# How post-selection affects device-independent claims under the fair sampling
assumption

Davide Orsucci  Quantum Optics Theory Group, Universität Basel, CH-4056 Basel,
Switzerland  Jean-Daniel Bancal  Quantum Optics Theory Group, Universität
Basel, CH-4056 Basel, Switzerland  Département de Physique Appliquée,
Université de Genève, CH-1211 Genève, Switzerland  Nicolas Sangouard  Quantum
Optics Theory Group, Universität Basel, CH-4056 Basel, Switzerland  Pavel
Sekatski  Quantum Optics Theory Group, Universität Basel, CH-4056 Basel,
Switzerland

###### Abstract

Device-independent certifications employ Bell tests to guarantee the proper
functioning of an apparatus from the sole knowledge of observed measurement
statistics, i.e. without assumptions on the internal functioning of the
devices. When these Bell tests are implemented with devices having too low
efficiency, one has to post-select the events that lead to successful
detections and thus rely on a fair sampling assumption. The question that we
address in this paper is what remains of a device-independent certification
under fair sampling. We provide an intuitive description of post-selections in
terms of _filters_ and define the fair sampling assumption as a property of
these filters, equivalent to the definition introduced in Ref. [1]. When this
assumption is fulfilled, the post-selected data is reproduced by an ideal
experiment where lossless devices measure a _filtered_ state which can be
obtained from the _actual_ state via local probabilistic maps. Trusted
conclusions can thus be obtained on the quantum properties of this filtered
state and the corresponding measurement statistics can reliably be used, e.g.,
for randomness generation or quantum key distribution. We also explore a
stronger notion of fair sampling leading to the conclusion that the post-
selected data is a fair representation of the data that would be obtained with
lossless detections. Furthermore, we show that our conclusions hold in cases
of small deviations from exact fair sampling. Finally, we describe setups
previously or potentially used in Bell-type experiments under fair sampling
and identify the underlying device-specific assumptions.

## Introduction

Measurement devices in many quantum experiments, and notably in quantum
optics, have a finite non-unit efficiency: such devices may refuse to provide
an outcome for the desired measurement, and produce a “no-click” event instead
[2]. This lack of detection can usually be explained in simple physical terms,
e.g., the photon to be measured is not always absorbed on the chip of the
detector. In a simplified physical model of the apparatus, the occurrence of a
no-click event is completely independent of the state of the quantum system to
be measured. Based on this model, one is tempted to ignore the no-click events
altogether and remove them from the measurement data, a.k.a. performing a
_post-selection_.

However, from a black-box perspective, a no-click event must be considered as
a measurement outcome, just like the others111A non-detection event is
obtained when a detector is supposed to click but, instead, it doesn’t. These
non-detections can be distinguished from the normal “silent” state of the
device by means of a trigger event: a classical signal indicating that a
quantum state has been sent to the detector within a certain time window..
This outcome has to be added to the alphabet corresponding to possible values
of the measured quantity [3, 4, 5]. The action of ignoring these events is not
always harmless: in the context of testing of Bell inequalities this is known
as the detection loophole, and there are explicit examples where the use of
post-selection can lead to wrong claims about the performed Bell test, e.g.
erroneously deducing the non-locality of a local model [5, 6, 7]. Such canny
local models are not abstract theoretical constructions, but have been
produced experimentally with simple optical elements and detectors that are
commonly used [8, 9, 10, 11, 12].

Nevertheless, an accurate physical understanding of the measurement device in
the lab may suggest that the no-click events are locally random and
independent of system state and measurement settings. If this is really the
case, data acquisition and post-selection is equivalent to directly generating
the data with an ideal measurement device that has unit efficiency but is
otherwise the same as the detector in the lab. Thus, any claim that could be
made in the ideal setting carries over when using the experimental post-
selected data. This logical step is known as fair sampling assumption (or _no-
enhancement assumption_ , at the origins [3, 4]). It is important to realize
that fair sampling is indeed an assumption and cannot be ensured from the
measurement probabilities alone – post-selected local models as mentioned
above can reproduce those perfectly.

Thanks to the advancement in quantum technologies we have recently witnessed
the first experimental violations of Bell inequalities free of loopholes [13,
14, 15, 16, 17], conclusively ruling out that locally causal models could
explain the correlations observed in quantum experiments. In the aftermath of
achieving this long-standing goal, renewed interest in Bell inequalities was
spurred by their technological applications within the framework of _device-
independent quantum information processing_ [18, 19]. Namely, the sole
violation of a Bell inequality can be used to certify devices, to guarantee
the randomness of measurement results or the security of a quantum key
distribution, without the need to know the internal functioning of the devices
used to perform the Bell test [20, 21, 22, 23, 24]. Device-independent quantum
information processing is at the verge of experimental feasibility [25].
However, performing Bell tests devoid of the detection loophole is still
challenging and, as such, most experiments at present still rely on fair
sampling. This naturally raises the following question: if assessing fair
sampling requires a detailed description of the way the measurement device
works, what remains of the device-independent framework in scenarios based on
Bell tests relying upon the fair sampling assumption? The aim of our work is
to provide a concrete answer to this question.

We first explain how a lossy detector can be understood in terms of filters:
we represent a finite-efficiency measurement as a two-step process where first
a filter is applied to the classical input and to the quantum state which
either accepts or rejects them (corresponding, respectively, to a successful
or failed detection) and subsequently a lossless measurement is performed on
the filtered quantum state. This is schematically illustrated in panel (a) of
Figure 1. An equivalent formulation based on the positive operator-valued
measure (POVM) description of the measurement devices is also provided; this
formulation allows to directly verify if a detector satisfies fair sampling,
given a full specification of its behaviour.

Figure 1: Mathematical model of a lossy device. In panel (a) we illustrate
that any device having finite efficiency can always be modelled as a filter
acting jointly on the classical setting $\mathsf{x}\in\mathsf{X}$ and on the
input quantum state $\rho\in S(\mathcal{H})$, followed by by an ideal lossless
measurement device. The filter returns a flag
$\mathsf{f}\in\\{{\checkmark},\textup{\,\o\,}\\}$ with a probability that
depends on $\mathsf{x}$ and $\rho$, and the ideal measurement is performed
only if $\mathsf{f}={\checkmark}$. In panel (b) we illustrate the _fair
sampling_ condition: the filter factorizes in two components $\mathcal{F}_{C}$
and $\mathcal{F}_{Q}$ which act independently on the classical and quantum
input, respectively, and a detection occurs only when both accept (indicated
by a $\wedge\equiv$ logical AND). In panel (c) we illustrate _strong fair
sampling_ : the component acting on the quantum input is proportional to the
identity channel.

Following the work of Berry et. al. [1], we then give a very general
formulation of fair sampling: to have fair sampling, it is sufficient to
require that the filter operates independently on the classical input and on
the quantum state, see panel (b) of Figure 1. Under this assumption the
sampling can be regarded as “fair”, since the post-selected data can be seen
as being generated from an ideal quantum experiment where lossless devices
measure a _filtered_ state222For instance, in quantum optics experiments the
vacuum component of photonic states is often ignored. This corresponds to
applying a filtering operation that projects the quantum state onto the non-
vacuum component.. This allows to reach trusted conclusions on the quantum
properties of this filtered state and the corresponding measurement statistics
can be used reliably for verifiable randomness generation [26, 27] or for
quantum key distribution [20, 21].

In this work, we also discuss a stronger version of the fair sampling
assumption, corresponding to what is usually assumed to be the very definition
of fair sampling (see e.g. [28, 29]). In strong fair sampling, filter
operating on the state is proportional to the identity, see panel (c) of
Figure 1. We show that, in this case, the post-selected statistics is an
unbiased representation of the statistics that would be obtained with unit-
efficiency detectors measuring the actual _experimental_ state.

The motivation of the present work is thus to bridge the gap between
theoretical studies of device-independent certifications and experimental
realizations thereof. To this end, we describe concrete examples of
experiments that can be used for device-independent quantum information
processing and show that, under fair sampling, these experiments allow one to
get semi-device-independent conclusions. In fact, even present-day experiments
frequently rely upon a fair sampling assumption [30, 31, 32, 33], and this
will likely continue to be necessary in future years; however, often there is
very little or no theoretical analysis of the consequences that post-selection
has in these experiments. Furthermore, we study what happens when small
deviations from exact fair sampling are present and we show that the results
are robust against small perturbations. Concretely, we provide a prescription
for the minimal information that in an actual quantum experiment should be
directly assessed in order to verify that a device (approximately) satisfies
fair sampling. Assuming that this minimal information is publicly available
and trusted, the certification of the device can be safely completed using a
standard device-independent protocol with post-selected data.

#### Paper structure

In Section 1 we describe the general setup and establish part of the notation.
In Section 2 we give the general mathematical formulation of lossy measurement
devices, as represented in panel (a) of Figure 1. In Section 3 we formalize
the definition of fair sampling as depicted in panel (b) of Figure 1
(equivalent to the definition introduced in reference [1]); we then prove
that, under this assumption, the post-selected data can be reproduced by an
ideal quantum experiment involving only lossless detectors. We then introduce
the stronger notion of fair sampling in Section 4, corresponding to panel (c)
of Figure 1. In Section 5 we discuss the use of fair sampling in cryptographic
settings. In Section 6 we give concrete examples of fair sampling based on a
quantum optics setup used to measure the polarization of photons. In Section 7
we investigate the robustness of the results against small perturbations from
exact fair sampling, and we show that the deviations in the post-selected data
are linear in the perturbations from the exact case. We go back to the optical
setup in Section 8 and analyse the consequence of approximate fair sampling in
this context. Finally, in Section 9 we show that even more general notions of
fair sampling are possible, considering cases of fair sampling that are state-
dependent. Supplementary material and further considerations are presented in
the Appendices.

##  1 Preliminary notions

##### Bell tests:

We consider a Bell test [34] involving two or more parties, which we will
denote as $A,B,\ldots,N,$ each party having a measurement device with
different possible measurement settings. A source produces particles which are
distributed to the individual parties. As customary in this context, it is
assumed that each party is in an isolated location, i.e., posterior to each
party receiving its particle, no further information can be exchanged among
them, at least not without their agreement.

##### Quantum framework:

We assume that quantum theory provides a valid description of the source and
measurements used in the Bell test. To each party $k\in\\{A,B,\ldots,N\\}$ is
associated a local Hilbert space $\mathcal{H}_{k}$ and the global Hilbert
space is
$\mathcal{H}:=\mathcal{H}_{A}\\!\otimes\\!\mathcal{H}_{B}\\!\otimes\\!\ldots\\!\otimes\\!\mathcal{H}_{N}$.
The particles are described by a quantum state $\Psi$ which is in general a
density operator (i.e., a mixed quantum state) $\Psi\in S(\mathcal{H})$, where
$S(\mathcal{H})$ denotes the class of positive semi-definite unit-trace
operators on $\mathcal{H}$. The local spaces are denoted as
$S(\mathcal{H}_{k})$, and we usually call $\rho$ a state that is received and
measured by a local detector, $\rho\in S(\mathcal{H}_{k})$. The measurement
devices are described in terms of general positive operator valued measures
(POVMs) and we assume that outcomes in a sequence of measurement rounds are
independent and identically distributed (i.i.d.). When one of the parties has
only partial information about a measurement device, she will possess a more
coarse-grained POVM description. We avoid the use of coarse-grained
descriptions, as these open the side to attacks, see Section 5.

##### Inputs and outputs:

Each party has full control over the setting $\mathsf{x}\in\mathsf{X}$ which
can be chosen, e.g., by turning a knob or via digital control. For each
measurement, there is a set of possible outcomes $\mathsf{a}\in\mathsf{A}$. In
the following, we will be mainly interested in the case where both the set of
inputs and of outputs are finite333This is customarily the setting used in
Bell tests, although cases with continuous outputs have also been considered
[35]. Note that the results we present do not depend on the size of the
classical input and output spaces; hence, they can be generalized to
continuous-variable cases.. For instance, in the standard Clauser-Horne-
Shimony-Holt (CHSH) inequality [3, 4] the set of inputs is
$\mathsf{X}=\\{0,1\\}$ and the possible outcomes are $\mathsf{A}=\\{+1,-1\\}$,
for party A’s device and similarly for party B’s device. We suppose that along
the set of “good” outcomes, labelled by an index $\mathsf{a}\in\mathsf{A}$,
there is a special label  ø  for the “no-click” event (i.e., a failed
detection in correspondence of a trigger signal) so that the entire alphabet
of outcomes becomes
$\mathsf{A}^{\prime}:=\\{\textup{\,\o\,}\\}\cup\mathsf{A}$.

##### Fair sampling for multiple parties:

In the following presentation we will often restrict our attention to a single
measurement device and to a local Hilbert space (so we can take, e.g.,
$\mathcal{H}\equiv\mathcal{H}_{A}$). In fact, fair sampling is an assumption
on a single apparatus, i.e., can be considered outside of the context of Bell
inequalities. For a Bell test involving $N$ measurement devices, we can say
that the setup satisfies fair sampling when each of the $N$ devices,
considered individually, satisfies the assumption. However, one obtains a
completely equivalent mathematical formulation by considering the $N$ devices
as a single collective measurement device split across $N$ locations, and then
requiring that this global multipartite device satisfies fair sampling. Hence,
all the considerations made for a single device will hold also for the global
setup, provided that the fair sampling assumption holds for each device.

##  2 Quantum model of a lossy detector

We now provide a description of a _measurement device_ having non-unit
efficiency. We show in particular that such a measurement can be seen as the
combination of a filter and an ideal unit-efficiency measurement. We then link
this description to the efficiency of the device.

###  2.1 POVM elements

We describe the measurement device $\mathcal{M}$ as a set of POVM elements
$\\{M_{\mathsf{a}}:=\sum_{\mathsf{x}\in\mathsf{X}}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\\!\otimes\\!M^{\mathsf{x}}_{\mathsf{a}}\\}$
“taking” both $\mathsf{x}$ and $\rho$ and returning the outcome $\mathsf{a}$
with a certain probability
${\Pr}_{\mathcal{M}}(\mathsf{a}\,|\,\mathsf{x},\rho)$:

$\displaystyle\mathcal{M}:\
S(\;\mathcal{H}_{\mathsf{X}}\;\otimes\mathcal{H}\,)\;$
$\displaystyle\to\;\mathcal{D}(\mathsf{A}^{\prime})$
$\displaystyle\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\,\rho~{}~{}~{}$
$\displaystyle\mapsto\;\big{[}\,{\Pr}_{\mathcal{M}}(\mathsf{a}\,|\,\mathsf{x},\rho)=\mathrm{Tr}(M_{\mathsf{a}}\,\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\\!\otimes\\!\rho)\,\big{]}_{\mathsf{a}\in\mathsf{A}^{\prime}}\;.$
(1)

$\mathcal{H}_{\mathsf{X}}$ is a Hilbert space with an orthonormal basis
$\\{\left|{\mathsf{x}}\right\rangle\\}_{\mathsf{x}\in\mathsf{X}}$ and
$\mathcal{D}(\mathsf{A}^{\prime})$ is the set of probability distributions on
$\mathsf{A}^{\prime}.$ For each $\mathsf{a}\in\mathsf{A}^{\prime}$, the
channel
$\mathcal{M}_{\mathsf{a}}^{\mathsf{x}}(\,\cdot\,):=\mathrm{Tr}(M^{\mathsf{x}}_{\mathsf{a}}\;\cdot\,)$
is a real-valued completely-positive (CP) map [36] satisfying
$\sum_{\mathsf{a}\in\mathsf{A}^{\prime}}\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)=1$
for all density operators $\rho.$ We introduce two flags
$\mathsf{f}={\checkmark}$ and $\mathsf{f}=\textup{\,\o\,}$ associated to two
complementary operators $M_{\checkmark}^{\mathsf{x}}$ and
$M_{\textup{\,\o\,}}^{\mathsf{x}}$ which satisfy

$\displaystyle\forall\mathsf{x}\in\mathsf{X}\qquad$ $\displaystyle
M_{\checkmark}^{\mathsf{x}}\;:=\;\sum_{\mathsf{a}\in\mathsf{A}}M_{\mathsf{a}}^{\mathsf{x}}\;,$
(2) so that $\displaystyle
M_{\checkmark}^{\mathsf{x}}+M_{\textup{\,\o\,}}^{\mathsf{x}}\;=\;\sum_{\mathsf{a}\in\mathsf{A}^{\prime}}M_{\mathsf{a}}^{\mathsf{x}}=\;\mathds{1}\;.$
(3)

As we will show later, a description of the operators
$M_{\checkmark}^{\mathsf{x}}$ (or equivalently of
$M_{\textup{\,\o\,}}^{\mathsf{x}}$) is all that is required to know if a
device satisfies fair sampling, while he full POVM description
$M_{\mathsf{a}}^{\mathsf{x}}$ is not necessary to this end.

###  2.2 Filters

We can now introduce the notion of filtering: for any lossy device
$\mathcal{M}$, one can always find a corresponding pair of completely-positive
trace-preserving (CPTP) maps

$\displaystyle\mathcal{F}:$
$\displaystyle~{}~{}~{}~{}S(\,\mathcal{H}_{\mathsf{X}}\otimes\mathcal{H}\,)~{}~{}~{}~{}\;\to\;S(\,\mathcal{H}_{\mathsf{f}}\otimes\mathcal{H}_{\mathsf{X}}\otimes\mathcal{H}\,)$
$\displaystyle\bar{\mathcal{M}}:$
$\displaystyle\;S(\,\mathcal{H}_{\mathsf{f}}\otimes\mathcal{H}_{\mathsf{X}}\otimes\mathcal{H}\,)\;\to\;\mathcal{D}(\mathsf{A}^{\prime})$
(4)

such that $\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$. Here,
$\mathcal{H}_{\mathsf{f}}=\mathrm{span}(\left|{\textup{\,\o\,}}\right\rangle,\left|{{\checkmark}}\right\rangle)$,
$\mathcal{F}$ represents the filter operation, and $\bar{\mathcal{M}}$ a
lossless measurement device. Notice that for mathematical consistency of the
equation $\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ the set of outputs of
$\bar{\mathcal{M}}$ must be equal to the set of outputs of $\mathcal{M}$ and
thus also include a “no-click” outcome. We therefore consider
$\bar{\mathcal{M}}$ to be lossless if a flag  ø  has the only effect of
“disabling” $\bar{\mathcal{M}}$, i.e., $\bar{\mathcal{M}}$ outputs  ø  if and
only if the filter $\mathcal{F}$ has already produced a  ø  flag.

We remark that the factorization
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ is not unique, since we have
freedom in redefining $\mathcal{F}$ and $\bar{\mathcal{M}}$. For instance, we
can consider any invertible linear operator $\mathcal{A}$ such that
$\bar{\mathcal{M}}^{\prime}:=\bar{\mathcal{M}}\circ\mathcal{A}^{-1}$ and
$\mathcal{F}^{\prime}:=\mathcal{A}\circ\mathcal{F}$ are both CPTP maps and
obtain
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}=\bar{\mathcal{M}}^{\prime}\circ\mathcal{F}^{\prime}$.
Observe that complete-positivity is surely preserved if $\mathcal{A}$ is a
unitary channel, thus the decomposition
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ is certainly not unique.

Given a measurement device $\mathcal{M}$ with POVM elements
$\\{M_{\mathsf{a}}\\},$ one can construct a decomposition
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ as follows444We have in this
case
$M_{\mathsf{a}}=\sum_{\mathsf{x}\in\mathsf{X}}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes
M^{\mathsf{x}}_{\mathsf{a}}$, but the construction of the decomposition
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ does not rely on this
property.. We have
$M_{\checkmark}:=\sum_{\mathsf{a}\in\mathsf{A}}M_{\mathsf{a}}$, so that
$M_{\checkmark}+M_{\textup{\,\o\,}}=\mathds{1}$. Then, we define the filter
$\mathcal{F}$ as a CPTP map having Kraus operators [36]:

$\displaystyle\begin{split}F_{\textup{\,\o\,}}\,&:=\;\left|{\textup{\,\o\,}}\right\rangle\otimes\sqrt{M_{\textup{\,\o\,}}}\,,\\\
F_{\checkmark}\,&:=\;\left|{{\checkmark}}\right\rangle\otimes\sqrt{M_{\checkmark}}\,,\end{split}$
(5)

which satisfy the completeness relation
$F_{\checkmark}^{\dagger}F_{\checkmark}+F_{\textup{\,\o\,}}^{\dagger}F_{\o}=\mathds{1}$.
Correspondingly, $\bar{\mathcal{M}}$ is defined as the measurement device
acting on the output of $\mathcal{F}$ (including the flag) and having POVM
elements:

$\displaystyle\forall\mathsf{a}\in\mathsf{A}\qquad\bar{M}_{\mathsf{a}}\;$
$\displaystyle=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes(M_{\checkmark})^{-1/2}\,M_{\mathsf{a}}\,(M_{\checkmark})^{-1/2}$
(6)

where the inverse square roots are defined on the support of the
operators555I.e., an Hermitian operator
$M=\sum_{j}p_{j}\left|{\psi_{j}}\right\rangle\\!\left\langle{\psi_{j}}\right|$,
with $p_{j}\geq 0$, is mapped to
$(M)^{-1/2}=\sum_{j}f(p_{j})\left|{\psi_{j}}\right\rangle\\!\left\langle{\psi_{j}}\right|$,
with $f(p)=1/\sqrt{p}$ if $p>0$ and $f(0)=0$.. One can immediately verify that
$\sum_{\mathsf{a}\in\mathsf{A}}\bar{M}_{\mathsf{a}}=\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\Pi_{\checkmark}$,
where $\Pi_{\checkmark}$ is the projector on the support of $M_{\checkmark}$.
This means that $\bar{\mathcal{M}}$ has unit efficiency when it receives
$\mathsf{f}={\checkmark}$ and a state $\rho$ in the support of
$M_{\checkmark}$.

From these definition, we can immediately show that
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ ensues. In fact, using the
shorthand
$\xi:=\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\rho$,
we have:

$\displaystyle\mathcal{F}(\xi)\;$
$\displaystyle=\;F_{\checkmark}\,\xi\,F_{\checkmark}^{\dagger}+F_{\textup{\,\o\,}}\,\xi\,F_{\textup{\,\o\,}}^{\dagger}$
$\displaystyle=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\sqrt{M_{\checkmark}}\,\xi\,\sqrt{M_{\checkmark}}\;+\;\left|{\textup{\,\o\,}}\right\rangle\\!\\!\left\langle{\textup{\,\o\,}}\right|\otimes\sqrt{M_{\textup{\,\o\,}}}\,\xi\,\sqrt{M_{\textup{\,\o\,}}}\;,$
(7)

and thus, using the cyclicity of the trace:

$\displaystyle{\Pr}_{\bar{\mathcal{M}}\circ\mathcal{F}}(\mathsf{a}\,|\,\xi)\;$
$\displaystyle=\;\mathrm{Tr}\\!\left[(M_{\checkmark})^{-1/2}M_{\mathsf{a}}(M_{\checkmark})^{-1/2}\,\sqrt{M_{\checkmark}}\,\xi\,\sqrt{M_{\checkmark}}\right]$
$\displaystyle=\;\mathrm{Tr}\big{(}\Pi_{\checkmark}M_{\mathsf{a}}\,\Pi_{\checkmark}\,\xi\big{)}\;=\;\mathrm{Tr}\big{(}M_{\mathsf{a}}\,\xi\big{)}$
$\displaystyle=\;{\Pr}_{\mathcal{M}}(\mathsf{a}\,|\,\xi)$ (8)

for all $\mathsf{a}\in\mathsf{A}$. Consequently, we also have
${\Pr}_{\bar{\mathcal{M}}\circ\mathcal{F}}(\textup{\,\o\,}\,|\,\xi)={\Pr}_{\mathcal{M}}(\textup{\,\o\,}\,|\,\xi)$.

###  2.3 Efficiency

The operator $M_{\checkmark}^{\mathsf{x}}$ allows to compute the efficiency
$\mathcal{E}$ of the device. The efficiency is, per definition, the
probability of obtaining a good outcome ($\mathsf{f}={\checkmark}$) when using
a setting $\mathsf{x}$ and an input quantum state $\rho$:

$\displaystyle\mathcal{E}(\mathsf{x},\rho)\;:=\;{\Pr}_{\mathcal{M}}(\checkmark\,|\,\mathsf{x},\rho)\;=\;\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\,\rho)\;=\;\mathrm{Tr}\big{[}F_{\checkmark}\,(\,\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\rho\,)\,F_{\checkmark}^{\dagger}\big{]}\;.$
(9)

The last equality shows that the efficiency $\mathcal{E}(\mathsf{x},\rho)$ can
be computed from just the specification of the filter, rather than requiring
the full POVM description of $\mathcal{M}$. Moreover, the efficiency of the
detector is a physical property which can be assessed experimentally, provided
access to a reliable source of quantum states $\rho\in S(\mathcal{H})$. Hence,
the efficiency does not depend upon the specific decomposition
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}=\bar{\mathcal{M}}^{\prime}\circ\mathcal{F}^{\prime}$.

The efficiency of the detector allows one to compute the post-selected outcome
probabilities

$\displaystyle{\Pr}_{\textup{p.s.}}(\mathsf{a}\,|\,\mathsf{x},\rho)\;:=\;\frac{{\Pr}(\mathsf{a}\,|\,\mathsf{x},\rho)}{{\Pr}({\checkmark}|\,\mathsf{x},\rho)}\;,$
(10)

where $\mathsf{a}\in\mathsf{A}$ are “good” outcomes, so that
$\sum_{\mathsf{a}\in\mathsf{A}}{\Pr}_{\textup{p.s.}}(\mathsf{a}|\mathsf{x},\rho)=1$.

##  3 The fair sampling assumption

The fair sampling assumption is a restriction on the physical models of lossy
detectors. We introduce a definition which is equivalent to the one introduced
by Berry et. al. [1]. We then discuss the consequences and the applications of
this definition.

######  Definition 1 (Fair sampling).

We say that a lossy measurement device $\mathcal{M}$ satisfies the (weak)
_fair sampling assumption_ if there exists a decomposition
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ (as specified in Section 2.2)
whereby the filter $\mathcal{F}$ factorizes in a part
$\mathcal{F}_{C}:S(\mathcal{H}_{\mathsf{X}})\to
S(\mathcal{H}_{\mathsf{f}}\otimes\mathcal{H}_{\mathsf{X}})$ acting on the
classical setting $\mathsf{x}$ and in a part
$\mathcal{F}_{Q}:S(\mathcal{H})\to
S(\mathcal{H}_{\mathsf{f}}\otimes\mathcal{H})$ acting on the quantum input
$\rho$. That is, we require:

$\displaystyle\mathcal{F}\big{(}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\rho\big{)}\;=\;\wedge\left[\;\mathcal{F}_{C}\big{(}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\big{)}\otimes\mathcal{F}_{Q}\big{(}\rho\big{)}\;\right]\;,$
(11)

where the function $\wedge$ (logical AND) acts only on the flags and it means
that the filter $\mathcal{F}$ returns ${\checkmark}$ if and only if both
$\mathcal{F}_{C}$ and $\mathcal{F}_{Q}$ return ${\checkmark}$.

A few remarks are now in order. First and foremost, the main reason for using
this definition stems from Proposition 2. There, we show the following strong
result: if the factorization of Eq. (11) holds, for a given (real) quantum
experiment involving lossy detectors and post-selection, there is another
(ideal) quantum experiment that involves lossless detectors (and possibly a
different quantum state) that exactly reproduces the post-selected statistics
of the real experiment. In other words, the post-selected statistics are
physical.

Second, in Definition 1 we have a decomposition of a filter in two sub-
filters. In more general scenarios involving $N$ devices and $2N$ sub-filters,
we assume that a successful round of the experiment is obtained only when all
filters return $\mathsf{f}={\checkmark}$. In fact, in most quantum experiments
a single detection failure is sufficient to invalidate the current observation
round666An exception is provided by experiments on loss-tolerant quantum error
correcting codes, which are specifically designed to retain quantum
information in a subspace even when some of the physical quantum systems are
lost..

Third, although it is not possible to verify if a measurement device satisfies
fair sampling in a device-independent way, fair sampling does impose
restrictions on the outcome probabilities. If these conditions are violated,
the experimenter can directly conclude that fair sampling does not hold. See
Appendix A for details.

Last, the classical filter is a stochastic map which, given an input
$\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|$ can
generate a probabilistic mixture of settings, i.e., the “real” setting is
$\sum_{\mathsf{x}^{\prime}}\Pr(\mathsf{x}^{\prime}|\mathsf{x})\left|{\mathsf{x}^{\prime}}\right\rangle\\!\\!\left\langle{\mathsf{x}^{\prime}}\right|$.
As a physical example, the experimenter may electronically set a rotation
angle $\theta_{0}$ of a polariser, but the polariser actually rotates by an
angle $\theta=\theta_{0}+\delta\theta$ (with $\delta\theta$ small and
stochastic) [37]. However, without loss of generality, we may assume that
$\mathcal{F}_{C}$ does not change the setting $\mathsf{x}$, i.e. it only
assigns different success probabilities to different settings. In fact, we can
simultaneously re-define $\bar{\mathcal{M}}$ and $\mathcal{F}_{C}$ so that the
definition of the physical device $\mathcal{M}$ remains unchanged while the
classical filter takes the form
$\mathcal{F}_{C,{\checkmark}}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)=\Pr({\checkmark}|\,\mathsf{x})\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|$,
as we show in Appendix B.

######  Proposition 1 (Equivalent formulations of fair sampling).

Consider a lossy measurement device $\mathcal{M}$ (as formalized in Section
2). The following three properties are equivalent:

  1. 1.

$\mathcal{M}$ satisfies fair sampling, as given in Definition 1.

  2. 2.

The efficiency of the device factorizes as:

$\displaystyle\mathcal{E}(\mathsf{x},\rho)\;=\;\mathcal{E}_{C}(\mathsf{x})\,\mathcal{E}_{Q}(\rho)$
(12)

for some real-valued functions $\mathcal{E}_{C}$ and $\mathcal{E}_{Q}$. This
is the definition that was given in Ref. [1].

  3. 3.

The POVM element $M_{\checkmark}$, associated to the set of good outcomes of
$\mathcal{M}$, factorizes as in

$\displaystyle M_{\checkmark}\;=\;M_{C,{\checkmark}}\otimes
M_{Q,{\checkmark}}$ (13)

where $M_{C,{\checkmark}}$ acts on the classical setting and
$M_{Q,{\checkmark}}$ acts on the quantum input. Moreover,
$M_{C,{\checkmark}}=\sum_{\mathsf{x}}\mathcal{E}_{C}(\mathsf{x})\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|$
is a diagonal matrix, equivalently,
$M_{\checkmark}^{\mathsf{x}}=\mathcal{E}_{C}(\mathsf{x})M_{Q,{\checkmark}}$
and therefore $M_{\checkmark}^{\mathsf{x}}$ and $M_{\checkmark}^{\mathsf{y}}$
are proportional for all $\mathsf{x},\mathsf{y}\in\mathsf{X}$.

###### Proof.

Eq. (11) immediately implies Eq. (12), since:

$\displaystyle\mathcal{E}(\mathsf{x},\rho)\;$
$\displaystyle=\;{\Pr}_{\bar{\mathcal{M}}\circ\mathcal{F}}({\checkmark}|\mathsf{x},\rho)\;=\;{\Pr}_{\mathcal{F}}({\checkmark}|\mathsf{x},\rho)$
$\displaystyle=\;{\Pr}_{\mathcal{F}_{C}}({\checkmark}|\mathsf{x})\,{\Pr}_{\mathcal{F}_{Q}}({\checkmark}|\rho)$
$\displaystyle\equiv\;\mathcal{E}_{C}(\mathsf{x})\,\mathcal{E}_{Q}(\rho)\;.$
(14)

To show that Eq. (12) implies Eq. (13), we use the fact that the efficiency
$\mathcal{E}:S(\mathcal{H}_{\mathsf{X}}\otimes\mathcal{H})\rightarrow[0,1]$ is
a CP linear map, hence $\mathcal{E}_{C}$ and $\mathcal{E}_{Q}$ are also linear
maps (over probabilistic mixtures of inputs) and we can assume, without loss
of generality, that both maps take values in $[0,1]$. Since any CP linear map
$\mathcal{E}$ taking value in $[0,1]$ can be written as
$\mathcal{E}(\,\cdot\,)=\mathrm{Tr}(M\,\cdot\,)$, for some matrix satisfying
$0\preccurlyeq M\preccurlyeq\mathds{1}$, we have:

$\displaystyle\mathcal{E}(\mathsf{x},\rho)\;$
$\displaystyle=\;\mathcal{E}_{C}(\mathsf{x})\,\mathcal{E}_{Q}(\rho)$
$\displaystyle\equiv\;\mathrm{Tr}(M_{C,{\checkmark}}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)\,\mathrm{Tr}(M_{Q,{\checkmark}}\rho)$
$\displaystyle=\;\mathrm{Tr}\big{[}\,(M_{C,{\checkmark}}\otimes
M_{Q,{\checkmark}})\,(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\rho)\,\big{]}\;,$
(15)

for some matrices $M_{C,{\checkmark}}$ and $M_{Q,{\checkmark}}$. Hence,
defining $M_{\checkmark}:=M_{C,{\checkmark}}\otimes M_{Q,{\checkmark}}$, Eq.
(13) holds. Moreover,
$M_{C,{\checkmark}}=\sum_{\mathsf{x}}\mathcal{E}_{C}(\mathsf{x})\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|$
is diagonal.

Finally, to show that Eq. (13) implies Eq. (11), we construct filters
$\mathcal{F}_{C}$ and $\mathcal{F}_{Q}$ as in Eq. (5), that is, via the Kraus
operators:

$\displaystyle\begin{split}F_{C,{\checkmark}}\,&:=\;\left|{{\checkmark}}\right\rangle\otimes\sqrt{M_{C,{\checkmark}}}\,,\\\
F_{Q,{\checkmark}}\,&:=\;\left|{\textup{\,\o\,}}\right\rangle\otimes\sqrt{M_{Q,{\checkmark}}}\,.\end{split}$
(16)

We then set
$\mathcal{F}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\rho):=\wedge\left[\;\mathcal{F}_{C}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)\otimes\mathcal{F}_{Q}(\rho)\;\right]$,
and defining the POVM elements of a lossless device $\bar{\mathcal{M}}$ as in
Section 2.2 allows one to show that
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ holds.

∎

We now provide a result that motivates using the factorization in Eq. (11) as
definition of fair sampling. The formalism of filters turns out to be very
handy: the essence of the proof Proposition 2 is illustrated in Figure 2.

Figure 2: Example of a two-party experiment (e.g., a test of a Bell
inequality). The picture schematically illustrates how the factorization
property of Definition 1 leads to fair sampling.

######  Proposition 2 (Probabilities achievable under fair sampling).

Consider any (real) experiment involving a source that produces an $N$-partite
quantum state $\Psi\in
S(\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\ldots\otimes\mathcal{H}_{N})$,
with $N\geq 1$. The state $\Psi$ is measured by a collection of $N$ lossy
detectors,
$\overrightarrow{\mathcal{M}}:=(\mathcal{M}_{1},\mathcal{M}_{2},\ldots,\mathcal{M}_{N})$,
each acting on a local sub-system of $\Psi$ and each satisfying fair sampling
as in Definition 1. Call
$\vec{\mathsf{x}}:=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{N})$ the
collection of $N$ settings of the devices in a given experimental run, and
call $\vec{\mathsf{a}}:=(\mathsf{a}_{1},\mathsf{a}_{2},\ldots,\mathsf{a}_{N})$
the collection of $N$ measurement outcomes777Each party $k$ may have a
different set of inputs and outputs, $\mathsf{x}_{k}\in\mathsf{X}_{k}$ and
$\mathsf{a}_{k}\in\mathsf{A}_{k}$., each of which could also be a “no-click”
event  ø . A round of the experiment is successful when all the detectors
click, and the post-selected statistics is obtained by restricting only to the
successful runs.

Then, there is an (ideal) experiment involving a source of a quantum state
$\Psi_{\checkmark}$ (defined on the same Hilbert space as $\Psi$) and $N$
lossless measurement devices
$\overrightarrow{\mathcal{M}}^{\checkmark}:=(\mathcal{M}_{1}^{\checkmark},\mathcal{M}_{2}^{\checkmark},\ldots,\mathcal{M}_{N}^{\checkmark})$
whose outcome statistics is equal to the post-selected statistics of the real
experiment. That is, we have:

$\displaystyle{\Pr}^{\textup{p.s.}}_{\overrightarrow{\mathcal{M}}}\big{(}\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi\big{)}\;\equiv\;\frac{{\Pr}_{\overrightarrow{\mathcal{M}}}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi)}{{\Pr}_{\overrightarrow{\mathcal{M}}}({\checkmark}|\,\vec{\mathsf{x}},\Psi)}\;=\;{\Pr}_{\overrightarrow{\mathcal{M}}^{\checkmark}}\big{(}\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi_{\checkmark}\big{)}\;,$
(17)

restricting to the settings $\vec{\mathsf{x}}$ such that
${\Pr}_{\overrightarrow{\mathcal{M}}}({\checkmark}|\,\vec{\mathsf{x}},\Psi)\neq
0$888A setting $\vec{\mathsf{y}}$ having zero acceptance probability
effectively can be erased from the set of allowed settings, since
$\vec{\mathsf{y}}$ does not appear in the post-selected data. Then, Eq. (17)
will hold on the restricted collection of settings.. Moreover, the state
$\Psi_{\checkmark}$ can be obtained probabilistically from $\Psi$ via
probabilistic local operations.

###### Proof.

We decompose each measurement device $\mathcal{M}_{k}$, acting on the $k$-th
subsystem, as a filter followed by a lossless measurement,
$\mathcal{M}_{k}=\bar{\mathcal{M}}_{k}\circ\mathcal{F}_{k}$. Moreover, each
filter $\mathcal{F}_{k}$ factorizes as:

$\displaystyle\mathcal{F}_{k}\big{(}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\rho\big{)}\;=\;\wedge\left[\;\mathcal{F}_{k,C}\big{(}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\big{)}\otimes\mathcal{F}_{k,Q}\big{(}\rho\big{)}\;\right]\;,$
(18)

and each sub-filter can herald either a success or a failure:

$\displaystyle\begin{split}\mathcal{F}_{k,C}\;&=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\mathcal{F}_{k,C,{\checkmark}}\,+\;\left|{\textup{\,\o\,}}\right\rangle\\!\\!\left\langle{\textup{\,\o\,}}\right|\otimes\mathcal{F}_{k,C,\textup{\,\o\,}}\\\
\mathcal{F}_{k,Q}\;&=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\mathcal{F}_{k,Q,{\checkmark}}\,+\;\left|{\textup{\,\o\,}}\right\rangle\\!\\!\left\langle{\textup{\,\o\,}}\right|\otimes\mathcal{F}_{k,Q,\textup{\,\o\,}}\;.\end{split}$
(19)

We then define the normalized quantum state:

$\displaystyle\Psi_{\checkmark}\;:=\;\frac{1}{\mathcal{E}_{Q}(\Psi)}\Big{(}{\bigotimes}_{k}\,\mathcal{F}_{k,Q,{\checkmark}}\Big{)}(\Psi)\;,$
(20)

where
$\mathcal{E}_{Q}(\Psi)=\mathrm{Tr}\big{[}\big{(}\bigotimes_{k}\mathcal{F}_{k,Q,{\checkmark}}\big{)}(\Psi)\big{]}$
is the probability that all the filters $\mathcal{F}_{k,Q}$ return
${\checkmark}$, and we assume $\mathcal{E}_{Q}(\Psi)>0$. The physical
interpretation of $\Psi_{\checkmark}$ is as follows: apply to the $N$ sub-
systems of $\Psi$ the filters $\mathcal{F}_{k,Q}$ and post-select on all of
them returning ${\checkmark}$ at the same time. That is to say, it is possible
to perform a heralded preparation of $\Psi_{\checkmark}$.

Analogously, we can define lossless measurement devices
$\mathcal{M}_{k}^{\checkmark}$ for each party $k$ through the maps
$(\mathcal{M}_{k}^{\checkmark})^{\mathsf{x}}_{\mathsf{a}}:S(\mathcal{H}_{k})\to[0,1]$,

$\displaystyle\big{(}\mathcal{M}_{k}^{\checkmark}\big{)}^{\mathsf{x}}_{\mathsf{a}}\,(\,\cdot\,)\;:=\;\frac{1}{\mathcal{E}_{k,C}(\mathsf{x})}\mathrm{Tr}\Big{\\{}\big{(}\bar{M}_{k}\big{)}_{\mathsf{a}}\;\big{[}\,\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\mathcal{F}_{k,C,{\checkmark}}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)\otimes\,\cdot\;\big{]}\Big{\\}}\;.$
(21)

Here $(\bar{M}_{k})_{\mathsf{a}}$ are the POVM elements associated to
$\bar{\mathcal{M}}_{k}$ as, e.g., in Eq. (6), and we assume
$\mathcal{E}_{k,C}(\mathsf{x})=\mathrm{Tr}[\,\mathcal{F}_{k,C,{\checkmark}}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)\,]>0$
for all parties999We can interpret $\mathcal{M}_{k}^{\checkmark}$ as follows.
We repeatedly send some value $\mathsf{x}$ to the filter $\mathcal{F}_{k,C}$,
until this accepts it. In this moment, the measurement device
$\mathcal{M}^{\checkmark}_{k}$ has been “switched on” and it is ready to
measure an incoming quantum state $\rho$ (without failure): this is a heralded
activation of $\mathcal{M}^{\checkmark}_{k}$.. We also remark that, as shown
in Appendix B, we can assume without loss of generality that the filters do
not change the setting,
$\mathcal{F}_{k,C,{\checkmark}}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)=\mathcal{E}_{k,C}(\mathsf{x})\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|$;
in this case Eq. (21) simply gives
$\mathcal{M}_{k}^{\checkmark}=\bar{\mathcal{M}}_{k}$.

As the final step of the proof, we have to compute the probability of
obtaining a collection of outcomes $\vec{\mathsf{a}}$ given inputs
$\vec{\mathsf{x}}$ when measuring the state $\Psi_{\checkmark}$ with the
devices $\overrightarrow{\mathcal{M}}^{\checkmark}$, and verify that Eq. (17)
is satisfied. This is a simple formal manipulation, which is given in
Appendix C.

∎

Proposition 2 is very powerful and general: it ensures that the post-selected
statistics obtained using lossy devices is equal to the data one could collect
with ideal lossless devices measuring the filtered state $\Psi_{\checkmark}$.
This has several interesting consequences.

For example, in a scenario where the post-selected data violates a Bell
inequality [3, 4], we can conclude under the fair sampling assumption that the
state $\Psi_{\checkmark}$ is Bell-correlated [38]. This implies in particular
that $\Psi$ itself contains hidden nonlocality [39]. Additional information
about the state $\Psi_{\checkmark}$ can also be obtained from existing device-
independent tools: if the Bell violation is high enough, the post-selected
data can be used in the framework of self-testing [18, 19], to certify
directly the exact structure of $\Psi_{\checkmark}$ or as a step aiming to
certify quantum operations including quantum gates [24] or entangling
measurements [40, 41, 42].

We remark, furthermore, that Proposition 2 can be generalized to cases in
which some devices are fully characterized, while others are not. In such
cases, one needs to appeal to fair sampling only for the not-fully-
characterized devices. For instance, in _quantum steering_ a trusted and an
untrusted quantum device jointly measure an entangled quantum state $\Psi$
[43]: here, we need to apply fair sampling only to the (single) untrusted
device.

Finally, we note that if $\Psi$ is a separable quantum state, then
$\Psi_{\checkmark}$ is also separable since it can be obtained from $\Psi$
through local probabilistic operations. Consequently, a separable quantum
state cannot violate any Bell inequality when using lossy detectors and post-
selection, if fair sampling holds. Similarly, under fair sampling it is
impossible to violate the quantum bound of a Bell operator with post-selected
statistics: in fact, the models and experiments showing that post-selection
can lead to incorrect conclusions [5, 6, 7, 8, 9, 10, 11, 12] do not satisfy
fair sampling.

##  4 The strong and homogeneous fair sampling assumptions

Now we consider two special cases of fair sampling, which we call _strong_ and
_homogeneous_ fair sampling. Incidentally, strong homogeneous fair sampling is
what, by and large, is implicitly assumed as the standard definition of fair
sampling [1, 28, 29].

######  Definition 2 (Strong fair sampling).

Consider a measurement device $\mathcal{M}$ that satisfies fair sampling, as
in Definition 1. The sub-filters $\mathcal{F}_{Q}$ and $\mathcal{F}_{C}$
herald either a failure or a success, hence they can be written as:

$\displaystyle\begin{split}\mathcal{F}_{C}\;&=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\mathcal{F}_{C,{\checkmark}}\;+\;\left|{\textup{\,\o\,}}\right\rangle\\!\\!\left\langle{\textup{\,\o\,}}\right|\otimes\mathcal{F}_{C,\textup{\,\o\,}}\\\
\mathcal{F}_{Q}\;&=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\mathcal{F}_{Q,{\checkmark}}\;+\;\left|{\textup{\,\o\,}}\right\rangle\\!\\!\left\langle{\textup{\,\o\,}}\right|\otimes\mathcal{F}_{Q,\textup{\,\o\,}}\;.\end{split}$
(22)

We say that $\mathcal{M}$ satisfies _strong_ fair sampling if there exists a
decomposition $\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ such that the
sub-filter $\mathcal{F}_{Q,{\checkmark}}$ is proportional to the identity
channel: $\mathcal{F}_{Q,{\checkmark}}=P_{Q,{\checkmark}}\textup{Id}$, where
$P_{Q,{\checkmark}}>0$ is the success probability. Moreover, if it is possible
to write $\mathcal{F}_{C,{\checkmark}}=P_{C,{\checkmark}}\textup{Id}$ for some
success probability $P_{C,{\checkmark}}>0$, we say that $\mathcal{M}$
satisfies _homogeneous_ fair sampling.

The notion of _homogeneous_ fair sampling is introduced in analogy with the
notion of _strong_ fair sampling and it means that the detection probability
is independent of $\mathsf{x}$. However, not much is to be gained from the
homogeneity property. This is because the classical input $\mathsf{x}$ is
under full control of the experimenter, therefore it is possible to directly
estimate the efficiency $\mathcal{E}_{C}(\mathsf{x})$ from the experimental
data (assuming that fair sampling holds) and one can compensate any
inhomogeneity in the efficiency. However, in quantum experiments it often
happens that devices that (approximately) satisfy fair sampling naturally also
possess the homogeneity property. This is the case, e.g., for the
_polarization analyser_ , a measurement device that we examine in Section 6.

In contrast, _strong_ fair sampling does give more stringent guarantees on the
experiment. In fact, the state $\Psi_{\checkmark}$ which reproduces the post-
selected statistics is obtained from the state $\Psi$ after the application of
the filters $\mathcal{F}_{k,Q}$. But if these filters are all proportional to
the identity channel, this implies $\Psi_{\checkmark}=\Psi$. That is, under
the strong fair sampling assumption, the post-selected statistics is a fair
representation of the statistics that would be obtained with unit-efficiency
detectors.

More in detail, consider again a $N$-local quantum experiment involving $N$
detectors each acting on a part of an entangled quantum state $\Psi$. If all
detectors
$\overrightarrow{\mathcal{M}}=(\mathcal{M}_{1},\mathcal{M}_{2},\ldots,\mathcal{M}_{N})$
satisfy strong fair sampling, then we have that the post-selected
probabilities are equal to those that one can obtain acting on the actual
experimental state $\Psi$ with some lossless devices
$\overrightarrow{\mathcal{M}}^{\checkmark}$:

$\displaystyle{\Pr}^{\textup{p.s.}}_{\overrightarrow{\mathcal{M}}}\big{(}\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi\big{)}\;\equiv\;\frac{{\Pr}_{\overrightarrow{\mathcal{M}}}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi)}{{\Pr}_{\overrightarrow{\mathcal{M}}}({\checkmark}|\,\vec{\mathsf{x}},\Psi)}\;=\;{\Pr}_{\overrightarrow{\mathcal{M}}^{\checkmark}}\big{(}\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi\big{)}\;.$
(23)

Consequently, if strong fair sampling holds, any claim on $\Psi$ based on the
post-selected probabilities will be as good as if the data was produced by
lossless devices. That is, the conclusions about non-locality, device-
independent entanglement, self-testing, and so on, will hold even if they are
based on post-selected data.

However, this strong notion of fair sampling comes at a cost: the conditions
specified in Definition 2 are very stringent and rarely met in actual quantum
experiments. We therefore recommend the usage of weak fair sampling as the
default definition to be used in experimental settings. The notion of weak
fair sampling is more widely applicable and, as argued in Section 3, it still
allows to obtain sharp conclusions based on the post-selected statistics.

Finally, we specialize Proposition 1 to the case of strong fair sampling.

######  Proposition 3 (Equivalent formulations of strong fair sampling).

Consider a lossy measurement device $\mathcal{M}$ which satisfies fair
sampling, as in Definition 1. Then, $\mathcal{M}$ satisfies _strong_ fair
sampling if and only if its efficiency is
$\mathcal{E}(\mathsf{x},\rho)=\mathcal{E}_{C}(\mathsf{x})$ or, equivalently,
if and only if the POVM element associated to ${\checkmark}$ has the form
$M_{\checkmark}=M_{C,{\checkmark}}\otimes\mathds{1}_{Q}$.

###### Proof.

It is an immediate consequence of specializing the proof of Proposition 1 to
the cases in which $\mathcal{F}_{Q}$ is proportional to the identity map.

∎

##  5 Fair sampling in cryptographic scenarios

Under fair sampling, the post-selected measurement results of a Bell test can
also be used for cryptographic tasks, e.g., for random number generation or
quantum key distribution [21, 26, 27]. The fair sampling assumption has to be
added to the standard _no-leakage_ assumption (i.e., no information about the
actual measurement is leaked to the adversary) as required also for fully
device-independent quantum cryptography [44]. Moreover, device-independent
certifications can be used as subroutines in more complex cryptograhic
protocols. If a certification is performed employing the fair sampling
assumption, then the larger cryptographic protocol will be secure provided
that such certification is _securely composable_ [45] and, naturally, provided
that the fair sampling assumption is valid to begin with.

It is important to note that different parties can have disparate POVM
descriptions of the same measurement device, reflecting the degree of
knowledge that each party has; in particular, an attacker might have access to
some hidden information about the internal functioning of the devices. This
also means that parties having different POVM descriptions of one and the same
detector might disagree on whether it satisfies fair sampling. We argue that
protocols involving lossy devices and post-selection are secure if fair
sampling holds unconditionally, and in particular also _according to the POVM
description possessed by the adversary_.

###  5.1 Explicit attack model

We consider the so-called _Makarov attack_ [46] as an example where fair
sampling appears to hold to the honest parties, but is violated according to
the fuller description possessed by the adversary. A detector having settings
$\mathsf{X}=\\{0,1\\}$ and outputs
$\mathsf{A}^{\prime}=\\{\textup{\,\o\,},+,-\\}$ purportedly performs the
following lossy measurement:

$\displaystyle\begin{array}[]{ll}M_{+}^{0}\;=\;\frac{1}{4}\left|{0}\right\rangle\\!\\!\left\langle{0}\right|&\quad
M_{+}^{1}\;=\;\frac{1}{4}\left|{+}\right\rangle\\!\\!\left\langle{+}\right|\\\
M_{-}^{0}\;=\;\frac{1}{4}\left|{1}\right\rangle\\!\\!\left\langle{1}\right|&\quad
M_{-}^{1}\;=\;\frac{1}{4}\left|{-}\right\rangle\\!\\!\left\langle{-}\right|\end{array}$
(26)

with
$\left|{\pm}\right\rangle:=(\left|{0}\right\rangle\pm\left|{1}\right\rangle)/\sqrt{2}$
and thus the no-click events are
$M_{\textup{\,\o\,}}^{0}=M_{\textup{\,\o\,}}^{1}=\frac{3}{4}\mathds{1}$. These
measurements are the ones canonically used in CHSH-Bell tests, apart from
having efficiency $1/4$, and they seem to satisfy (strong and homogeneous)
fair sampling. However, the device is maliciously designed by the adversary:
in actuality, it draws uniformly a random variable $r\in\\{1,2,3,4\\}$, which
is unknown to the user, and performs a different measurement depending on $r$.
Specifically:

$\displaystyle\begin{array}[]{r|cccc}&r=1&r=2&r=3&r=4\\\ \hline\cr\\\\[-9.0pt]
M_{+}^{0}&\left|{0}\right\rangle\\!\\!\left\langle{0}\right|&0&0&0\\\\[1.0pt]
M_{-}^{0}&0&\left|{1}\right\rangle\\!\\!\left\langle{1}\right|&0&0\\\\[1.0pt]
M_{\textup{\,\o\,}}^{0}&\left|{1}\right\rangle\\!\\!\left\langle{1}\right|&\left|{0}\right\rangle\\!\\!\left\langle{0}\right|&\mathds{1}&\mathds{1}\\\\[2.0pt]
\hline\cr
M_{+}^{1}&0&0&\left|{+}\right\rangle\\!\\!\left\langle{+}\right|&0\\\\[1.0pt]
M_{-}^{1}&0&0&0&\left|{-}\right\rangle\\!\\!\left\langle{-}\right|\\\\[2.0pt]
M_{\textup{\,\o\,}}^{1}&\mathds{1}&\mathds{1}&\left|{-}\right\rangle\\!\\!\left\langle{-}\right|&\left|{+}\right\rangle\\!\\!\left\langle{+}\right|\\\\[2.0pt]
\end{array}$ (35)

so that tracing out the value of $r$ results in the POVM of Eq. (26). In a
cryptographic protocol, the parties must communicate when the detection has
succeeded (over a classical authenticated channel). However, this piece of
information, together with the knowledge of $r$101010The adversary can know
the value of $r$ even under the no-leakage assumption. For instance, a long
list of random choices of $r$ could have been stored in a hidden memory inside
the device, or they could be obtained algorithmically by a pseudo-random
number generator., allows the adversary to infer the setting $\mathsf{x}$ and
the outcome $\mathsf{a}$ for each successful detection, completely
compromising the security. We emphasize that, in this example, from the point
of view of the adversary the POVMs describing the detector have the elements

$\displaystyle\begin{array}[]{ll}M_{+}^{0}\;=\;\left|{0}\right\rangle\\!\\!\left\langle{0}\right|\otimes\left|{1}\right\rangle\\!\\!\left\langle{1}\right|_{r}&\quad
M_{+}^{1}\;=\;\left|{+}\right\rangle\\!\\!\left\langle{+}\right|\otimes\left|{3}\right\rangle\\!\\!\left\langle{3}\right|_{r}\\\
M_{-}^{0}\;=\;\left|{1}\right\rangle\\!\\!\left\langle{1}\right|\otimes\left|{2}\right\rangle\\!\\!\left\langle{2}\right|_{r}&\quad
M_{-}^{1}\;=\;\left|{-}\right\rangle\\!\\!\left\langle{-}\right|\otimes\left|{4}\right\rangle\\!\\!\left\langle{4}\right|_{r},\end{array}$
(38)

which act on the the quantum input of the detector but also on the random
variable $r$ which is chosen uniformly at random by the attacker. Manifestly,
the POVMs above violate the fair sampling assumption, which corrupts the
security guarantees. However, if the variable $r$ is held by the environment
(as a physical realization of the loss channel) and can not be accessed by the
adversary, her POVM description of the detector is no better than the one of
Eq. (26).

Moreover, with two detectors that implement Makarov’s attack, it is possible
to fake the violation of the CHSH inequality using only separable states, and
even saturate the algebraic bound of the operator [7]. That is, calling
$\mathsf{x},\mathsf{y}\in\\{0,1\\}$ the two inputs and
$\mathsf{a},\mathsf{b}\in\\{+,-\\}$ the post-selected outputs, these always
satisfy $\mathsf{a}\,\mathsf{b}=(-1)^{\mathsf{x}\,\mathsf{y}}$. This can be
realized by sending to the two detectors a state $\rho_{r_{A},r_{B}}$ which
explicitly depends on the hidden parameters $r_{A}$ and $r_{B}$ of the first
and second device. A possible choice for $\rho_{r_{A},r_{B}}$ is given in the
following table:

$\displaystyle\begin{array}[]{l|cccc}&r_{B}=1&r_{B}=2&r_{B}=3&r_{B}=4\\\
\hline\cr\\\\[-9.0pt] r_{A}=1&[0,0]&\mathrm{vac}&[0,+]&\mathrm{vac}\\\\[1.0pt]
r_{A}=2&\mathrm{vac}&[1,1]&\mathrm{vac}&[1,-]\\\\[1.0pt]
r_{A}=3&[+,0]&\mathrm{vac}&\mathrm{vac}&[+,-]\\\\[1.0pt]
r_{A}=4&\mathrm{vac}&[-,1]&[-,+]&\mathrm{vac}\end{array}$ (45)

where $[p,q]$ is a shorthand for the separable state
$\left|{p}\right\rangle\\!\\!\left\langle{p}\right|\otimes\left|{q}\right\rangle\\!\\!\left\langle{q}\right|$,
while “vac” denotes the vacuum state which, naturally, does not trigger the
detector; taking into account the detector efficiencies and the vacuum
component, the overall detection probability is $1/32=1/4\times 1/4\times
1/2$. Of course, surpassing the quantum bound of the CHSH operator would
reveal that something suspicious is happening. But by adding some white noise
in the input states, the expectation value of the CHSH operator can be brought
in the region between the classical and quantum limit.

In conclusion, we have to impose that the provided POVM description represents
the “true” behaviour of the measurement device and that therefore we can be
confident that fair sampling holds also according to the adversary’s
description of the device. In this way, Proposition 2 can be immediately
applied to prove security against this adversary, to whom the post-selected
statistics are as good as if it were produced by an ideal device. It may seem
rather artificial to impose fair sampling in this form, but we argue that
there are cryptographic scenarios where this assumption might be plausible.
For example, imagine that a trusted company manufactures and sells lossy
measurement devices that are guaranteed to work properly, but no information
about the internal functioning is provided to the end-users (e.g., as a way to
preserve industrial secrets) so that they can only model the device as a black
box. A third-party attacker might be able to study a copy of the device,
reverse-engineer its functioning, obtain a full POVM description, and discover
that fair sampling holds indeed. This would be a setting were Proposition 2
can be put to use and security guarantees recovered, without further
assumptions on the structure or calibration of the measurements.

###  5.2 Extending to non i.i.d. settings

In this Section we briefly discuss what happens when we do not assume that the
measurement statistics are independent and identically distributed (i.i.d.),
as done in the rest of the paper. Under no-leakage, the most general attack
strategy is obtained when the device internally stores all the quantum states
that have been received (and measured) in the past, so that the measurement
outcome in a certain round can depend on all the states (and measurement
results) at previous rounds. Consequently, the device acts on a Hilbert space
whose dimension increases in time and it has a different POVM description at
every round. This enables very sophisticated and convoluted attack strategies
that are very far from the contexts where fair sampling is a reasonable
assumption. Nonetheless, considering fair sampling in non i.i.d. settings may
be necessary when the measurement devices are not stable in time, e.g. as
consequence of thermal drift.

Consider now running a Bell experiment where the state preparation and the
measurements are repeated for $T$ rounds, in order to collect statistics, and
remove the i.i.d. assumption, i.e., the devices may behave differently in each
of the $T$ rounds (and thus the outcomes are not identically distributed) and
there may be memory effect (so that the outcomes are not independent). Are the
results of Proposition 2 applicable in these cases?

We remark that the fair sampling assumption can be extended to the non i.i.d.
case in different ways. To keep the analysis simple, here we propose a
redefinition of fair sampling which allows to extend the results of
Proposition 2 in a straightforward manner. More general definitions of fair
sampling may still be viable, but they would require a more careful analysis.

######  Definition 3 (Fair sampling in non i.i.d. settings).

Consider a lossy measurement device $\mathcal{M}^{(T)}$ that is used for $T$
rounds in a protocol or experiment. In a general formulation,
$\mathcal{M}^{(T)}$ consists of $T$ devices
$\mathcal{M}_{1},\ldots,\mathcal{M}_{T}$ where each device $\mathcal{M}_{t}$
has a setting $\mathsf{x}_{t}$, an outcome $\mathsf{a}_{t}$ and measures a
(entangled) state $\rho_{t}\in S(\mathcal{H}_{t})$; moreover,
$\mathcal{M}_{t}$ can access an internal memory storing the classical and
quantum information it has received or produced during all previous rounds.
Each lossy device $\mathcal{M}_{t}$ admits a decomposition
$\mathcal{M}_{t}=\bar{\mathcal{M}}_{t}\circ\mathcal{F}_{t}$, where
$\bar{\mathcal{M}}_{t}$ is a lossless device and $\mathcal{F}_{t}$ a filter.
Both $\bar{\mathcal{M}}_{t}$ and $\mathcal{F}_{t}$ can act on all the
information available at round $t$.

We say that $\mathcal{M}^{(T)}$ satisfies fair sampling if and only if each
filter $\mathcal{F}_{t}$ acts only on $\rho_{t}$ and $\mathsf{x}_{t}$ (i.e.,
it acts as the identity channel on the internal memory of the device) and
moreover $\mathcal{F}_{t}$ factorizes as
$\mathcal{F}_{t}=\wedge[\mathcal{F}_{t,C}\otimes\mathcal{F}_{t,Q}]$, as in Eq.
(11).

Similarly as done in Proposition 2, we now consider a Bell experiment
involving $N$ physically separated non-communicating devices, used for $T$
consecutive rounds, which collectively measure a global quantum state $\Psi$
having $N\times T$ subsystems. A round $t$ of the experiment is successful if
and only if all the $N$ devices outputs a ${\checkmark}$ flag simultaneously;
we then collect the sequence of successes/failures in a list
$\vec{f}=\\{\mathsf{f}_{1},\ldots,\mathsf{f}_{T}\\}$ with
$\mathsf{f}_{t}\in\\{\textup{\,\o\,},{\checkmark}\\}$ for each $t$. Notice
that the probability that all rounds are successful is exponentially small in
$T$ and will essentially never occur when $T$ is large, as is required for
having a significant statistical sample. Thus, we suppose that only
$T^{\prime}$ rounds are successful (with $T^{\prime}\leq T$) and introduce a
filtered state $\Psi_{\vec{\mathsf{f}}}$ which is obtained by applying all the
$N\times T^{\prime}$ local filters corresponding to the successful rounds and
tracing out the systems involved in unsuccessful rounds.

Following the line of reasoning given in Proposition 2 we can now see that the
post-selected measurement statistics of the real experiment is exactly
reproduced by the measurement statistics of an ideal experiment where a
filtered state $\Psi_{\vec{\mathsf{f}}}$ is measured by lossless
devices111111Notice that the filtered state $\Psi_{\vec{\mathsf{f}}}$ now
explicitly depends on the sequence of flags $\vec{\mathsf{f}}$ obtained in the
experiment. This is not an issue, since in a device-independent protocol the
state $\Psi$ is arbitrary to begin with.. Therefore, in this non i.i.d.
version of fair sampling we are able to certify in a device-independent way
the properties of a quantum state $\Psi_{\vec{\mathsf{f}}}$ that could have
been obtained with local probabilistic operations from the state that was
present in the lab. Hence, the security proofs that have been developed for
certifiable random number generation [26, 27, 47, 48, 49] and device-
independent quantum key distribution [21, 50, 51, 52] immediately extend to
these devices that satisfy the non i.i.d. version of fair sampling.

##  6 Fair sampling in quantum optics experiments

In this Section we provide a concrete example of a measurement apparatus
commonly used in quantum optics, with the intent of showing a possible real-
world application of our formalism and also to better understand the
conditions that are required to satisfy weak and strong fair sampling,
illuminating also how to deal with the case where losses are present prior to
the measurement. Specifically, we consider a _polarization analyser_ , a
device which allows measuring the polarization of incoming photons in
arbitrary directions [53, 54] and the classical setting is the measurement
angle $(\mathsf{x}\equiv\theta)$. The layout of the apparatus is schematically
depicted in Figure 3.

###  6.1 Description of a polarization analyser

We assume that the photons entering the apparatus are described by bosonic
operators $b^{\dagger}_{H}$ and $b^{\dagger}_{V}$ for the horizontal and
vertical polarization, respectively. These operators satisfy canonical
commutation relations
$[b_{H},b_{H}^{\dagger}]=[b_{V},b_{V}^{\dagger}]=\mathds{1}$. A polarizing
beam-splitter (PBS) is followed by two non-photon-number-resolving (NPNR)
detectors $D_{1}$ and $D_{2}$ having POVM elements

$\displaystyle
D_{1}:\begin{cases}E_{\textup{\,\o\,}}^{\theta}=(1-\eta_{1})^{b_{\theta}^{\dagger}b_{\theta}}~{}~{}~{}~{}~{}\,=R_{1}^{\hat{N}_{\theta}}\\\
E_{\checkmark}^{\theta}=\mathds{1}-(1-\eta_{1})^{b_{\theta}^{\dagger}b_{\theta}}=\mathds{1}-R_{1}^{\hat{N}_{\theta}}\end{cases}~{}D_{2}:\begin{cases}E_{\textup{\,\o\,}}^{\theta^{\bot}}=(1-\eta_{2})^{b_{\theta^{\bot}}^{\dagger}b_{\theta^{\bot}}}~{}~{}~{}~{}~{}\,=R_{2}^{\hat{N}_{\theta^{\bot}}}\\\
E_{\checkmark}^{\theta^{\bot}}=\mathds{1}-(1-\eta_{2})^{b_{\theta^{\bot}}^{\dagger}b_{\theta^{\bot}}}=\mathds{1}-R_{2}^{\hat{N}_{\theta^{\bot}}}.\end{cases}$
(46)

Here, $\eta_{1}=1-R_{1}$ and $\eta_{2}=1-R_{2}$ are the efficiencies of
detectors $D_{1}$ and $D_{2}$. Note also that
$\hat{N}_{\theta}=b_{\theta}^{\dagger}b_{\theta}$ and
$\hat{N}_{\theta^{\bot}}=b_{\theta^{\bot}}^{\dagger}b_{\theta^{\bot}}$ are the
number operators associated to photons having $\theta$ and $\theta^{\bot}$
polarization, respectively. The angles $\theta,\theta^{\bot}$ are chosen using
a variable polarization rotator (PR) preceding the PBS and are linked to the
horizontal/vertical polarization via

$\displaystyle\bigg{(}\begin{array}[]{c}b_{\theta}^{\dagger}\\\
b_{\theta^{\bot}}^{\dagger}\end{array}\bigg{)}\;:=\;\bigg{(}\begin{array}[]{rr}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\bigg{)}\bigg{(}\begin{array}[]{c}b_{H}^{\dagger}\\\
b_{V}^{\dagger}\end{array}\bigg{)}\;.$ (53)

If we assume, for the moment being, that the two detectors have equal
efficiency $\eta_{1}=\eta_{2}=\eta=1-R,$ the four POVM elements of the
polarization analyser are:

$\displaystyle\begin{split}M_{\textup{\,\o\,}}^{\theta}\,:=\;E_{\textup{\,\o\,}}^{\theta}\\!\otimes\\!E_{\textup{\,\o\,}}^{\theta^{\bot}}\;&=\;R^{\hat{N}_{\theta}}\,R^{\hat{N}_{\theta^{\bot}}}\
=\ R^{\hat{N}}\\\
M_{1}^{\theta}\;:=\;E_{\textup{\,\o\,}}^{\theta}\\!\otimes\\!E_{\checkmark}^{\theta^{\bot}}\;&=\;R^{\hat{N}_{\theta}}\,\big{(}\mathds{1}-R^{\hat{N}_{\theta^{\bot}}}\big{)}\\\
M_{2}^{\theta}\;:=\;E_{\checkmark}^{\theta}\\!\otimes\\!E_{\textup{\,\o\,}}^{\theta^{\bot}}\;&=\;\big{(}\mathds{1}-R^{\hat{N}_{\theta}}\big{)}\,R^{\hat{N}_{\theta^{\bot}}}\\\
M_{1\&2}^{\theta}\;:=\;E_{\checkmark}^{\theta}\\!\otimes\\!E_{\checkmark}^{\theta^{\bot}}\;&=\;\big{(}\mathds{1}-R^{\hat{N}_{\theta}}\big{)}\,\big{(}\mathds{1}-R^{\hat{N}_{\theta^{\bot}}}\big{)}\;.\end{split}$
(54)

Here, the tensor product represents the bipartition across detectors $D_{1}$
and $D_{2}$. In the first line we have used
$[\hat{N}_{\theta},\hat{N}_{\theta^{\bot}}]=0$ and introduced
$\hat{N}=\hat{N}_{\theta}+\hat{N}_{\theta^{\bot}}.$

Figure 3: A polarization analyser. The polarization of the incoming photons is
transformed by a variable polarization rotator (PR); then, the horizontal and
vertical components are separated through a polarizing beam-splitter (PBS) in
two optical paths; finally, the photons are collected by two non-photon-
number-resolving (NPNR) detectors $D_{1}$ and $D_{2}$.

###  6.2 Fair sampling with polarization analysers

We now show that the polarization analyser previously described (with
detectors having equal efficiencies $\eta$ and no dark counts) satisfies the
fair sampling assumption. We consider the general case where an arbitrary
number of photons can enter the apparatus and, in particular, the input state
$\rho$ may have a zero-photon (i.e. vacuum) component. Notice that no-click
events are caused by two distinct physical processes: either the photons are
lost before reaching the apparatus, or some photons reached it, but were not
detected. Distinguishing between these two processes is not always possible
but also, especially in a device-independent context, not necessary. When
dealing with states having a vacuum component, no-clicks would thus occur even
while using lossless detectors. Consequently, the ideal experiment reproducing
the post-selected data (where no  ø  are present) both requires a lossless
detector and a filtered state, in which the vacuum component has been
discarded.

The relevant Hilbert space is the Fock space consisting of photons that are
only distinguishable in their polarization degree of freedom:

$\displaystyle\mathcal{H}\;=\;\mathrm{span}\big{\\{}\,\big{(}b_{\theta}^{\dagger}\big{)}^{n_{\theta}}\big{(}b_{\theta^{\bot}}^{\dagger}\big{)}^{n_{\theta^{\bot}}}\left|{\mathrm{vac}}\right\rangle\;\big{|}\;n_{\theta},n_{\theta^{\bot}}\in\mathds{N}\,\big{\\}}\;,$
(55)

where $n_{\theta}$ and $n_{\theta^{\bot}}$ are the number of photons polarized
in direction $\theta$ and in the orthogonal direction $\theta^{\bot}$,
respectively. In this Hilbert space, the POVM elements of the polarization
analyser are given by121212Here, we have the possibility that both NPNR
detectors click at the same time, corresponding to the POVM element
$M_{1\&2}^{\theta}$. If it is required that the device has only two possible
measurement outcomes, we can arbitrarily re-assign this outcome to either
$M_{1}^{\theta}$ or $M_{2}^{\theta}$.

$\displaystyle M_{\textup{\,\o\,}}^{\theta}\;$
$\displaystyle=\;R^{\hat{N}}\;=\;\left|{\mathrm{vac}}\right\rangle\\!\\!\left\langle{\mathrm{vac}}\right|+\sum_{n=1}^{\infty}R^{n}\;\Pi_{n}$
(56) $\displaystyle M_{\checkmark}^{\theta}\;$
$\displaystyle=\;M_{1}^{\theta}+M_{2}^{\theta}+M_{1\&2}^{\theta}\;=\;\mathds{1}-R^{\hat{N}}$
(57)

where $\Pi_{n}$ is the ($\theta$-invariant) projector onto the $n$-photon
sector of the Fock space. The expression on the right hand side of Eq. (57) is
invariant under change of rotation angle $\theta$: hence, we are in a
(homogeneous) fair sampling situation.

We also remark that $M_{\checkmark}^{\theta}$ has no overlap with the vacuum,
$\left\langle{\mathrm{vac}}\right|M_{\checkmark}^{\theta}\left|{\mathrm{vac}}\right\rangle=0$,
which immediately implies that the filtered state

$\displaystyle\rho_{{\checkmark}}\;=\;\frac{1}{\mathcal{E}_{Q}(\rho)}\sqrt{M_{\checkmark}^{\theta}}\,\rho\,\sqrt{M_{\checkmark}^{\theta}}$
(58)

also does not have overlap with the vacuum. As a special case, notice that
when $\eta=1$ (the detector are lossless) we have
$M_{\checkmark}^{\theta}=\mathds{1}-\left|{\mathrm{vac}}\right\rangle\\!\\!\left\langle{\mathrm{vac}}\right|$
and therefore post-selecting is equivalent to using a filtered state where the
vacuum component has been projected out.

###  6.3 Strong fair sampling with polarization analysers

We now look under which conditions the _strong_ version of fair sampling
holds. From Eq. (57) it follows that strong fair sampling is formally verified
whenever we restrict the Hilbert space to the $k$-photon sector (for any fixed
integer $k>0$). However, we consider only the case $k=1$, which already
corresponds to a very stringent experimental requirement. That is, we imagine
that in each round of the experiment exactly one photon enters the detector.
Thus we have:

$\displaystyle\mathcal{H}\;=\;\mathrm{span}\big{\\{}\,\left|{\theta}\right\rangle=b_{\theta}^{\dagger}\left|{\mathrm{vac}}\right\rangle,\;|{\theta^{\bot}}\rangle=b_{\theta^{\bot}}^{\dagger}\left|{\mathrm{vac}}\right\rangle\,\big{\\}}\;,$
(59)

with $\theta$ and $\theta^{\bot}$ any two orthogonal polarization directions.
This yields:

$\displaystyle M_{\textup{\,\o\,}}^{\theta}\;$
$\displaystyle=\;R^{\left|{\theta}\right\rangle\left\langle{\theta}\right|}R^{|{\theta^{\bot}}\rangle\langle{\theta^{\bot}}|}\;=\;R^{\mathds{1}}\;=\;(1-\eta)\,\mathds{1}$
$\displaystyle M_{\checkmark}^{\theta}\;$
$\displaystyle=\;\mathds{1}-M_{\textup{\,\o\,}}^{\theta}\;=\;\eta\,\mathds{1}\;.$
(60)

The expression on the right hand side of Eq. (60) is manifestly independent of
$\theta$ and is proportional to the identity: the (homogeneous) strong fair
sampling assumption holds in this case.

###  6.4 Experimental applicability

As discussed, the polarization analyser satisfies strong fair sampling only
when the input quantum state consists of exactly one photon. This is a very
demanding condition that is not met in any experiment that we know of, as
currently there is no technology capable of producing individual photons with
high fidelity and (almost) unit efficiency [55, 56]. To this end, the best we
can achieve is through the _heralded_ preparation of a photon, e.g., by first
producing a photon pair and exploit the fact that one of the two photons has
triggered a detector to infer the presence of the second one in a separated
optical path. Even in this case, though, the applicability of strong fair
sampling is debatable, since losses are always present in practice. To this
end, we show in Section 9 how to extend the results to input states that can
only be approximated as a single-photon states.

When the input quantum is not restricted to the single-photon sector, we have
shown that the polarization analyser satisfies (weak) fair sampling. This
happens, e.g., when the state has a (significant) overlap with the vacuum.
However, since the polarization analyser can click only when one or more
photons are present in the apparatus, the filtered state does not have overlap
with the vacuum.

We could then consider sources that (up to an excellent degree of accuracy)
emit at most one photon at a time. In these cases, the _filtered_ state will
have exactly one photon entering the polarization analyser. We could also
consider sources that could emit multiple photons at a time, as is the case in
spontaneous up- and down-conversion processes in non-linear crystals. In these
cases, the filtered quantum state does not contain the vacuum, but it does
contain multi-photon components. Noticing that the more photons enter the
device, the higher is the probability of a detection we deduce that,
correspondingly, the filtered state $\rho_{\checkmark}$ has more weight on
large photon-number components, compared to the experimental state $\rho$.

Finally, we can give a concrete recommendation for proof-of-principle
demonstrations of device-independent protocols relying on post-selected data.
The measurement device should be extensively tested and calibrated.
Subsequently, one should publish the best available estimation of the
efficiency of the device (the POVM elements $M_{\checkmark}^{\theta}$),
together with the uncertainty to which they are known. From an abstract
perspective, this is less information than having the individual POVM
description of $M_{1}^{\theta},M_{1}^{\theta}$, and $M_{1\&2}^{\theta}$, which
would be required in device-specific protocols (such as, say, the BB84
protocol). Moreover, in some cases the estimation of
$M_{\checkmark}^{\mathsf{x}}$ might be experimentally easier than the
estimation of all $M_{\mathsf{a}}^{\mathsf{x}}$ elements, for instance when
the number of possible outcomes (the size of the set $\mathsf{A}$) is large.
Provided that the fair sampling condition is satisfied, this approach provides
the means to demonstrate the fundamental elements of device-independent
protocols with post-selected data, avoiding the stringent requirement of
having a extremely high detector efficiency.

##  7 Approximate fair sampling

Here we investigate what happens when the fair sampling assumption is not
satisfied exactly. Intuitively, we expect that the probability distributions
that can be obtained after post-selection are close to those that can be
obtained measuring the filtered state $\rho_{\checkmark}$ with a lossless
device. And indeed, a small deviation from the exact fair sampling condition
induces a small perturbation in the post-selected probability distribution
(with linear dependence, to the leading order). In turn, the fact that the
post-selected probabilities are close to the ideal ones implies, e.g., that
the experimental expectation values of Bell operators are also close to the
ideal value.

###  7.1 Bound on the total variation distance

As a starting point, we recall that $M_{C,{\checkmark}}$ is diagonal, hence we
have
$M_{\checkmark}=\sum_{\mathsf{x}}\mathcal{E}_{C}(\mathsf{x})\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes
M_{Q,{\checkmark}}$ or, equivalently,
$M_{\checkmark}^{\mathsf{x}}=\mathcal{E}_{C}(\mathsf{x})M_{Q,{\checkmark}}$.
We can rewrite this as:

$\displaystyle\mathcal{E}_{C}(\mathsf{x})\,\Pi_{\checkmark}\;=\;M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\mathsf{x}}M_{Q,{\checkmark}}^{-1/2}$
(61)

where $\Pi_{\checkmark}$ is the projector onto the support of
$M_{Q,{\checkmark}}$ (and on the support of
$M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\mathsf{x}}M_{Q,{\checkmark}}^{-1/2}$
as well) and thus we have

$\displaystyle\mathcal{E}_{C}(\mathsf{x})\;=\;\left|\left|M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\mathsf{x}}M_{Q,{\checkmark}}^{-1/2}\right|\right|\;.$
(62)

Here and in the following, the norms are operator norms. We then consider
small deviations from fair sampling as expressed in Eq. (61).

######  Proposition 4 (Approximate fair sampling).

Consider a lossy device $\mathcal{M}$ having POVM elements
$M_{\mathsf{a}}^{\mathsf{x}}$ and
$M_{\checkmark}^{\mathsf{x}}=\sum_{\mathsf{a}\in\mathsf{A}}M^{\mathsf{x}}_{\mathsf{a}}$.
We suppose that there exists a positive semi-definite operator
$M_{Q,{\checkmark}}\succcurlyeq 0$ such that each
$M_{\checkmark}^{\mathsf{x}}$ satisfies, for some $\epsilon\in[0,1)$:

$\displaystyle\left|\left|\,\Pi_{\checkmark}-\frac{M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\mathsf{x}}M_{Q,{\checkmark}}^{-1/2}}{\left|\left|M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\mathsf{x}}M_{Q,{\checkmark}}^{-1/2}\right|\right|}\,\right|\right|\;\leq\;\epsilon$
(63)

where $\Pi_{\checkmark}$ is the projector on the support of
$M_{Q,{\checkmark}}$. We also assume
$\Pi_{\checkmark}M_{\checkmark}^{\mathsf{x}}\Pi_{\checkmark}=M_{\checkmark}^{\mathsf{x}}$
for all $\mathsf{x}\in\mathsf{X}$ and
$\Pi_{\checkmark}\rho\,\Pi_{\checkmark}\neq 0$ for the experimentally relevant
input states.

Then, there exists a normalized quantum state $\rho_{\checkmark}$, which can
be probabilistically filtered from $\rho$, and there exists a lossless device
$\bar{\mathcal{M}}$ such that:

$\displaystyle\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2}\left|\,\frac{\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)}{\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)}-\mathrm{Tr}(\bar{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark})\,\right|\;\leq\;\frac{\epsilon}{1-\epsilon}$
(64)

When the measurement device $\mathcal{M}$ satisfies inequality (63) we say
that the $\epsilon$-_approximate_ (weak) fair sampling assumption holds.

###### Proof.

See Appendix D. ∎

A few remarks are in order. First, in the limit $\epsilon\to 0$ approximate
fair sampling becomes the condition for (exact) fair sampling. Second, the
left hand side of inequality (64) is the total variation distance between the
post-selected probability distribution
${\Pr}_{\mathcal{M}}^{\mathrm{p.s.}}(\mathsf{a}|\mathsf{x},\rho)$ and the
distribution
${\Pr}_{\bar{\mathcal{M}}}(\mathsf{a}|\mathsf{x},\rho_{\checkmark})$ obtained
by measuring the filtered state $\rho_{\checkmark}$ with a lossless
detector131313The total variation distance is the classical analogue of the
trace distance for density matrices. It expresses the expected
distinguishability between two distributions using random samples from one of
the distributions.. Finally, we can specialize the Proposition to approximate
_strong_ fair sampling by consider the special condition
$M_{Q,{\checkmark}}=\mathds{1}$. In this case, Eq. (63) simply becomes:

$\displaystyle\left|\left|\,\mathds{1}-\frac{M_{\checkmark}^{\mathsf{x}}}{\left|\left|M_{\checkmark}^{\mathsf{x}}\right|\right|}\,\right|\right|\;\leq\;\epsilon$
(65)

and the filtered state is equal to the experimental state,
$\rho_{\checkmark}=\rho$.

###  7.2 Approximate fair sampling in multipartite settings

Proposition 4 links an inequality regarding the POVM of a measurement device
to an inequality on the probability distributions that can be observed using
such a device. In a multipartite quantum experiment this probability
distribution corresponds to the marginal distribution that is observed by a
single detector, ignoring the measurement outcomes of all other devices
involved in the experiment. Therefore, Proposition 4 does not (directly) bound
the deviation for the global distribution over all possible outcome
configurations of the devices. However, this shortcoming can be easily
amended, using the following simple idea.

Consider a quantum experiment where $N$ parties are involved, measuring a
multipartite state $\Psi$ with lossy devices
$\mathcal{M}_{1},\ldots,\mathcal{M}_{N}$ which approximately satisfy fair
sampling. Then, we can equivalently consider the multipartite measurement
device $\overrightarrow{\mathcal{M}}$ consisting of the collection of these
$N$ devices, whose settings (and outputs) consists in the collection of
settings (and outputs) of the individual devices. It is straightforward to
verify that this joint device then satisfies
$\epsilon_{\mathrm{tot}}$-approximate fair sampling; the total approximation
error $\epsilon_{\mathrm{tot}}$ is essentially given by the sum of the
approximation errors for the individual devices. Therefore, for this
multipartite device Proposition 4 directly applies. That is, it is possible to
bound to the deviation of the post-selected probability distribution from the
result of measuring a filtered state $\Psi_{\checkmark}$ with $N$ lossless
devices. More details can be found in Appendix E. As a further consequence,
suppose we want to measure the expectation value of a $N$-partite Bell
operator $B$. Using the (slightly skewed) post-selected distribution the
relative error on $\langle B\rangle$ is upper bounded by
$2\epsilon_{\mathrm{tot}}$ – for the proof, see Appendix F.

Finally, we remark that if the $N$ separated devices satisfy approximate
_strong_ fair sampling, then the entire multipartite setting also satisfies
approximate strong fair sampling; this implies that the outcome statistics for
the collection of devices is close (in total variation distance) to what could
be obtained measuring the actual experimental state $\Psi$ with lossless
devices. In this case, the global approximation error
$\epsilon_{\mathrm{tot}}$ for the multipartite device is also roughly equal
the sum of the errors for individual devices.

##  8 Approximate fair sampling in quantum optics

In Section 6 we have shown that a polarization analyser satisfies fair
sampling provided that the two NPNR detectors have equal efficiency $\eta$.
Now, we will consider the case where the efficiencies of the NPNR detectors
are very similar, but not exactly equal. In this case, we show that the
polarization analyser satisfies _approximate_ fair sampling; moreover, if the
input state contains exactly one photon, approximate strong fair sampling
holds.

###  8.1 Polarization analyser with unequal detectors

We generalize the quantum modelling of the polarization analyser given in
Section 6 to the case in which the two detectors have similar (but not exactly
equal) efficiencies, $\eta_{1}$ and $\eta_{2}$. We assume without loss of
generality $\eta_{2}\geq\eta_{1}$ and we write $R=R_{2}\leq
R_{1}=R\,(1+\delta)$, with $\delta\geq 0$. This results in the following four
POVM elements:

$\displaystyle\begin{split}M_{\textup{\,\o\,}}^{\theta}\,:=\;E_{\textup{\,\o\,}}^{\theta}\\!\otimes\\!E_{\textup{\,\o\,}}^{\theta^{\bot}}\;&=\;R^{\hat{N}_{\theta}}(1+\delta)^{\hat{N}_{\theta}}\,R^{\hat{N}_{\theta^{\bot}}}\;=\;R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}\\\
M_{1}^{\theta}\;:=\;E_{\textup{\,\o\,}}^{\theta}\\!\otimes\\!E_{\checkmark}^{\theta^{\bot}}\;&=\;R^{\hat{N}_{\theta}}(1+\delta)^{\hat{N}_{\theta}}\,\big{(}\mathds{1}-R^{\hat{N}_{\theta^{\bot}}}\big{)}\\\
M_{2}^{\theta}\;:=\;E_{\checkmark}^{\theta}\\!\otimes\\!E_{\textup{\,\o\,}}^{\theta^{\bot}}\;&=\;\big{[}\mathds{1}-R^{\hat{N}_{\theta}}(1+\delta)^{\hat{N}_{\theta}}\big{]}\,R^{\hat{N}_{\theta^{\bot}}}\\\
M_{1\&2}^{\theta}\;:=\;E_{\checkmark}^{\theta}\\!\otimes\\!E_{\checkmark}^{\theta^{\bot}}\;&=\;\big{[}\mathds{1}-R^{\hat{N}_{\theta}}(1+\delta)^{\hat{N}_{\theta}}\big{]}\,\big{(}\mathds{1}-R^{\hat{N}_{\theta^{\bot}}}\big{)}\;.\end{split}$
(66)

###  8.2 Approximate fair sampling with a polarization analyser

We now consider photonic states which have a (possibly large) component of
vacuum and a certain probability of emitting more than one photon, as common
in experiments that use up- or down-conversion processes. The Hilbert space is

$\displaystyle\mathcal{H}\;=\;\mathrm{span}\big{\\{}\,\big{(}b_{\theta}^{\dagger}\big{)}^{n_{\theta}}\big{(}b_{\theta^{\bot}}^{\dagger}\big{)}^{n_{\theta^{\bot}}}\left|{\mathrm{vac}}\right\rangle\;\big{|}\;n_{\theta},n_{\theta^{\bot}}\in\mathds{N}\,\big{\\}}\;.$
(67)

The operator $M_{\checkmark}^{\theta}$ has an explicit (small) dependence from
the setting $\theta$:

$\displaystyle\begin{split}M_{\textup{\,\o\,}}^{\theta}\;&=\;R^{\hat{N}}\,(1+\delta)^{\hat{N}_{\theta}}\\\
M_{\checkmark}^{\theta}\;&=\;\mathds{1}-R^{\hat{N}}\,(1+\delta)^{\hat{N}_{\theta}}\;.\end{split}$
(68)

We now have to find a suitable positive semi-definite operator
$M_{Q,{\checkmark}}$ and a projector $\Pi_{\checkmark}$ in order to show that
weak fair sampling is approximately satisfied. We choose:

$\displaystyle M_{Q,{\checkmark}}$ $\displaystyle:=\;\mathds{1}-R^{\hat{N}}$
(69) $\displaystyle\Pi_{\checkmark}\;$
$\displaystyle:=\;\mathds{1}-\left|{\mathrm{vac}}\right\rangle\\!\\!\left\langle{\mathrm{vac}}\right|\;.$
(70)

In Appendix G we show that, with these definition, the condition

$\displaystyle\left|\left|\,\Pi_{\checkmark}-\frac{M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\theta}M_{Q,{\checkmark}}^{-1/2}}{\left|\left|M_{Q,{\checkmark}}^{-1/2}M_{\checkmark}^{\theta}M_{Q,{\checkmark}}^{-1/2}\right|\right|}\,\right|\right|\;$
$\displaystyle=\;\frac{1-\eta}{\eta}\,\delta$ (71)

holds. Thus, we can directly apply the results on approximate weak fair
sampling derived in Proposition 4. The maximum deviation in the post-selected
probability and those obtainable from $\rho_{\checkmark}$ is of order
$\delta\,(1-\eta)/\eta$, in total variation distance.

We can also characterize the filtered state $\rho_{\checkmark}$ that allows to
(approximately) reproduce the post-selected probabilities:

$\displaystyle\rho_{\checkmark}\;=\;\frac{1}{\mathcal{E}_{Q}(\rho)}\sqrt{M_{Q,{\checkmark}}}\,\rho\,\sqrt{M_{Q,{\checkmark}}}\;=\;\frac{1}{\mathcal{E}_{Q}(\rho)}\sqrt{\mathds{1}-R^{\hat{N}}}\,\rho\,\sqrt{\mathds{1}-R^{\hat{N}}}\;.$
(72)

In particular, the state $\rho_{\checkmark}$ has no overlap with the vacuum.

###  8.3 Approximate strong fair sampling with a polarization analyser

In this Section, we restrict to cases where exactly one photon is entering the
apparatus, i.e., we assume that we are employing an (almost) perfect single-
photon source. The Hilbert space is

$\displaystyle\mathcal{H}\;=\;\mathrm{span}\big{\\{}\,\left|{\theta}\right\rangle=b_{\theta}^{\dagger}\left|{\mathrm{vac}}\right\rangle,\;|{\theta^{\bot}}\rangle=b_{\theta^{\bot}}^{\dagger}\left|{\mathrm{vac}}\right\rangle\,\big{\\}}\;,$
(73)

and the POVM elements associated to a failed/successful detection are:

$\displaystyle\begin{split}M_{\textup{\,\o\,}}^{\theta}\;&=\;R\,(1+\delta)^{\left|{\theta}\right\rangle\left\langle{\theta}\right|}\;=\;(1-\eta)\,\big{(}\mathds{1}+\delta\left|{\theta}\right\rangle\\!\\!\left\langle{\theta}\right|\big{)}\\\
M_{\checkmark}^{\theta}\;&=\;\mathds{1}-M_{\textup{\,\o\,}}^{\theta}~{}~{}~{}~{}~{}~{}~{}\>=\;\eta\,\mathds{1}-(1-\eta)\delta\left|{\theta}\right\rangle\\!\\!\left\langle{\theta}\right|\;.\end{split}$
(74)

The POVM element $M_{\checkmark}^{\theta}$ has an explicit (small) dependence
on $\theta$. In fact, we can explicitly show that the condition for
approximate strong fair sampling holds. Assuming $\delta\geq 0$, we obtain

$\displaystyle\forall\theta\qquad\big{|}\big{|}{M_{\checkmark}^{\theta}}\big{|}\big{|}\;=\;\left|\left|\,\eta\,\mathds{1}-(1-\eta)\delta|{\theta^{\bot}}\rangle\\!\langle{\theta^{\bot}}|\,\right|\right|\;=\;\eta\;,$
(75)

and then

$\displaystyle\left|\left|\mathds{1}-\frac{M_{\checkmark}^{\theta}}{\left|\left|M_{\checkmark}^{\theta}\right|\right|}\right|\right|\;=\;\left|\left|\mathds{1}-\frac{M_{\checkmark}^{\theta}}{\eta}\right|\right|\;=\;\left|\left|\frac{1-\eta}{\eta}\,\delta\,|{\theta^{\bot}}\rangle\\!\langle{\theta^{\bot}}|\right|\right|\;=\;\frac{1-\eta}{\eta}\,\delta\;.$
(76)

This result directly allows us to employ Proposition 4, specialized to the
case of strong fair sampling. We can conclude that the post-selected
statistics obtained with this device can have a deviation (in total variation
distance) at most of order $\delta\,(1-\eta)/\eta$ from the statistics
obtained measuring the single-photon state with a lossless device.

As previously discussed, single photon sources are not achievable with current
technology, heralded photon preparation being the method that comes the
closest to this goal. Therefore, in the next Section we analyse what happens
in the case in which the input state is only approximately a single-photon
state and has a small vacuum and multi-photon component.

##  9 Assumptions on the input state

In this final Section, we consider the case in which the quantum state has a
small component not belonging to the reference Hilbert space. We show that the
outcome statistics is not overly affected by the presence of such
perturbations. Next, we extend the fair sampling assumption to the case where
the factorization of Eq. (11) holds only for the specific state $\Psi$ that is
employed in the (multipartite) quantum experiment. We call this the _state-
dependent_ fair sampling assumption.

###  9.1 Effects of imperfections in state preparation

Consider an imperfectly prepared quantum state having the form

$\displaystyle\hat{\rho}\ $ $\displaystyle\in\
S(~{}~{}~{}\mathcal{H}~{}~{}~{}\oplus~{}~{}~{}\mathcal{H}^{\bot}~{}~{})$
$\displaystyle\hat{\rho}\;$
$\displaystyle=\,\left(\begin{array}[]{cc}(1-\epsilon^{\prime})\,\rho&c\\\
c^{\dagger}&\epsilon^{\prime}\,\rho^{\bot}\end{array}\right)\;,$ (79)

where $\rho$ and $\rho^{\bot}$ are normalized quantum states and $c$,
$c^{\dagger}$ are the coherence terms. The Hilbert space $\mathcal{H}$
represents the set of states for which the measurement device has been
calibrated, and $\mathcal{H}^{\bot}$ is an orthogonal space containing all
other degrees of freedom. The fair sampling assumption holds for states
$\rho\in S(\mathcal{H})$, while the behaviour of the measurement device for an
input state $\rho^{\bot}\in S(\mathcal{H}^{\bot})$ can be arbitrary.

As a concrete example, consider a photonic state $\hat{\rho}$ entering a
polarization analyser. Strong fair sampling holds when the input consists of
exactly one photon, but the experimental state $\hat{\rho}$ also possess a
small component $\epsilon^{\prime}\rho^{\bot}$ which overlaps with the vacuum
and with multi-photon states. As a second example, a polarization analyser
satisfies (weak) fair sampling when $\hat{\rho}$ consists of photons that are
only distinguishable in polarization. That is, the photons must have the
correct frequency, waveform, spatial mode, and so on, consistent with the
specifications and calibration of the measurement device. However, the
experimental state $\hat{\rho}$ may have a small component
$\epsilon^{\prime}\rho^{\bot}$ of photons that have different properties, for
which the behaviour of the device is not known. If $\epsilon^{\prime}$ is
small, the deviation in the post-selected statistics will also be small;
quantitatively, we have the following result.

######  Proposition 5.

Consider a quantum state $\hat{\rho}\in
S(\mathcal{H}\oplus\mathcal{H}^{\bot})$ as in Eq. (79). Consider a lossy
detector having POVM elements $\\{\hat{M}_{\mathsf{a}}^{\mathsf{x}}\\}$ acting
on $\mathcal{H}\oplus\mathcal{H}^{\bot}$ and define
$M_{\mathsf{a}}^{\mathsf{x}}:=\Pi\hat{M}_{\mathsf{a}}^{\mathsf{x}}\Pi$,
$\hat{M}_{\checkmark}^{\mathsf{x}}:=\sum_{\mathsf{a}\in\mathsf{A}}\hat{M}_{\mathsf{a}}^{\mathsf{x}}$
and
$M_{\checkmark}^{\mathsf{x}}:=\sum_{\mathsf{a}\in\mathsf{A}}M_{\mathsf{a}}^{\mathsf{x}}$.
Then, the following trace distance bound holds:

$\displaystyle\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2}\left|\,\frac{\mathrm{Tr}(\hat{M}_{\mathsf{a}}^{\mathsf{x}}\hat{\rho})}{\mathrm{Tr}(\hat{M}_{\checkmark}^{\mathsf{x}}\hat{\rho})}-\frac{\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)}{\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)}\,\right|\;\leq\;\frac{2\left(\left|\left|c\right|\right|_{\mathrm{Tr}}+\epsilon^{\prime}\right)}{\max\left\\{\mathrm{Tr}(\hat{M}_{\checkmark}^{\mathsf{x}}\hat{\rho})\,,\,\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)\right\\}}\;,$
(80)

where
$\left|\left|c\right|\right|_{\mathrm{Tr}}:=\mathrm{Tr}(\sqrt{c^{\dagger}c})$
is the trace norm of $c$, which satisfies
$\left|\left|c\right|\right|_{\mathrm{Tr}}\leq\sqrt{\epsilon^{\prime}(1-\epsilon^{\prime})}$.

###### Proof.

See Appendix H. ∎

In general, the deviation in the post-selected statistics can be of order
$\sqrt{\epsilon^{\prime}}$. However, if the state has no coherence terms
($c=0$), then the deviation is only of order $\epsilon^{\prime}$. Thermal
light is an instance of this case, being completely incoherent in the photon-
number basis.

We also remark that in the realistic case where both device calibration and
state preparation are not perfect, the bounds given in Proposition 5 and in
Proposition 4 can be combined. The deviation from the ideal case is at most
the sum of the bounds given in Eq. (64) and in Eq. (80), as can be shown
applying the triangle inequality.

###  9.2 State-dependent fair sampling

We suppose that the measurement device
$\mathcal{M}_{1}=\bar{\mathcal{M}}_{1}\circ\mathcal{F}_{1}$ used by the first
party of a Bell-like experiment does not satisfy fair sampling (not even
approximately). We require, instead, that the application of the successful
branch
$\mathcal{F}_{1,{\checkmark}}:S(\mathcal{H}_{\mathsf{X}}\otimes\mathcal{H}_{1})\to
S(\mathcal{H}_{\mathsf{X}}\otimes\mathcal{H}_{1})$ to the first component of
the state $\Psi\in
S(\mathcal{H}_{1}\\!\otimes\\!\cdots\\!\otimes\\!\mathcal{H}_{N})$ results in:

$\displaystyle\forall\mathsf{x}\in\mathsf{X}\qquad\big{(}\mathcal{F}_{1,{\checkmark}}\otimes\mathrm{Id}_{\mathrm{R}}\big{)}\,\big{(}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\otimes\Psi\big{)}\;=\;\widetilde{\mathcal{F}}_{1,C,{\checkmark}}\big{(}\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|\big{)}\otimes\mathcal{E}_{1,Q}(\Psi)\,\Psi_{1,{\checkmark}}$
(81)

where $\mathrm{Id}_{\mathrm{R}}$ is the identity channel on the rest of the
parties, $\Psi_{\checkmark}$ is a (normalized) quantum state that can be
locally filtered from $\Psi$, and $\mathcal{E}_{1,Q}(\Psi)$ is proportional to
the probability of successfully filtering the state $\Psi_{1,{\checkmark}}$
from $\Psi$. That is, there exists a CP map
$\widetilde{\mathcal{F}}_{1,Q,{\checkmark}}$ such that:

$\displaystyle\Psi_{1,{\checkmark}}\;=\;\frac{1}{\mathcal{E}_{1,Q}(\Psi)}\,\big{(}\widetilde{\mathcal{F}}_{1,Q,{\checkmark}}\otimes\mathrm{Id}_{\mathrm{R}}\big{)}\,\big{(}\Psi\big{)}\;.$
(82)

Then, the same results as in Proposition 2 apply in this case. Consider, in
fact, the post-selection corresponding to first device clicking (we ignore the
post-selection applied by the other parties for now). Then, the post-selected
probabilities that one observes in the realized quantum experiment can be
reproduced by an ideal experiment where the first device is lossless, defined
as in Eq. (21), and the state that has been distributed to the $N$ parties is
the filtered state $\Psi_{\checkmark}$.

We can then apply the same reasoning to each of the parties involved in the
experiment. Suppose that all the measurement devices satisfy _state-dependent_
fair sampling; then, the global post-selection (where the devices click
contemporaneously) produces probabilities that are equal to those one would
obtain when all parties use lossless devices, defined as in Eq. (21), acting
on the normalized quantum state:

$\displaystyle\Psi_{\checkmark}\;=\;\frac{1}{\mathcal{E}_{1,Q}(\Psi)\cdots\mathcal{E}_{N,Q}(\Psi)}\,\big{(}\widetilde{\mathcal{F}}_{1,Q,{\checkmark}}\otimes\cdots\otimes\widetilde{\mathcal{F}}_{N,Q,{\checkmark}}\big{)}\,\big{(}\Psi\big{)}\;,$
(83)

where each $\widetilde{\mathcal{F}}_{k,Q,{\checkmark}}$ is a local extraction
map that depends on $\mathcal{M}_{k}$ and on $\Psi$.

We can also consider approximate versions of state-dependent version fair
sampling, taking into account detector and state preparation imperfections.
The results previously derived for approximate fair sampling would also hold
in this case.

## Discussion

In this paper, we have analysed the effect of post-selection on the outcome
statistics of a measurement apparatus in the context of Bell-like experiments.
To this end, we have employed a general and fully quantum model of the
measurement device, described in terms of POVM elements. We have identified a
condition, equivalent to the one of Ref. [1], which allows the measurement
statistics to exactly correspond to the statistics of some ideal quantum
experiment, involving _ideal_ lossless detectors and a _filtered_ quantum
state. The filtered state is (possibly) different from the actual experimental
state but can be obtained from it through local probabilistic filters. In
simple terms, this condition amount to the fact that the device acts
independently on the classical and quantum inputs it receives, conditioned on
having had a successful detection. Differently from what is done in most of
the previous literature, it is this condition that we identify as the (weak)
fair sampling assumption.

We argue that the weak notion of fair sampling is the most useful and
applicable one, especially in the context of quantum optics. In fact, in
quantum optics experiments the produced photonic states have a component of
vacuum, but the common praxis is to characterize properties (such as fidelity,
purity, or expectation values of observables) of a quantum state where the
vacuum component has been projected out. This exactly corresponds to the
filtering operation induced by fair sampling.

To make the result of this work applicable in real experimental settings, we
have investigated the effect of small deviations from exact fair sampling. We
have shown that imperfections in the calibration of the device result in a
deviation in the post-selected probabilities that scales essentially linearly
in the size of the miscalibration, but is amplified by a factor inversely
proportional to the efficiency of the device. For a polarization analyser to
satisfy fair sampling the two photon detectors are required to have the same
efficiency, which could be enforced by artificially decreasing the efficiency
of one of the detectors. Our analysis of approximate fair sampling takes care
of any remaining efficiency mismatch. We have also considered the effect of
imperfect state preparations, which lead the input state to have a small
component not belong to the Hilbert space where the measurement device
satisfies fair sampling. In this case, the presence of off-diagonal coherence
terms in the density matrix can make the error in the statistics to scale
(unfavourably) as the square root of the state preparation error.

We have also argued that the measurement devices can be securely used in
cryptographic protocols, provided that they satisfy fair sampling. It is
paramount in this case that fair sampling holds also from the perspective of
the adversary. If, for instance, the adversary possesses a more complete
description (a.k.a., a purification) of the measurement devices where fair
sampling does not hold, explicit attack strategies can be exhibited.

## Acknowledgements

We thank Matthias Bock and Christoph Becher for discussions which motivated us
to work on the effect of post-selections on device-independent claims. We
acknowledge funding by the Swiss National Science Foundation (SNSF) through
the Grant PP00P2-179109 and by the Army Research Laboratory Center for
Distributed Quantum Information via the project SciNet.

## References

  * [1] D. W. Berry, H. Jeong, M. Stobińska, and T. C. Ralph, _Fair-sampling assumption is not necessary for testing local realism_ , Physical Review A 81(1), 012109 (2010).
  * [2] R. H. Hadfield, _Single-photon detectors for optical quantum information applications_ , Nature Photonics 3(12), 696–705 (2009).
  * [3] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, _Proposed experiment to test local hidden-variable theories_ , Physical Review Letters 23(15), 880–884 (1969).
  * [4] J. F. Clauser, and M. A. Horne, _Experimental consequences of objective local theories_ , Physical Review D 10(2), 526–535 (1974).
  * [5] P. M. Pearle, _Hidden-variable example based upon data rejection_ , Physical Review D 2(8), 1418–1425 (1970).
  * [6] A. Garg, and N. D. Mermin, _Detector inefficiencies in the Einstein-Podolsky-Rosen experiment_ , Physical Review D 35(12), 3831–3835 (1987).
  * [7] S. Massar, and S. Pironio, _Violation of local realism versus detection efficiency_ , Physical Review A 68(6), 062109 (2003).
  * [8] E. Pomarico, B. Sanguinetti, P. Sekatski, H. Zbinden, and N. Gisin, _Experimental amplification of an entangled photon: what if the detection loophole is ignored?_ New Journal of Physics 13(6), 063031 (2011).
  * [9] I. Gerhardt, Q. Liu, A. Lamas-Linares, J. Skaar, V. Scarani, V. Makarov, and C. Kurtsiefer, _Experimentally faking the violation of Bell’s inequalities_ , Physical Review Letters 107(17), 170404 (2011).
  * [10] J. Romero, D. Giovannini, D. S. Tasca, S. M. Barnett, and M. J. Padgett, _Tailored two-photon correlation and fair-sampling: a cautionary tale_ , New Journal of Physics 15(8), 083047 (2013).
  * [11] J. Jogenfors, A. M. Elhassan, J. Ahrens, M. Bourennane, and J. Å. Larsson. _Hacking the Bell test using classical light in energy-time entanglement-based quantum key distribution_ , Science Advances 1(11), 1500793 (2015).
  * [12] J. Jogenfors, _Breaking the Unbreakable: Exploiting Loopholes in Bell’s Theorem to Hack Quantum Cryptography_ , Linköping University Electronic Press (2017).
  * [13] B. Hensen et. al., _Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres_ , Nature 526, 682 (2015).
  * [14] L. K. Shalm et. al., _Strong loophole-free test of local realism_ , Physical Review Letters 115(25), 250402 (2015).
  * [15] M. Giustina et. al., _Significant-loophole-free test of Bell’s theorem with entangled photons_ , Physical Review Letters 115(25), 250401 (2015).
  * [16] W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. Ortegel, M. Rau, and H. Weinfurter, _Event-ready Bell test using entangled atoms simultaneously closing detection and locality loopholes_ , Physical Review Letters 119(1), 010402 (2017).
  * [17] M. H. Li, C. Wu, Y. Zhang, W. Z. Liu, B. Bai, Y. Liu, W. Zhang, Q. Zhao, H. Li, Z. Wang, L. You, W. J. Munro, J. Yin, J. Zhang, C.-Z. Peng, X. Ma, Q. Zhang, J. Fan, and J.-W. Pan, _Test of local realism into the past without detection and locality loopholes_ , Physical Review Letters 121(8), 080404 (2018).
  * [18] D. Mayers, and A. Yao, _Self testing quantum apparatus_ , arXiv:quant-ph/0307205.
  * [19] I. Šupić, and J. Bowles, _Self-testing of quantum systems: a review_ , arxiv:quant-ph/1904.10042.
  * [20] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, _Device-independent security of quantum cryptography against collective attacks_ , Physical Review Letters 98(23), 230501 (2007).
  * [21] S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, _Device-independent quantum key distribution secure against collective attacks_ , New Journal of Physics 11(4), 045021 (2009).
  * [22] M. McKague, T. H. Yang, and V. Scarani, _Robust self-testing of the singlet_ , Journal of Physics A: Mathematical and Theoretical 45(45), 455304 (2012).
  * [23] J. Kaniewski, _Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities_ , Physical Review Letters 117(7), 070402 (2016).
  * [24] P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, _Certifying the building blocks of quantum computers from Bell’s theorem_ , Physical Review Letters 121(18), 180505 (2018).
  * [25] J.-D. Bancal, K. Redeker, P. Sekatski, W. Rosenfeld, and N. Sangouard, _Device-independent certification of an elementary quantum network link_ , arXiv:quant-ph/1812.09117.
  * [26] R. Colbeck, and A. Kent, _Private randomness expansion with untrusted devices_ , Journal of Physics A: Mathematical and Theoretical 44(9), 095305 (2011).
  * [27] A. Acín, and L. Masanes, _Certified randomness in quantum physics_ , Nature 540, 213 (2016).
  * [28] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, _Violation of Bell’s inequality under strict Einstein locality conditions_ , Physical Review Letters 81(23), 5039 (1998).
  * [29] D. L. Moehring, M. J. Madsen, B. B. Blinov, and C. Monroe, _Experimental Bell inequality violation with an atom and a photon_ , Physical Review Letters 93(9), 090410 (2004).
  * [30] S. Gómez, A. Mattar, I. Machuca, E. S. Gómez, D. Cavalcanti, O. J. Farías, A. Acín, and G. Lima, _Experimental investigation of partially entangled states for device-independent randomness generation and self-testing protocols_ , Physical Review A 99(3), 032108 (2019).
  * [31] K. T. Goh, C. Perumangatt, Z. X. Lee, A. Ling, and V. Scarani, _Experimental comparison of tomography and self-testing in certifying entanglement_ , Physical Review A 100(2), 022305 (2019).
  * [32] E. Polino, I. Agresti, D. Poderini, G. Carvacho, G. Milani, G. B. Lemos, R. Chaves, and F. Sciarrino, _Device independent certification of a quantum delayed choice experiment_ , arXiv:quant-ph/1806.00211.
  * [33] I. Agresti, D. Poderini, L. Guerini, M. Mancusi, G. Carvacho, L. Aolita, D. Cavalcanti, R. Chaves, and F. Sciarrino, _Experimental device-independent certified randomness generation with an instrumental causal structure_ , arXiv:quant-ph/1905.02027.
  * [34] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, _Bell nonlocality_ , Reviews of Modern Physics 86(2), 419–478 (2014).
  * [35] Q. Y. He, E. G. Cavalcanti, M. D. Reid, and P. D. Drummond, _Bell inequalities for continuous-variable measurements_ , Physical Review A 81(6), 062106 (2010).
  * [36] M. A. Nielsen, and I. Chuang, Quantum computation and quantum information, Cambridge University Press (2002).
  * [37] D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, _Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses_ , Physical Review A 86(6), 062325 (2012).
  * [38] R. Schmied, J.-D. Bancal, B. Allard, M. Fadel, V. Scarani, P. Treutlein, and N. Sangouard, _Bell Correlations in a Bose-Einstein Condensate_ , Science 352, 441 (2016).
  * [39] N. Gisin, _Hidden quantum nonlocality revealed by local filters_ , Physics Letters A 210(3), 151-156 (1996).
  * [40] R. Rabelo, M. Ho, D. Cavalcanti, N. Brunner, and V. Scarani, _Device-independent certification of entangled measurements_ , Physical Review Letters 107(5), 050502 (2011).
  * [41] J.-D. Bancal, N. Sangouard, and P. Sekatski, _Noise-resistant device-independent certification of Bell state measurements_ , Physical Review Letters 121(25), 250506 (2018).
  * [42] M. O. Renou, J. Kaniewski, and N. Brunner, _Self-testing entangled measurements in quantum networks_ , Physical Review Letters 121(25), 250507 (2018).
  * [43] D. Cavalcanti, and P. Skrzypczyk, _Quantum steering: a review with focus on semidefinite programming_ , Reports on Progress in Physics 80(2), 024001 (2016).
  * [44] J. Barrett, R. Colbeck, and A. Kent, _Memory attacks on device-independent quantum cryptography_ , Physical Review Letters 110(1), 010503 (2013).
  * [45] J. Müller-Quade, and R. Renner, _Composability in quantum cryptography_ , New Journal of Physics 11(8), 085006 (2009).
  * [46] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, _Hacking commercial quantum cryptography systems by tailored bright illumination_ , Nat. Photonics 4(10), 686-689 (2010).
  * [47] S. Pironio, A. Acín, S. Massar, A. B. de La Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, _Random numbers certified by Bell’s theorem_ , Nature 464, 1021 (2010).
  * [48] Z. Cao, H. Zhou, X. Yuan, and X. Ma, _Source-independent quantum random number generation_ , Physical Review X 6(1), 011020 (2016).
  * [49] C. A. Miller, and Y. Shi, _Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices_ , Journal of the ACM 63(4), 33 (2016).
  * [50] C. C. W. Lim, C. Portmann, M. Tomamichel, R. Renner, and N. Gisin, _Device-independent quantum key distribution with local Bell test_ , Physical Review X 3(3), 031006 (2013).
  * [51] U. Vazirani, and T. Vidick, _Fully device independent quantum key distribution_ , Physical Review Letters 113(14), 140501 (2014).
  * [52] R. Arnon-Friedman, R. Renner, and T. Vidick, _Simple and tight device-independent security proofs_ , SIAM Journal on Computing 48(1), 181-225 (2019).
  * [53] A. Aspect, J. Dalibard, and G. Roger, _Experimental test of Bell’s inequalities using time-varying analyzers_ , Physical Review Letters 49(25), 1804 (1982).
  * [54] Scully, M.O. and Zubairy, M.S., _Quantum optics_ , Cambridge University Press (1999).
  * [55] F. Flamini, N. Spagnolo, and F. Sciarrino, _Photonic quantum information processing: a review_ , Reports on Progress in Physics 82(1), 016001 (2018).
  * [56] S. Slussarenko, and G. J. Pryde, _Photonic quantum information processing: a concise review_ , arXiv:quant-ph/1907.06331.

## Appendices

###  A Necessary conditions for fair sampling

Fair sampling is formulated referring to ideal lossless devices, so it is not
experimentally testable. However, fair sampling does imply restrictions in the
correlations that can be observed, and these provide some consistency checks
that the experimenter can perform using the classical outcomes alone. Note
that fair sampling gives an independent set of equations from those obtained
from the _no-signalling_ condition.

The local quantum state that the one party measures in a multipartite
experiment is

$\displaystyle\rho\;=\;\Psi_{\vec{\mathsf{a}}_{R}}^{\vec{\mathsf{x}}_{R}}$
(84)

where the indices imply that the state may depend on the measurement settings
$\vec{\mathsf{x}}_{R}$ and outcomes $\vec{\mathsf{a}}_{R}$ obtained by the
other parties, a phenomenon known as _quantum steering_. Fair sampling for the
device of a given party requires that the efficiency factorizes in a component
dependent on $\mathsf{x}$ and in a component dependent on
$\rho=\Psi_{\vec{\mathsf{a}}_{R}}^{\vec{\mathsf{x}}_{R}}$, see Proposition 1.
Hence:

$\displaystyle\mathcal{E}(\mathsf{x},\vec{\mathsf{x}}_{R},\vec{\mathsf{a}}_{R})\;=\;\mathcal{E}_{C}(\mathsf{x})\;\mathcal{E}_{Q}(\vec{\mathsf{x}}_{R},\vec{\mathsf{a}}_{R})\;.$
(85)

If impose the _strong_ version of fair sampling, the dependence on $\rho$
disappears, see Proposition 3. Hence we have:

$\displaystyle\mathcal{E}(\mathsf{x},\vec{\mathsf{x}}_{R},\vec{\mathsf{a}}_{R})\;=\;\mathcal{E}_{C}(\mathsf{x})\;.$
(86)

Obviously, these are not _sufficient_ conditions to have fair sampling. Even
when all devices have constant efficiency for all parties,
$\mathcal{E}(\mathsf{x},\vec{\mathsf{x}}_{R},\vec{\mathsf{a}}_{R})=const$, the
detection loophole is still open.

###  B Filter diagonal form

We consider a lossy measurement device
$\mathcal{M}=\bar{\mathcal{M}}\circ\mathcal{F}$ which satisfies fair sampling,
$\mathcal{F}=\mathcal{F}_{C}\otimes\mathcal{F}_{Q}$, and we consider the
successful branch $\mathcal{F}_{C,{\checkmark}}$ of $\mathcal{F}_{C}$, which
is a (non normalized) probabilistic map. Here, we show that we can use the
freedom in defining the filters to bring $\mathcal{F}_{C,{\checkmark}}$ to a
diagonal form $\mathcal{T}_{C,{\checkmark}}$ which does not change
$\mathsf{x}$.

The general form of the probabilistic filter $\mathcal{F}_{C,{\checkmark}}$
is:

$\displaystyle\mathcal{F}_{C,{\checkmark}}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)\;$
$\displaystyle=\;\sum_{\mathsf{x}^{\prime}\in\mathsf{X}}\Pr(\mathsf{x}^{\prime}|\,\mathsf{x})\,|{\mathsf{x}^{\prime}}\rangle\\!\langle{\mathsf{x}^{\prime}}|\;,$
(87)

where $\Pr(\mathsf{x}^{\prime}|\mathsf{x})$ are sub-normalized probabilities,
i.e.,
$\Pr({\checkmark}|\,\mathsf{x})=\sum_{\mathsf{x}^{\prime}\in\mathsf{X}}\Pr(\mathsf{x}^{\prime}|\,\mathsf{x})\leq
1$. We define a new filter $\mathcal{T}_{C,{\checkmark}}$ and a new device
$\bar{\mathcal{W}}$ (with POVM elements $\bar{W}^{\mathsf{x}}_{\mathsf{a}}$)
as:

$\displaystyle\mathcal{T}_{C,{\checkmark}}(\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|)\;:=\;\Pr({\checkmark}|\,\mathsf{x})\left|{\mathsf{x}}\right\rangle\\!\\!\left\langle{\mathsf{x}}\right|$
(88)
$\displaystyle\bar{W}_{\mathsf{a}}^{\mathsf{x}}\;:=\;\frac{1}{\Pr({\checkmark}|\,\mathsf{x})}\sum_{\mathsf{x}^{\prime}\in\mathsf{X}}\Pr(\mathsf{x}^{\prime}|\,\mathsf{x})\,\bar{M}_{\mathsf{a}}^{\mathsf{x}^{\prime}}\;.$
(89)

We then have
$\bar{\mathcal{M}}\circ(\mathcal{F}_{C,{\checkmark}}\otimes\mathrm{Id}_{Q})=\bar{\mathcal{W}}\circ(\mathcal{T}_{C,{\checkmark}}\otimes\mathrm{Id}_{Q})$141414In
the computation in Eq. (B) we omit the flag register
$\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|$ for
sake of clarity.:

$\displaystyle{\Pr}_{\bar{\mathcal{M}}\circ(\mathcal{F}_{C,{\checkmark}}\otimes\mathrm{Id}_{Q})}(\mathsf{a}\,|\,\mathsf{x},\rho)\;$
$\displaystyle=\;\mathrm{Tr}\Big{(}\,{\sum}_{\mathsf{x}^{\prime}}\,\Pr(\mathsf{x}^{\prime}|\,\mathsf{x})\,\bar{M}_{\mathsf{a}}^{\mathsf{x}^{\prime}}\rho\,\Big{)}$
$\displaystyle=\;\mathrm{Tr}\Big{(}\Pr({\checkmark}|\,\mathsf{x})\;\bar{W}_{\mathsf{a}}^{\mathsf{x}}\,\rho\,\Big{)}$
$\displaystyle=\;{\Pr}_{\bar{\mathcal{W}}\circ(\mathcal{T}_{C,{\checkmark}}\otimes\mathrm{Id}_{Q})}(\mathsf{a}\,|\,\mathsf{x},\rho)\;,$
(90)

and one can immediately verify that the device $\bar{\mathcal{W}}$ has unit
efficiency:

$\displaystyle\forall\mathsf{x}\in\mathsf{X}\qquad\sum_{\mathsf{a}\in\mathsf{A}}\bar{W}_{\mathsf{a}}^{\mathsf{x}}\;=\;\sum_{\mathsf{x}^{\prime}\in\mathsf{X}}\frac{\Pr(\mathsf{x}^{\prime}|\,\mathsf{x})}{\Pr({\checkmark}|\,\mathsf{x})}\,\sum_{\mathsf{a}\in\mathsf{A}}\bar{M}_{\mathsf{a}}^{\mathsf{x}^{\prime}}\;=\;\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|\otimes\mathds{1}\;.$
(91)

###  C Complementary computations for Proposition 2

In the following computation we plug the state $\Psi_{\checkmark}$ of Eq. (20)
into the measurement devices $\mathcal{M}_{k}^{\checkmark}$ defined in Eq.
(21), and verify that Eq. (17) holds.

$\displaystyle{\Pr}_{\overrightarrow{\mathcal{M}}^{\checkmark}}\big{(}\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi_{\checkmark}\big{)}\;=\;\big{(}\,\overrightarrow{\mathcal{M}}^{\checkmark}\big{)}^{\vec{\mathsf{x}}}_{\vec{\mathsf{a}}}\
(\Psi_{\checkmark})$ (92)
$\displaystyle=\;\frac{1}{\mathcal{E}_{C}(\vec{\mathsf{x}})\,\mathcal{E}_{Q}(\Psi)}\mathrm{Tr}\Big{\\{}\Big{(}{\bigotimes}_{k}\bar{M}_{k}\Big{)}_{\vec{\mathsf{a}}}\;\Big{[}\,(\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|)^{\otimes
N}\otimes\mathcal{F}_{C,{\checkmark}}(\left|{\vec{\mathsf{x}}}\right\rangle\\!\\!\left\langle{\vec{\mathsf{x}}}\right|)\otimes\mathcal{F}_{Q,{\checkmark}}(\Psi)\,\Big{]}\,\Big{\\}}$
(93)
$\displaystyle=\;\frac{1}{\mathcal{E}_{C}(\vec{\mathsf{x}})\,\mathcal{E}_{Q}(\Psi)}\mathrm{Tr}\Big{\\{}\Big{(}{\bigotimes}_{k}\bar{M}_{k}\Big{)}_{\vec{\mathsf{a}}}\;\Big{[}\,(\left|{{\checkmark}}\right\rangle\\!\\!\left\langle{{\checkmark}}\right|)^{\otimes
N}\otimes\mathcal{F}_{{\checkmark}}(\left|{\vec{\mathsf{x}}}\right\rangle\\!\\!\left\langle{\vec{\mathsf{x}}}\right|\otimes\Psi)\,\Big{]}\,\Big{\\}}$
(94)
$\displaystyle=\;\frac{1}{\mathcal{E}_{C}(\vec{\mathsf{x}})\,\mathcal{E}_{Q}(\Psi)}\Big{(}{\bigotimes}_{k}\bar{\mathcal{M}}_{k}\circ\mathcal{F}_{k}\Big{)}_{\vec{\mathsf{a}}}\;(\left|{\vec{\mathsf{x}}}\right\rangle\\!\\!\left\langle{\vec{\mathsf{x}}}\right|\otimes\Psi)$
(95)
$\displaystyle=\;\frac{1}{\mathcal{E}_{C}(\vec{\mathsf{x}})\,\mathcal{E}_{Q}(\Psi)}\Big{(}{\bigotimes}_{k}\mathcal{M}_{k}\Big{)}_{\vec{\mathsf{a}}}\;(\left|{\vec{\mathsf{x}}}\right\rangle\\!\\!\left\langle{\vec{\mathsf{x}}}\right|\otimes\Psi)$
(96)
$\displaystyle=\;\frac{{\Pr}_{\overrightarrow{\mathcal{M}}}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}},\Psi)}{{\Pr}_{\overrightarrow{\mathcal{M}}}({\checkmark}|\,\vec{\mathsf{x}},\Psi)}\;.$
(97)

In line (93), we have introduced
$\mathcal{F}_{C,{\checkmark}}:=\bigotimes_{k}\mathcal{F}_{k,C,{\checkmark}}$
and
$\mathcal{F}_{C,{\checkmark}}:=\bigotimes_{k}\mathcal{F}_{k,C,{\checkmark}}$.
In line (94), we defined
$\mathcal{F}_{\checkmark}:=\mathcal{F}_{C,{\checkmark}}\otimes\mathcal{F}_{Q,{\checkmark}}$.
In line (95), we have used the property that each device
$\bar{\mathcal{M}}_{k}$ returns a valid outcome $\mathsf{a}_{k}$ only when the
filter $\mathcal{F}_{k}$ returns ${\checkmark}$. Finally, in line (96) and
(97) we have used the definitions
$\mathcal{M}_{k}=\bar{\mathcal{M}}_{k}\circ\mathcal{F}_{k}$ and
$\Pr_{\overrightarrow{\mathcal{M}}}({\checkmark}|\,\vec{\mathsf{x}},\Psi)=\mathcal{E}(\vec{\mathsf{x}},\Psi)=\mathcal{E}_{C}(\vec{\mathsf{x}})\,\mathcal{E}_{Q}(\Psi)$.

###  D Proof of Proposition 4

We now give a constructive proof of Proposition 4, meaning that we provide an
explicit construction of a lossless device $\bar{\mathcal{M}}$ for which
inequality (64) holds. From preliminary numerical investigations we conjecture
that the bound is not tight and could be further improved. Moreover, an
explicit optimization over $\bar{\mathcal{M}}$ (for a given $\mathcal{M}$)
could provide better bounds.

We introduce some handy notation. Given an operator
$M_{Q,{\checkmark}}\succcurlyeq 0$, we define:

$\displaystyle\begin{split}&\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}\;:=\;\sqrt{M_{Q,{\checkmark}}^{-1}}\,M_{\mathsf{a}}^{\mathsf{x}}\,\sqrt{M_{Q,{\checkmark}}^{-1}}\\\
&\widetilde{M}_{\checkmark}^{\mathsf{x}}\;:=\;\sqrt{M_{Q,{\checkmark}}^{-1}}\,M_{\checkmark}^{\mathsf{x}}\,\sqrt{M_{Q,{\checkmark}}^{-1}}\;,\end{split}$
(98)

so that condition (63) can be compactly written as

$\displaystyle\left|\left|\,\Pi_{\checkmark}-\frac{\widetilde{M}_{\checkmark}^{\mathsf{x}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\,\right|\right|\;\leq\;\epsilon\;,$
(99)

where we remind that $\Pi_{\checkmark}$ is the projector on the support of
$M_{Q,{\checkmark}}$ and the inverse square roots are defined on its support;
then, the support of $\widetilde{M}_{\checkmark}^{\mathsf{x}}$ is contained in
the support of $M_{Q,{\checkmark}}$. We also require
$\Pi_{\checkmark}M_{\checkmark}^{\mathsf{x}}\Pi_{\checkmark}=M_{\checkmark}^{\mathsf{x}}$;
it then follows
$\Pi_{\checkmark}M_{\mathsf{a}}^{\mathsf{x}}\Pi_{\checkmark}=M_{\mathsf{a}}^{\mathsf{x}}$
for all $\mathsf{x}$ and for all $\mathsf{a}$.

Now we define the filtered state $\rho_{\checkmark}$ as

$\displaystyle\rho_{\checkmark}\;:=\;\frac{1}{\mathcal{E}_{Q}(\rho)}\sqrt{M_{Q,{\checkmark}}}\,\rho\,\sqrt{M_{Q,{\checkmark}}}$
(100)

with $\mathcal{E}_{Q}(\rho)=\mathrm{Tr}(M_{Q,{\checkmark}}\rho)$. Notice that
$\rho_{\checkmark}$ can be filtered probabilistically from $\rho$ and,
moreover,
$\Pi_{\checkmark}\rho_{\checkmark}\Pi_{\checkmark}=\rho_{\checkmark}$.

The definition of the lossless detector $\bar{\mathcal{M}}$ is more laborious.
We first introduce:

$\displaystyle\Delta^{\mathsf{x}}\;:=\;\Pi_{\checkmark}-\frac{\widetilde{M}_{\checkmark}^{\mathsf{x}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}$
(101)

and from inequality (99) it follows that:

$\displaystyle
0\preccurlyeq\Delta^{\mathsf{x}}\preccurlyeq\epsilon\,\Pi_{\checkmark}\;.$
(102)

We can then define $\bar{M}_{\mathsf{a}}^{\mathsf{x}}$ as:

$\displaystyle\bar{M}_{\mathsf{a}}^{\mathsf{x}}\;:=\;\frac{\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}+\Delta_{\mathsf{a}}^{\mathsf{x}}\;,$
(103)

where $\\{\Delta_{\mathsf{a}}^{\mathsf{x}}\\}$ is an arbitrary set of positive
semi-definite operators that satisfy
$\sum_{\mathsf{a}\in\mathsf{A}}\Delta_{\mathsf{a}}^{\mathsf{x}}=\Delta^{\mathsf{x}}$
for all $\mathsf{x}$. Explicitly, we may choose
$\Delta_{\mathsf{a}}^{\mathsf{x}}=\Delta^{\mathsf{x}}/|\mathsf{A}|$, where
$|\mathsf{A}|$ is the size of $\mathsf{A}$, so that we have:

$\displaystyle\bar{M}_{\mathsf{a}}^{\mathsf{x}}\;=\;\frac{\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}+\frac{1}{|\mathsf{A}|}\left(\Pi_{\checkmark}-\frac{\widetilde{M}_{\checkmark}^{\mathsf{x}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\right)$
(104)

and one can immediately verify that
$\bar{M}_{\mathsf{a}}^{\mathsf{x}}\succcurlyeq 0$ and
$\sum_{\mathsf{a}\in\mathsf{A}}\bar{M}_{\mathsf{a}}^{\mathsf{x}}=\Pi_{\checkmark}$.

With all the definitions in place, we proceed to verify inequality (64).
First, notice that:

$\displaystyle\frac{\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)}{\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)}\;=\;\frac{\mathrm{Tr}(\Pi_{{\checkmark}}M_{\mathsf{a}}^{\mathsf{x}}\,\Pi_{{\checkmark}}\rho)}{\mathrm{Tr}(\Pi_{{\checkmark}}M_{\checkmark}^{\mathsf{x}}\,\Pi_{{\checkmark}}\rho)}\;=\;\frac{\mathrm{Tr}\big{(}\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark}\big{)}}{\mathrm{Tr}\big{(}\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark}\big{)}}\;,$
(105)

where we have used
$\Pi_{\checkmark}M_{\mathsf{a}}^{\mathsf{x}}\Pi_{\checkmark}=M_{\mathsf{a}}^{\mathsf{x}}$
and $\Pi_{{\checkmark}}=M_{Q,{\checkmark}}^{1/2}\,M_{Q,{\checkmark}}^{-1/2}$.
Then we have:

$\displaystyle\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2}\left|\,\frac{\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)}{\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)}-\mathrm{Tr}(\bar{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark})\,\right|\;=\;\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2}\left|\,\frac{\mathrm{Tr}\big{(}\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark}\big{)}}{\mathrm{Tr}\big{(}\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark}\big{)}}-\mathrm{Tr}(\bar{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark})\,\right|$
(106) $\displaystyle=\ $
$\displaystyle\frac{1}{2\,\mathrm{Tr}\big{(}\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark}\big{)}}\sum_{\mathsf{a}\in\mathsf{A}}\left|\,\mathrm{Tr}\big{(}\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark}\big{)}-\mathrm{Tr}\big{(}\bar{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark}\big{)}\mathrm{Tr}\big{(}\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark}\big{)}\,\right|$
(107) $\displaystyle\leq\ $
$\displaystyle\frac{1}{2\,(1-\epsilon)\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\sum_{\mathsf{a}\in\mathsf{A}}\max_{\sigma_{\mathsf{a}}\in\\{0,1\\}}\left|\,\mathrm{Tr}\left\\{\left[\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}-\left(\frac{\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}+\Delta_{\mathsf{a}}^{\mathsf{x}}\right)(1-\sigma_{\mathsf{a}}\epsilon)\,\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}\right]\rho_{\checkmark}\right\\}\,\right|$
(108) $\displaystyle=\ $
$\displaystyle\frac{1}{2\,(1-\epsilon)\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\sum_{\mathsf{a}\in\mathsf{A}}\max_{\sigma_{\mathsf{a}}\in\\{0,1\\}}\left|\,\mathrm{Tr}\left\\{\left[\phantom{\Big{|}}\sigma_{\mathsf{a}}\epsilon\,\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}-\Delta_{\mathsf{a}}^{\mathsf{x}}\,(1-\sigma_{\mathsf{a}}\epsilon)\,\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}\,\right]\rho_{\checkmark}\right\\}\,\right|$
(109) $\displaystyle\leq\ $
$\displaystyle\frac{1}{2\,(1-\epsilon)\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\left[\,\sum_{\mathsf{a}\in\mathsf{A}}\epsilon\,\mathrm{Tr}\big{(}\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}\rho_{\checkmark}\big{)}+\sum_{\mathsf{a}\in\mathsf{A}}\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}\,\mathrm{Tr}\big{(}\Delta_{\mathsf{a}}^{\mathsf{x}}\,\rho_{\checkmark}\big{)}\,\right]$
(110) $\displaystyle=\ $
$\displaystyle\frac{1}{2\,(1-\epsilon)\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\left[\phantom{\Big{|}}\epsilon\,\mathrm{Tr}\big{(}\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark}\big{)}+\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}\,\mathrm{Tr}(\Delta^{\mathsf{x}}\,\rho_{\checkmark})\,\right]$
(111) $\displaystyle\leq\ $
$\displaystyle\frac{1}{2\,(1-\epsilon)\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}}\left[\phantom{\Big{|}}\epsilon\,\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}+\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}\,\epsilon\,\right]$
(112) $\displaystyle=\ $ $\displaystyle\frac{\epsilon}{1-\epsilon}\;.$ (113)

In line (108), we used
$(1-\epsilon)\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}\leq\mathrm{Tr}(\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark})\leq\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}$
and the definition of $\bar{M_{\mathsf{a}}^{\mathsf{x}}}$. In line (110), we
employed the triangle inequality, then maximized two terms separately over
$\sigma_{\mathsf{a}}\in\\{0,1\\}$, and also used the fact that
$\widetilde{M}_{\mathsf{a}}^{\mathsf{x}}$ and
$\Delta_{\mathsf{a}}^{\mathsf{x}}$ are positive semi-definite to get rid of
the absolute values. In line (111), the definitions of
$\widetilde{M}_{\checkmark}^{\mathsf{x}}$ and $\Delta^{\mathsf{x}}$. Finally,
in line (112),
$\mathrm{Tr}(\widetilde{M}_{\checkmark}^{\mathsf{x}}\rho_{\checkmark})\leq\big{|}\big{|}\widetilde{M}_{\checkmark}^{\mathsf{x}}\big{|}\big{|}$
and $\left|\left|\Delta^{\mathsf{x}}\right|\right|\leq\epsilon$.

###  E Approximate fair sampling in multipartite settings

Suppose that in an $N$-local quantum experiment each party holds a measurement
device $\mathcal{M}_{k}$ with POVM elements
$M^{\mathsf{x}_{k}}_{k,{\checkmark}}$ which approximately satisfies fair
sampling, as in Eq. (63). With the same conventions as in Appendix D, we
introduce:

$\displaystyle\Delta_{k}^{\mathsf{x}_{k}}\;:=\;\Pi_{k,{\checkmark}}-\frac{\widetilde{M}_{k,{\checkmark}}^{\mathsf{x}_{k}}}{\big{|}\big{|}\widetilde{M}_{k,{\checkmark}}^{\mathsf{x}_{k}}\big{|}\big{|}}$
(114)

so that the approximate fair sampling condition is expressed as
$\left|\left|\Delta_{k}^{\mathsf{x}_{k}}\right|\right|\leq\epsilon_{k}$, for
some small $\epsilon_{k}$. Equivalently, we can write:

$\displaystyle\widetilde{M}_{k,{\checkmark}}^{\mathsf{x}_{k}}\;=\;\big{|}\big{|}\widetilde{M}_{k,{\checkmark}}^{\mathsf{x}_{k}}\big{|}\big{|}\,(\Pi_{k,{\checkmark}}-\Delta_{k}^{\mathsf{x}_{k}})\;.$
(115)

We define also the following operators for a setting
$\vec{\mathsf{x}}:=(\mathsf{x}_{1},\ldots,\mathsf{x}_{N})$:

$\displaystyle\widetilde{M}_{\checkmark}^{\vec{\mathsf{x}}}\;$
$\displaystyle:=\;\widetilde{M}_{1,{\checkmark}}^{\mathsf{x}_{1}}\otimes\cdots\otimes\widetilde{M}_{N,{\checkmark}}^{\mathsf{x}_{N}}$
(116) $\displaystyle\Pi_{\checkmark}^{\vec{\mathsf{x}}}\;$
$\displaystyle:=\;\Pi_{1,{\checkmark}}^{\mathsf{x}_{1}}\otimes\cdots\otimes\Pi_{N,{\checkmark}}^{\mathsf{x}_{N}}$
(117)

We can finally define an operator $\Delta^{\vec{\mathsf{x}}}$ which expresses
how much the collection of $N$ measurement devices deviates from the fair
sampling assumption:

$\displaystyle\Delta^{\vec{\mathsf{x}}}\;:=$
$\displaystyle\;\Pi_{\checkmark}-\frac{\widetilde{M}_{\checkmark}^{\vec{\mathsf{x}}}}{\big{|}\big{|}\widetilde{M}_{\checkmark}^{\vec{\mathsf{x}}}\big{|}\big{|}}$
$\displaystyle\;=$
$\displaystyle\;\Pi_{\checkmark}-\frac{\widetilde{M}_{1,{\checkmark}}^{\mathsf{x}_{1}}}{\big{|}\big{|}{\widetilde{M}_{1,{\checkmark}}^{\mathsf{x}_{1}}}\big{|}\big{|}}\otimes\cdots\otimes\frac{\widetilde{M}_{N,{\checkmark}}^{\mathsf{x}_{N}}}{\big{|}\big{|}{\widetilde{M}_{N,{\checkmark}}^{\mathsf{x}_{N}}}\big{|}\big{|}}$
$\displaystyle\;=$