x_{1}},(-1)^{x_{2}},(-1)^{x_{3}}\right]/\sqrt{3}$.
Bob1 and Bob2 perform unsharp measurements (5) of $\sigma_{x}$, $\sigma_{y}$
and $\sigma_{z}$ whereas Bob3 performs projective (sharp) measurements of the
same observables.

Figure 2: Optical set-up used to reveal contextuality sharing. See text for
details. Q and H represent quarter-wave plates (QWPs) and half-wave plates
(HWPs).

In the experiment we peform unsharp measurements on the polarisation state of
a single photon using shifted Sagnac interferometers, as shown in Bob1 and
Bob2 in Fig. (2). A HWP is placed in each path of the interferometer, rotated
to $\theta_{i}/2$ in the horizontal path and $\pi/4-\theta_{i}/2$ in the
vertical path to control the sharpness of the measurement. A HWP and QWP
before and after the interferometer are used to select the basis of the
measurement. The measurement outcome is encoded in the output path, i.e.
outcome $b_{i}=0$ ($b_{i}=1$) corresponds to the detection of the photon in
output path 1 (2, beam blocked in figure). In the sequential scenario we
choose to consider only one path at a time for Bob1 and Bob2 to simplify the
set-up. By adding an additional rotation to the HWPs or QWPs before and after
Bob, we can select the output we want to analyse Exp1 ; Exp2 . The results of
Bob1 and Bob2’s unsharp measurements are therefore obtained at Bob3, comprised
of a PBS and single photon detectors D1 and D2. For example, if we consider
output 1 at Bob1 and Bob2, a click in either detector at Bob3 tells us that
Bob1 and Bob2 had the outcome $b_{1}=0$ and $b_{2}=0$. We analyse the counts
in Bob3 corresponding to all possible combinations of output ports to realise
a full measurement. This protocol relies on a stable photon generation rate.
Details of measurement angles are given in Appendix (section D). This set-up
can be used to perform projective measurements ($\eta=1$, $\theta_{i}=0$), no
measurement ($\eta=0$, $\theta_{i}=\pi/4$), or an intermediate-strength
measurement, where the the sharpness (strength) of the measurement is tuned by
varying $\theta_{i}$.

The full set-up is shown in Fig. 2. We generate heralded single photons at 780
nm via spontaneous parametric down-conversion (SPDC) using a single type-I
beta barium borate (BBO) crystal of thickness 2 mm pumped by 390 nm femto-
second laser pulses. The idler photon is detected by an APD single-photon
detector, Dtrigger, and is used as a trigger. The single photons are coupled
into single-mode fibres (SMF) after passing through a narrowband 3 nm
interference filter (F) to define the spatial and spectral properties of the
photons. After filtering, the signal photon is prepared into one of Alice’s
eight states, using a polariser, two QWPs and a HWP (angles given in Appendix
(section D). The unsharp measurements of Bob1 and Bob2 correspond to
$\theta_{1}=24.95\degree$ ($\eta_{1}=0.6441$) and $\theta_{2}=20.10\degree$
($\eta_{2}=0.7637$) respectively, which ideally produce
$\mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{A}_{3}=0.6859>2/3$ with
$\mathcal{A}_{k}=\mathcal{A}_{k}^{(3)}$.

Results.— In order to test each of the three preparation noncontextuality
inequalities (between Alice and each of the three Bobs), we require 24
marginal probabilities (the ‘winning’ answers $b_{k}=x_{y_{k}}$) corresponding
to the three measurement bases and Alice’s eight preparations. To reduce the
Poissonian error, each Bob collects approximately 34 million counts for each
of these 24 settings. Our experimental values can be found in Appendix
(section E). These lead to three preliminary values of
$\mathcal{A}_{1}^{\text{pre}}=0.687\pm 0.001$,
$\mathcal{A}_{2}^{\text{pre}}=0.675\pm 0.001$, and
$\mathcal{A}_{3}^{\text{pre}}=0.681\pm 0.001$.

Data analysis.— Due to small yet unavoidable experimental imperfections, e.g.
waveplate imperfections and offsets in the rotation of the waveplates, it is
impossible to perfectly satisfy the operational indistinguishability relations
required to test preparation contextuality. This problem can be overcome by
suitable post-processing methods MP16 . As described in Appendix (section F),
we have used a relaxed variant of these methods to enforce the
indistinguishability relations relevant to a test of inequality (1) on our
experimental data. This comes at the cost of the observed values
$\left(\mathcal{A}_{1}^{\text{pre}},\mathcal{A}_{2}^{\text{pre}},\mathcal{A}_{3}^{\text{pre}}\right)$
decreasing in a manner corresponding to how well the statistics approximates
said relations. Due to the high visibility and precision of the experimental
set-up, we find only a small decrease in the three correlation witnesses:

$\displaystyle\mathcal{A}_{1}^{\text{post}}=0.683\pm 0.001$
$\displaystyle\mathcal{A}_{2}^{\text{post}}=0.670\pm 0.001$
$\displaystyle\mathcal{A}_{3}^{\text{post}}=0.677\pm 0.001$

all of which violate inequality (1).

Conclusions.— We have theoretically developed and experimentally demonstrated
the sharing of preparation contextual correlations in scenarios that require
no entanglement. In addition to such correlations being possible to share
between any number of observers, we found that this can be done in a strongly
noise-robust manner. This distinguishes shared preparation contextuality from
known results in e.g. shared Bell nonlocality in which the fragility to noise
of sequential demonstrations scales super-exponentially SG16 . This fragility
poses a significant experimental hurdle and has hitherto limited
demonstrations to two sequential violations of Bell inequalities Exp1 ; Exp2 .
We experimentally observed three sequential demonstrations of preparation
contextuality. Optical set-ups of this spirit (see also Refs Exp1 ; Exp2 ) are
promising candidates for a variety of sequential correlation tests. Finally,
an interesting question is to understand which forms of quantum correlations
can be shared between indefinitely many observers in a noise-robust manner.

Acknowledgements.— This work was supported by the project “Photonic Quantum
Information” (Knut and Alice Wallenberg Foundation, Sweden), the Swedish
Research Council, and the Swiss National Science Foundation (Starting grant
DIAQ, grant No. 200020_165843, and the National Centre of Competence in
Research “Quantum Systems and Technology”).

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  * (24) A. Tavakoli, A. Hameedi, B. Marques, and M. Bourennane, Quantum random access codes using single d-Level systems, Phys. Rev. Lett. 114, 170502 (2015).
  * (25) A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, Optimal bounds for parity-oblivious random access codes, New J. Phys. 18, 045003 (2016).
  * (26) One could alternatively consider the Bobs’ measurement devices inducing the noise. However, this is less detrimental than noisy preparations. The reason is that if Alice’s preparations are noisy the correlations due to all Bobs’ measurements are weaker, whereas if instead one (or many) of the Bobs sometimes fail to perform the intended measurement, the state relayed to the next Bob retains a higher degree of coherence and leads to him observering stronger correlations.

##  Appendix A Proof of Lemma

In this section, we prove the lemma of the main text. In the considered
scenario, Alice receives a random input $x\in\\{0,1\\}^{n}$ and prepares the
associated state

$\displaystyle\rho_{x}=\operatorname{tr}_{\text{A}}\left[\left(\openone+A_{x}\right)\otimes\openone\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right],$ (9)

where $\phi_{\text{max}}^{\otimes\lfloor n/2\rfloor}$ is ${\lfloor
n/2\rfloor}$ copies of the two-qubit maximally entangled state, and the
partial trace is taken over all the first qubits in each pair. Consider that
the sequence of Bobs, labelled by $\\{1,2,...,m-1\\}$, apply measurements of
intermediate sharpness to the state above, each denoted by
$\eta_{k}=\sin\theta_{k}$. We proceed to prove that the average state
$\tilde{\rho}_{x}^{(m)}$ received by Bobm will be of the form

$\displaystyle\tilde{\rho}^{(m)}_{x}$
$\displaystyle=\operatorname{tr}_{\text{A}}\left[\left(\openone+v_{m}A_{x}\right)\otimes\openone\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right],$ (10)

where $v_{m}$ (the “visibility" of the state) is given by

$\displaystyle v_{m}$ $\displaystyle=v_{m-1}f_{m-1}=\prod_{j=1}^{m-1}f_{j},$
(11) $\displaystyle\text{where}\quad f_{j}$
$\displaystyle=\frac{1+(n-1)\cos\theta_{j}}{n}.$ (12)

We call $f_{j}$ the “quality factor" of the measurement of the $j^{th}$ Bob.
The visibility of the first Bob is $v_{1}=1$, since he possesses the
undisturbed state received directly from Alice.

The proof is inductive. For the first Bob, the statement holds trivially.
Consider that it holds true for $m-1$ Bobs, so that the average state
$\tilde{\rho}_{x}^{(m)}$ received by Bobm is given by (10). Then using the
Kraus operators stated in the main text, the average state
$\tilde{\rho}_{x}^{(m+1)}$ (averaging over all Bobm’s possible and
equiprobable inputs, and with no knowledge of his outcome), is given by

$\displaystyle\tilde{\rho}_{x}^{(m+1)}$
$\displaystyle=\frac{1}{n}\sum_{y,b}K_{y}^{b}\tilde{\rho}_{x}^{(m)}(K_{y}^{b})^{\dagger}=\frac{1}{n}\sum_{y,b}\operatorname{tr}_{\text{A}}\left[\left(\openone+v_{m}A_{x}\right)\otimes
K^{b}_{y}\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\;\openone\otimes(K_{y}^{b})^{\dagger}\right],$ (13)

where the Kraus operators are acting on the part of the Hilbert space
complementary to that being traced out. First, using the property of the
maximally entangled state that $\left(\openone\otimes
O\right)\phi_{\text{max}}\left(\openone\otimes
O^{\dagger}\right)=\left(O^{T}\otimes\openone\right)\phi_{\text{max}}\left(O^{*}\otimes\openone\right)$,
and then using the cyclicity of the trace, we obtain

$\displaystyle\tilde{\rho}_{x}^{(m+1)}$
$\displaystyle=\frac{1}{n}\sum_{y,b}\operatorname{tr}_{\text{A}}\left[\left(\openone+v_{m}A_{x}\right)(K^{b}_{y})^{T}\otimes\openone\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\;(K^{b}_{y})^{\dagger T}\otimes\openone\right]$ (14)
$\displaystyle=\frac{1}{n}\sum_{y,b}\operatorname{tr}_{\text{A}}\left[(K^{b}_{y})^{\dagger
T}\left(\openone+v_{m}A_{x}\right)(K^{b}_{y})^{T}\otimes\openone\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right].$ (15)

Splitting the above into the sum of the two terms from the
$(\openone+v_{m}A_{x})$, the contribution of the $\openone$ part is

$\displaystyle\frac{1}{n}\sum_{y,b}\operatorname{tr}_{\text{A}}\left[(K^{b}_{y})^{\dagger
T}(K^{b}_{y})^{T}\otimes\openone\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right]=\frac{1}{n}\sum_{y}\operatorname{tr}_{\text{A}}\left[\openone\otimes\openone\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right]=\operatorname{tr}_{\text{A}}\left[\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right],$ (16)

where we have used that the $K^{b}_{y}$ are Hermitian and that measurements
are complete i.e., $\sum_{b}(K^{b}_{y})^{\dagger T}(K^{b}_{y})^{T}=\openone$.

For the term involving $A_{x}$, we calculate the sum using the Kraus operators
from the main text, denoting by $\eta_{m}=\sin\theta_{m}$ the strength of the
measurement of Bob $m$,

$K_{y}^{b}=\sqrt{\frac{1+\eta_{m}}{2}}\Pi_{n,y}^{b}+\sqrt{\frac{1-\eta_{m}}{2}}\Pi_{n,y}^{\bar{b}}=\left(\frac{\cos\frac{\theta_{m}}{2}\openone+(-1)^{b}\sin\frac{\theta_{m}}{2}G_{n,y}}{\sqrt{2}}\right),$
(17)

which results in

$\displaystyle\frac{1}{n}\sum_{y,b}(K^{b}_{y})^{\dagger
T}A_{x}(K^{b}_{y})^{T}$
$\displaystyle=\frac{1}{n}\sum_{y,b}\left(\frac{\cos\frac{\theta_{m}}{2}\openone+(-1)^{b}\sin\frac{\theta_{m}}{2}G_{n,y}}{\sqrt{2}}\right)A_{x}\left(\frac{\cos\frac{\theta_{m}}{2}\openone+(-1)^{b}\sin\frac{\theta_{m}}{2}G_{n,y}}{\sqrt{2}}\right)$
$\displaystyle=\frac{1}{2n}\sum_{y,b}\cos^{2}\left(\frac{\theta_{m}}{2}\right)A_{x}+(-1)^{b}\cos\left(\frac{\theta_{m}}{2}\right)\sin\left(\frac{\theta_{m}}{2}\right)\left\\{G_{n,y},A_{x}\right\\}+\sin^{2}\left(\frac{\theta_{m}}{2}\right)G_{n,y}A_{x}G_{n,y}.$
$\displaystyle=\frac{1}{n}\sum_{y}\left(\frac{1+\cos\theta_{m}}{2}\right)A_{x}+\left(\frac{1-\cos\theta_{m}}{2}\right)G_{n,y}A_{x}G_{n,y}$
$\displaystyle=\left(\frac{1+\cos\theta_{m}}{2}\right)A_{x}+\left(\frac{1-\cos\theta_{m}}{2}\right)\frac{1}{n}\sum_{y}G_{n,y}A_{x}G_{n,y}.$
(18)

We may now use the expansion
$A_{x}=\frac{1}{\sqrt{n}}\sum_{i}(-1)^{x_{i}}G_{n,i}$, and the anti-
commutation relation $\\{G_{n,j},G_{n,k}\\}=2\delta_{j,k}\openone$ from
IndexGame  to simplify

$\displaystyle\frac{1}{n}\sum_{y}G_{n,y}A_{x}G_{n,y}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i}(-1)^{x_{i}}\frac{1}{n}\sum_{y}G_{n,y}G_{n,i}G_{n,y}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i}(-1)^{x_{i}}\frac{1}{n}\sum_{y}\left(2\delta_{i,y}G_{n,y}-G_{n,i}\right)$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i}(-1)^{x_{i}}\frac{1}{n}(2-n)G_{n,i}$
$\displaystyle=\frac{2-n}{n}A_{x}.$ (19)

Inserting this into Eq. (18), we obtain

$\displaystyle\frac{1}{n}\sum_{y,b}(K^{b}_{y})^{\dagger
T}A_{x}(K^{b}_{y})^{T}$ $\displaystyle=f_{m}A_{x},$ (20)
$\displaystyle\text{where}\quad f_{m}$
$\displaystyle=\left(\frac{1+(n-1)\cos\theta_{m}}{n}\right)=\left(\frac{1+(n-1)\sqrt{1-\eta^{2}}}{n}\right),$
(21)

is the quality factor of the measurement of Bobm. Combining this with Eq. (16)
to find the final expression for the average state after Bobm’s measuremet Eq.
(15), we find

$\displaystyle\tilde{\rho}_{x}^{(m+1)}$
$\displaystyle=\operatorname{tr}_{\text{A}}\left[\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right]+\operatorname{tr}_{\text{A}}\left[v_{m}f_{m}A_{x}\otimes\openone\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right]$ (22)
$\displaystyle=\operatorname{tr}_{\text{A}}\left[\left(\openone+v_{m}f_{m}A_{x}\right)\otimes\openone\;\phi_{\text{max}}^{\otimes\lfloor
n/2\rfloor}\right],$ (23)

which proves the desired relation ((10) - (12)).

##  Appendix B The number of Bobs that can share the preparation contextuality
enabled by Alice’s ensemble using unsharp measurements based on the strategy
from IndexGame  is exactly $n$.

The figure of merit to witness non-contextuality between Alice and Bobk is

$\displaystyle\mathcal{A}_{k}=\frac{1}{2}\left(1+\frac{v_{k}\eta_{k}}{\sqrt{n}}\right),$
(24)

and so the condition on $\eta_{k}$ so that Bobk violates the constraint of
noncontextuality therefore reads

$\displaystyle\eta_{k}=\sin\theta_{k}>\frac{1}{v_{k}\sqrt{n}}.$ (25)

To construct the longest possible sequence of violations, consider that each
Bob tunes the strength of his measurement so as to just violate the non-
contextuality inequality, but not more, so that
$\eta_{k}=\sin\theta_{k}=1/(v_{k}\sqrt{n})$. In order for this to be possible,
it must be that $v_{k}\geq 1/\sqrt{n}$, because $\eta$ is at most $1.$ At what
point does it become impossible for the next Bob to violate the preparation
noncontextuality inequality? We proceed to prove that there can be _exactly_
$n$ violations in such a sequence for the considered class of strategies.

Consider the quality factor $f_{k}$ of the the measurement of Bobk. We can
find upper and lower bounds on $f_{k}^{2}$ in the following manner. First, for
an upper bound,

$\displaystyle f_{k}$
$\displaystyle=\frac{1+(n-1)\cos\theta_{k}}{n}>\cos\theta_{k},$ (26)
$\displaystyle\therefore f_{k}^{2}$
$\displaystyle>\cos^{2}\theta_{k}=1-\sin^{2}\theta_{k}=1-\frac{1}{nv_{k}^{2}}.$
(27)

For the lower bound,

$\displaystyle f_{k}^{2}$
$\displaystyle<f_{k}^{2}+4\frac{(n-1)}{n^{2}}\sin^{4}\frac{\theta_{k}}{2}=1-\frac{(n-1)}{n}\sin^{2}\theta_{k}=1-\frac{(n-1)}{n^{2}v_{k}^{2}}.$
(28)

But since the visibility $v_{k+1}$ of the next Bob is given by
$v_{k+1}=v_{k}f_{k}$, we can bound the next visibility as

$\displaystyle
v_{k}^{2}\left(1-\frac{1}{nv_{k}^{2}}\right)<v^{2}_{k+1}<v_{k}^{2}\left(1-\frac{(n-1)}{n^{2}v_{k}^{2}}\right),$
(29)

from which the decrease in the visibility squared is both bounded on both
sides, by

$\displaystyle\frac{1}{n}<v_{k}^{2}-v_{k+1}^{2}<\frac{n-1}{n^{2}}.$ (30)

Proceeding from the first Bob, who has visibility $v_{1}=1$, we can use the
upper bound to find that $v_{n}^{2}>1/n$, and the lower bound to find that
$v_{n+1}^{2}<1/n$. Since $1/\sqrt{n}$ is the critical visibility to violate
the preparation noncontextuality inequality, it follows that Bobn can violate
the inequality (as all of the Bobs before him), but that Bobn+1 and later Bobs
cannot.

##  Appendix C Sharing noise-robust preparation contextuality in an anonymous
setting

Consider a quantum strategy in which the set of possible measurements
performed by each Bob is the same, i.e., they all perform equally unsharp
measurements. This is useful in a scenario in which each Bob does not know
their position in the sequence, and thus the optimal strategy is for all of
them to pick equally sharp measurements.

If it is the case that the first $k$ Bobs in the sequence violate the
preparation noncontextuality inequality, then the weakest violation will be by
the last Bob. The condition for the $k$’th Bob to just saturate the
preparation noncontextuality inequality reads

$\displaystyle\sin\theta=\frac{1}{v_{k-1}\sqrt{n}},$ (31)

where $\eta=\sin\theta$ is the strength of all of the Bobs’ measurements, and
the visibility is given by,

$v_{k-1}=\left(\frac{1+(n-1)\cos\theta}{n}\right)^{k-1}.$ (32)

Solving the equation for the value of $k$ returns

$k=1-\frac{\log\sin\theta+\frac{1}{2}\log
n}{\log\left(1+(n-1)\cos\theta\right)-\log n}.$ (33)

Consider that the strength of the measurement is chosen to be
$\sin\theta=\sqrt{e/n}$ (where $n\geq 3$). In any case, we are interested in
the scaling for large $n$, for which we may approximate
$\cos\theta=\sqrt{1-e/n}\approx 1-e/2n$. Substituting this in the above, and
further approximating $\log(1-x)\approx-x$ for $|x|\ll 1$, one gets

$\displaystyle k\approx 1+\frac{n^{2}}{(n-1)e}=O\left(\frac{n}{e}\right).$
(34)

One can show that this is the optimal scaling by the Maclaurin expansion of
(33) for small $\theta$, and differentiating to find the optimal value of
$\theta$.

Thus we find that even in the anonymous setting where each Bob is unaware of
their position in the sequence, the maximum number of observers able to share
the contextuality enabled by Alice’s ensemble by all performing equally
unsharp measurements scales as

$k_{\text{max}}\approx\frac{n}{e}.$ (35)

Note that this scaling is the same as obtained in the main text for the non-
anonymous setting, up to a pre-factor of $1/e$.

##  Appendix D Experimental settings

The angles used for Alice’s state preparation are given below:

State | Pol. ($\degree$) | QA1 ($\degree$) | HA1 ($\degree$) | QA2 ($\degree$)
---|---|---|---|---
000 | 27.37 | 45 | -33.75 | 45
001 | 27.37 | 45 | -11.25 | 45
010 | 27.37 | 45 | -56.25 | 45
011 | 27.37 | 45 | -78.75 | 45
100 | 62.63 | 45 | -33.75 | 45
101 | 62.63 | 45 | -11.25 | 45
110 | 62.63 | 45 | -56.25 | 45
111 | 62.63 | 45 | -78.75 | 45
Table 1: Angles for the polarizer, QWPs and HWP for the preparation of Alice’s
states.

The settings of the HWPs and QWPs used for the unsharp measurements in Bob1
and Bob2 are as follows. Note these settings are independent of the sharpness
of the measurement, which is determined by the angle of the HWPs inside the
interferometer.

Measurement | Output Port | HBi1 ($\degree$) | QBi1 ($\degree$) | HBi2($\degree$) | QBi2 $\degree$)
---|---|---|---|---|---
$\sigma_{x}$ | 1 | 22.5 | 0 | 90 | 22.5
$\sigma_{x}$ | 2 | 67.5 | 0 | 90 | 67.5
$\sigma_{y}$ | 1 | 0 | -45 | 45 | 0
$\sigma_{y}$ | 2 | 0 | 45 | 135 | 0
$\sigma_{z}$ | 1 | 0 | 0 | 90 | 0
$\sigma_{z}$ | 2 | 45 | 0 | 90 | 45
Table 2: Waveplate settings for measurement and output selection of Bob1 and
Bob2.

##  Appendix E Experimental results

The experimental marginal probabilities corresponding to the outcomes that
satisfy $b_{i}=x_{y_{i}}$ (the ‘winning’ answer in the communication game) for
Bob1 and Bob2’s unsharp measurements and Bob3’s projective measurements of
$\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ on each of Alice’s preparations
are shown in the following three tables:

| Bob1
---|---
State | $\sigma_{x}$ | $\sigma_{y}$ | $\sigma_{z}$
000 | $0.7369\pm 0.0003$ | $0.7044\pm 0.0003$ | $0.6593\pm 0.0002$
001 | $0.6473\pm 0.0002$ | $0.7257\pm 0.0003$ | $0.7079\pm 0.0003$
010 | $0.6900\pm 0.0003$ | $0.6727\pm 0.0002$ | $0.6571\pm 0.0002$
011 | $0.6879\pm 0.0003$ | $0.6501\pm 0.0002$ | $0.7005\pm 0.0003$
100 | $0.6911\pm 0.0003$ | $0.6195\pm 0.0002$ | $0.7180\pm 0.0003$
101 | $0.6813\pm 0.0003$ | $0.6464\pm 0.0002$ | $0.6779\pm 0.0003$
110 | $0.6400\pm 0.0002$ | $0.7471\pm 0.0003$ | $0.7125\pm 0.0003$
111 | $0.7242\pm 0.0003$ | $0.7132\pm 0.0003$ | $0.6755\pm 0.0002$
Table 3: Experimental marginal probabilities for Bob1. | Bob2
---|---
State | $\sigma_{x}$ | $\sigma_{y}$ | $\sigma_{z}$
000 | $0.6997\pm 0.0003$ | $0.6422\pm 0.0002$ | $0.6851\pm 0.0003$
001 | $0.6586\pm 0.0002$ | $0.6785\pm 0.0002$ | $0.6746\pm 0.0002$
010 | $0.6537\pm 0.0002$ | $0.7088\pm 0.0003$ | $0.6715\pm 0.0002$
011 | $0.6896\pm 0.0003$ | $0.6824\pm 0.0003$ | $0.6572\pm 0.0002$
100 | $0.7106\pm 0.0003$ | $0.6370\pm 0.0002$ | $0.6775\pm 0.0002$
101 | $0.6446\pm 0.0002$ | $0.6792\pm 0.0003$ | $0.6868\pm 0.0003$
110 | $0.6553\pm 0.0002$ | $0.7000\pm 0.0003$ | $0.6752\pm 0.0002$
111 | $0.6787\pm 0.0002$ | $0.6666\pm 0.0002$ | $0.6853\pm 0.0002$
Table 4: Experimental marginal probabilities for Bob2. | Bob3
---|---
State | $\sigma_{x}$ | $\sigma_{y}$ | $\sigma_{z}$
000 | $0.7044\pm 0.0003$ | $0.6470\pm 0.0002$ | $0.6786\pm 0.0003$
001 | $0.6661\pm 0.0002$ | $0.6886\pm 0.0003$ | $0.6854\pm 0.0003$
010 | $0.6582\pm 0.0002$ | $0.7113\pm 0.0003$ | $0.6702\pm 0.0002$
011 | $0.7011\pm 0.0003$ | $0.6783\pm 0.0003$ | $0.6746\pm 0.0003$
100 | $0.6975\pm 0.0003$ | $0.6558\pm 0.0002$ | $0.6915\pm 0.0003$
101 | $0.6655\pm 0.0003$ | $0.7049\pm 0.0003$ | $0.6891\pm 0.0003$
110 | $0.6469\pm 0.0002$ | $0.6942\pm 0.0003$ | $0.6881\pm 0.0003$
111 | $0.7027\pm 0.0003$ | $0.6512\pm 0.0002$ | $0.6853\pm 0.0003$
Table 5: Experimental marginal probabilities for Bob3.

##  Appendix F Enforcing strict operational equivalences on experimental data

Tests of preparation contextuality require that the observed probabilities
satisfy an equivalence relation. In the specific preparation noncontextuality
inequalities considered in the main text, that equivalence relation follows
from the indistinguishability relation imposed on Alice’s quantum
preparations, i.e. that she hides the value of the parity $r\cdot x$ for every
string $r\in\\{0,1\\}^{n}$ with $|r|\geq 2$. This is an operational
equivalence relation that is expressed in terms of probabilities as follows,

$\forall r,\forall M:\sum_{r\cdot x=0}p(P_{x}|b,M)=\sum_{r\cdot
x=1}p(P_{x}|b,M).$ (36)

Evidently, due to unavoidable experimental imperfections, such a constraint
can never be exactly satisfied. This necessitates data processing methods to
contend with the problem. Ref. MP16  developed a method for post-processing
outcome statistics that approximately satisfies an operational equivalence
constraint into data that strictly satisfies said constraint. The price to pay
for this mapping is that the value of the witness after post-processing is
worse than what is originally measured. Roughly speaking, the closer the
unprocessed outcome statistics is to satisfying the operational equivalence
constraint, the smaller the decrease in the witness value due to the post-
processing scheme.

We have applied a simplified variant (which assumes that the experiment is
accurately described by quantum theory) of the method of MP16  to enforce
operational equivalence in each of the three sequential tests of preparation
contextuality. We describe how it applies to the experimental results of the
pair Alice-Bob1. Since the the outcomes are binary, the full distribution
$p(b_{1}|x,y_{1})$ can be described by only considering $p(b_{1}=0|x,y_{1})$.
We can write this distribution as eight vectors
$\mathbf{P}_{x}=[p(0|x,1),p(0|x,2),p(0|x,3)]$. The vectors $\mathbf{P}_{x}$
will not perfectly satisfy the operational equivalence constraint (36).
Therefore, we aim to map them to other distributions $\mathbf{P}^{\prime}_{x}$
which perfectly satisfy (36). This can be done by noting that an experiment in
which $\\{\mathbf{P}_{x}\\}$ is realised, also constitutes an effective
realisation of all distributions in the convex hull of $\\{\mathbf{P}_{x}\\}$
(due to linearity). Hence, we set

$\mathbf{P}^{\prime}_{x}=\sum_{x^{\prime}}\omega_{x}^{x^{\prime}}\mathbf{P}_{x^{\prime}},$
(37)

where for $\forall x$ $\\{\omega_{x}^{x^{\prime}}\\}_{x^{\prime}}$ is a
probability distribution. We search a set of distributions $\\{\omega_{x}\\}$
that maximises the witness of preparation contextuality while also enforcing
(36). This problem is solved with a linear program

$\displaystyle\qquad\qquad\mathcal{A}_{1}^{\text{post}}=\max_{\\{\omega\\}}\mathcal{A}_{1}^{\text{pre}}[\\{\mathbf{P}_{x}^{\prime}\\}]$
$\displaystyle\text{such that }\forall
r\in\\{011,101,110,111\\}\qquad\sum_{r\cdot
x=0}\mathbf{P}^{\prime}_{x}=\sum_{r\cdot x=1}\mathbf{P}^{\prime}_{x}.$ (38)

In addition, we can employ the quantity $F=\sum_{x}\omega_{x}^{x}$ as a
measure of the closeness of the observed and post-processed data. Moreover,
this procedure can be straightforwardly adapted to the experimental results
obtained for Alice-Bob2 and Alice-Bob3. The minor difference is that the
preparation procedure for e.g. Alice-Bob2 effectively becomes the average
state relayed by Bob1 to Bob2. Thus, change the definition of vectors
$\mathbf{P}_{x}$ to instead apply to the distributions $p(b_{2}=0|x,y_{2})$
and $p(b_{3}=0|x,y_{3})$ respectively and proceed in analogy with the above.

Solving the above linear program, we have obtained the following results for
the three demonstrations of preparation contextuality.

$\displaystyle\mathcal{A}_{1}^{\text{pre}}=0.687$
$\displaystyle\mathcal{A}_{1}^{\text{post}}=0.683$ $\displaystyle F=0.9690$
(39) $\displaystyle\mathcal{A}_{2}^{\text{pre}}=0.675$
$\displaystyle\mathcal{A}_{2}^{\text{post}}=0.670$ $\displaystyle F=0.9537$
(40) $\displaystyle\mathcal{A}_{3}^{\text{pre}}=0.681$
$\displaystyle\mathcal{A}_{3}^{\text{post}}=0.677$ $\displaystyle F=0.9700$
(41)