Tight Bounds for the Pearle-Braunstein-Caves Chained Inequality Without   the Fair-Coincidence Assumption

Jonathan Jogenfors; Jan-{\AA}ke Larsson

arXiv:1706.06596·quant-ph·August 16, 2017

Tight Bounds for the Pearle-Braunstein-Caves Chained Inequality Without the Fair-Coincidence Assumption

Jonathan Jogenfors, Jan-{\AA}ke Larsson

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TL;DR

This paper derives tight bounds on the coincidence probability for the Pearle-Braunstein-Caves chained Bell inequality, eliminating the coincidence-time loophole without assuming fair coincidences, which enhances the reliability of Bell test experiments.

Contribution

It provides the minimum coincidence probability bounds for the PBC inequality for any number of settings, without relying on the fair-coincidence assumption.

Findings

01

Derived tight bounds for the PBC inequality's coincidence probability.

02

Identified critical coincidence probability thresholds to close the loophole.

03

Applicable to quantum communication protocols like Quantum Key Distribution.

Abstract

In any Bell test, loopholes can cause issues in the interpretation of the results, since an apparent violation of the inequality may not correspond to a violation of local realism. An important example is the coincidence-time loophole that arises when detector settings might influence the time when detection will occur. This effect can be observed in many experiments where measurement outcomes are to be compared between remote stations because the interpretation of an ostensible Bell violation strongly depends on the method used to decide coincidence. The coincidence-time loophole has previously been studied for the Clauser-Horne-Shimony-Holt (CHSH) and Clauser-Horne (CH) inequalities, but recent experiments have shown the need for a generalization. Here, we study the generalized "chained" inequality by Pearle-Braunstein-Caves (PBC) with two or more settings per observer. This…

Tables1

Table 1. Table 1: Critical coincidence probabilities γ c r i t , N subscript 𝛾 𝑐 𝑟 𝑖 𝑡 𝑁 \gamma_{crit,N} and detection probabilities η c r i t , N subscript 𝜂 𝑐 𝑟 𝑖 𝑡 𝑁 \eta_{crit,N} for a loophole-free violation of the PBC equality for 2,3,4, and 5 measurement settings. Note that N = 2 𝑁 2 N=2 corresponds to the special CHSH case.

$N$	$γ_{c r i t, N}$	$η_{c r i t, N}$
2 (CHSH)	$87.87 %$	$82.84 %$
3	$89.32 %$	$86.99 %$
4	$90.96 %$	$89.61 %$
5	$92.26 %$	$91.37 %$
$N$	Increases with $N$

Equations80

A_{i, j} : Λ \to λ \mapsto B_{i, j} : Λ \to λ \mapsto V A_{i, j} (λ) V B_{i, j} (λ)

A_{i, j} : Λ \to λ \mapsto B_{i, j} : Λ \to λ \mapsto V A_{i, j} (λ) V B_{i, j} (λ)

A_{i, j} (λ) = B_{i, j} (λ) = A_{i, l} (λ) B_{k, j} (λ)

A_{i, j} (λ) = B_{i, j} (λ) = A_{i, l} (λ) B_{k, j} (λ)

V = {x \in R; - 1 \leq x \leq + 1} .

V = {x \in R; - 1 \leq x \leq + 1} .

\begin{split}S_{N}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}&\big{|}E(A_{1}B_{1})+E(A_{2}B_{1})\big{|}\\ +&\big{|}E(A_{2}B_{2})+E(A_{3}B_{2})\big{|}+\cdots\\ +&\big{|}E(A_{N}B_{N})-E(A_{1}B_{N})\big{|}\end{split}

\begin{split}S_{N}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}&\big{|}E(A_{1}B_{1})+E(A_{2}B_{1})\big{|}\\ +&\big{|}E(A_{2}B_{2})+E(A_{3}B_{2})\big{|}+\cdots\\ +&\big{|}E(A_{N}B_{N})-E(A_{1}B_{N})\big{|}\end{split}

S_{N} \leq 2 N - 2

S_{N} \leq 2 N - 2

S_{N} = 2 N cos (\frac{π}{2 N}) .

S_{N} = 2 N cos (\frac{π}{2 N}) .

T_{i, j} : Λ \to λ \mapsto T_{i, j}^{'} : Λ \to λ \mapsto R T_{i, j} (λ) R T_{i, j}^{'} (λ),

T_{i, j} : Λ \to λ \mapsto T_{i, j}^{'} : Λ \to λ \mapsto R T_{i, j} (λ) R T_{i, j}^{'} (λ),

Λ_{i} = \mbox d e f {λ : ∣ T_{a_{i}, b_{i}} (λ) - T_{a_{i}, b_{i}}^{'} (λ) ∣ < Δ T} .

Λ_{i} = \mbox d e f {λ : ∣ T_{a_{i}, b_{i}} (λ) - T_{a_{i}, b_{i}}^{'} (λ) ∣ < Δ T} .

γ_{N} = \mbox d e f i in f P (Λ_{i}) .

γ_{N} = \mbox d e f i in f P (Λ_{i}) .

E (X_{i} ∣ Λ_{i}) = \mbox d e f \int_{Λ_{i}} X_{i} (λ) d P (λ)

E (X_{i} ∣ Λ_{i}) = \mbox d e f \int_{Λ_{i}} X_{i} (λ) d P (λ)

X_{i} = \mbox d e f A_{a_{i}} B_{b_{i}}

X_{i} = \mbox d e f A_{a_{i}} B_{b_{i}}

\begin{split}S_{C,N}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\big{|}E(X_{1}|\Lambda_{1})+E(X_{2}|\Lambda_{2})\big{|}+\cdots\\ +\big{|}E(X_{2N-1}|\Lambda_{2N-1})-E(X_{2N}|\Lambda_{2N})\big{|}\end{split}

\begin{split}S_{C,N}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\big{|}E(X_{1}|\Lambda_{1})+E(X_{2}|\Lambda_{2})\big{|}+\cdots\\ +\big{|}E(X_{2N-1}|\Lambda_{2N-1})-E(X_{2N}|\Lambda_{2N})\big{|}\end{split}

S_{C, N} \leq \frac{4 N - 2}{γ _{N}} - 2 N

S_{C, N} \leq \frac{4 N - 2}{γ _{N}} - 2 N

Λ_{I} = \mbox d e f i = 1 ⋂ 2 N Λ_{j},

Λ_{I} = \mbox d e f i = 1 ⋂ 2 N Λ_{j},

\begin{split}&\big{|}E(A_{1}B_{1}|\Lambda_{I})+E(A_{2}B_{1}|\Lambda_{I})\big{|}+\cdots\\ +&\big{|}E(A_{N}B_{N}|\Lambda_{I})-E(A_{1}B_{N}|\Lambda_{I})\big{|}\leq 2N-2\end{split}

\begin{split}&\big{|}E(A_{1}B_{1}|\Lambda_{I})+E(A_{2}B_{1}|\Lambda_{I})\big{|}+\cdots\\ +&\big{|}E(A_{N}B_{N}|\Lambda_{I})-E(A_{1}B_{N}|\Lambda_{I})\big{|}\leq 2N-2\end{split}

\delta\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\inf_{i}\frac{P\Big{(}\bigcap_{j=1}^{2N}\Lambda_{j}\Big{)}}{P(\Lambda_{i})}=\inf_{i}P\bigg{(}\bigcap_{j\neq i}\Lambda_{j}\bigg{|}\Lambda_{i}\bigg{)}

\delta\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\inf_{i}\frac{P\Big{(}\bigcap_{j=1}^{2N}\Lambda_{j}\Big{)}}{P(\Lambda_{i})}=\inf_{i}P\bigg{(}\bigcap_{j\neq i}\Lambda_{j}\bigg{|}\Lambda_{i}\bigg{)}

P\bigg{(}\bigcap_{j\neq i}\Lambda_{j}\bigg{|}\Lambda_{i}\bigg{)}\geq 2N-2+\sum_{j\neq i}P(\Lambda_{j}|\Lambda_{i})

P\bigg{(}\bigcap_{j\neq i}\Lambda_{j}\bigg{|}\Lambda_{i}\bigg{)}\geq 2N-2+\sum_{j\neq i}P(\Lambda_{j}|\Lambda_{i})

P (Λ_{j} ∣ Λ_{i}) = \frac{P ( Λ _{i} \cap Λ _{j} )}{P ( Λ _{i} )} = \frac{P ( Λ _{i} ) + P ( Λ _{j} ) - P ( Λ _{i} \cup Λ _{j} )}{P ( Λ _{i} )} \geq 1 + \frac{P ( Λ _{j} ) - 1}{P ( Λ _{i} )} \geq 1 + \frac{γ _{N} - 1}{γ _{N}} = 2 - \frac{1}{γ _{N}} .

P (Λ_{j} ∣ Λ_{i}) = \frac{P ( Λ _{i} \cap Λ _{j} )}{P ( Λ _{i} )} = \frac{P ( Λ _{i} ) + P ( Λ _{j} ) - P ( Λ _{i} \cup Λ _{j} )}{P ( Λ _{i} )} \geq 1 + \frac{P ( Λ _{j} ) - 1}{P ( Λ _{i} )} \geq 1 + \frac{γ _{N} - 1}{γ _{N}} = 2 - \frac{1}{γ _{N}} .

\begin{split}&P\bigg{(}\bigcap_{j\neq i}\Lambda_{j}\bigg{|}\Lambda_{i}\bigg{)}\\ &\geq 2N-2+(2N-1)\left(2-\frac{1}{\gamma_{N}}\right)\\ &=2N-\frac{2N-1}{\gamma_{N}},\end{split}

\begin{split}&P\bigg{(}\bigcap_{j\neq i}\Lambda_{j}\bigg{|}\Lambda_{i}\bigg{)}\\ &\geq 2N-2+(2N-1)\left(2-\frac{1}{\gamma_{N}}\right)\\ &=2N-\frac{2N-1}{\gamma_{N}},\end{split}

δ \geq 2 N - \frac{2 N - 1}{γ _{N}},

δ \geq 2 N - \frac{2 N - 1}{γ _{N}},

\begin{split}&S_{C,N}=\Big{|}E(X_{1}|\Lambda_{1})-\delta E(X_{1}|\Lambda_{I})\\ &+\delta E(X_{1}|\Lambda_{I})+E(X_{2}|\Lambda_{2})\\ &-\delta E(X_{2}|\Lambda_{I})+\delta E(X_{2}|\Lambda_{I})\Big{|}+\cdots\\ &+\Big{|}E(X_{2N-1}|\Lambda_{2N-1})-\delta E(X_{2N-1}|\Lambda_{I})\\ &+\delta E(X_{2N-1}|\Lambda_{I})-E(X_{2N}|\Lambda_{2N})\\ &+\delta E(X_{2N}|\Lambda_{I})-\delta E(X_{2N}|\Lambda_{I})\Big{|}\\ \leq&\delta\Big{(}\big{|}E(X_{1}|\Lambda_{I})+E(X_{2}|\Lambda_{I})|+\cdots\\ &+|E(X_{2N-1}|\Lambda_{I})-E(X_{2N}|\Lambda_{I})\big{|}\Big{)}\\ &+\sum_{i=1}^{2N}\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\\ \leq&\delta S_{N}+\sum_{i=1}^{2N}\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\end{split}

\begin{split}&S_{C,N}=\Big{|}E(X_{1}|\Lambda_{1})-\delta E(X_{1}|\Lambda_{I})\\ &+\delta E(X_{1}|\Lambda_{I})+E(X_{2}|\Lambda_{2})\\ &-\delta E(X_{2}|\Lambda_{I})+\delta E(X_{2}|\Lambda_{I})\Big{|}+\cdots\\ &+\Big{|}E(X_{2N-1}|\Lambda_{2N-1})-\delta E(X_{2N-1}|\Lambda_{I})\\ &+\delta E(X_{2N-1}|\Lambda_{I})-E(X_{2N}|\Lambda_{2N})\\ &+\delta E(X_{2N}|\Lambda_{I})-\delta E(X_{2N}|\Lambda_{I})\Big{|}\\ \leq&\delta\Big{(}\big{|}E(X_{1}|\Lambda_{I})+E(X_{2}|\Lambda_{I})|+\cdots\\ &+|E(X_{2N-1}|\Lambda_{I})-E(X_{2N}|\Lambda_{I})\big{|}\Big{)}\\ &+\sum_{i=1}^{2N}\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\\ \leq&\delta S_{N}+\sum_{i=1}^{2N}\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\end{split}

\displaystyle\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\leq 1-\delta

\displaystyle\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\leq 1-\delta

\begin{split}\big{|}&E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\\ \leq&\big{|}P(\Lambda_{*}|\Lambda_{i})E(X_{i}|\Lambda_{*})\big{|}\\ &+\big{|}P(\Lambda_{I}|\Lambda_{i})E(X_{i}|\Lambda_{I})-\delta E(X_{i}|\Lambda_{I})\big{|}\\ \leq&P(\Lambda_{*}|\Lambda_{i})E(|X_{i}|\big{|}\Lambda_{*})\\ &+\Big{(}P(\Lambda_{I}|\Lambda_{i})-\delta\Big{)}E(|X_{i}|\big{|}\Lambda_{I})\\ \leq&P(\Lambda_{*}|\Lambda_{i})+P(\Lambda_{I}|\Lambda_{i})-\delta=1-\delta\end{split}

\begin{split}\big{|}&E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\\ \leq&\big{|}P(\Lambda_{*}|\Lambda_{i})E(X_{i}|\Lambda_{*})\big{|}\\ &+\big{|}P(\Lambda_{I}|\Lambda_{i})E(X_{i}|\Lambda_{I})-\delta E(X_{i}|\Lambda_{I})\big{|}\\ \leq&P(\Lambda_{*}|\Lambda_{i})E(|X_{i}|\big{|}\Lambda_{*})\\ &+\Big{(}P(\Lambda_{I}|\Lambda_{i})-\delta\Big{)}E(|X_{i}|\big{|}\Lambda_{I})\\ \leq&P(\Lambda_{*}|\Lambda_{i})+P(\Lambda_{I}|\Lambda_{i})-\delta=1-\delta\end{split}

\begin{split}\sum_{i=1}^{2N}\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\leq 2N(1-\delta)\end{split}

\begin{split}\sum_{i=1}^{2N}\big{|}E(X_{i}|\Lambda_{i})-\delta E(X_{i}|\Lambda_{I})\big{|}\leq 2N(1-\delta)\end{split}

2 N cos (\frac{π}{2 N}) = \frac{4 N - 2}{γ _{crit, N}} - 2 N

2 N cos (\frac{π}{2 N}) = \frac{4 N - 2}{γ _{crit, N}} - 2 N

γ_{crit, N} = \frac{2 N - 1}{2 N} (1 + tan^{2} (\frac{π}{4 N})) .

γ_{crit, N} = \frac{2 N - 1}{2 N} (1 + tan^{2} (\frac{π}{4 N})) .

S_{C, N} = 2 N cos (\frac{π}{2 N}) .

S_{C, N} = 2 N cos (\frac{π}{2 N}) .

ϕ (i) = \mbox d e f a_{i} \frac{π}{2 N}

ϕ (i) = \mbox d e f a_{i} \frac{π}{2 N}

ϕ (i) = \mbox d e f b_{i} \frac{π}{2 N} .

ϕ (i) = \mbox d e f b_{i} \frac{π}{2 N} .

γ_{N} = (2 N - 1 + p) /2 N .

γ_{N} = (2 N - 1 + p) /2 N .

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Full text

Tight Bounds for the Pearle-Braunstein-Caves Chained Inequality Without

the Fair-Coincidence Assumption

Jonathan Jogenfors

Institutionen för systemteknik, Linköpings Universitet, 581 83 Linköping, Sweden

Jan-Åke Larsson

Institutionen för systemteknik, Linköpings Universitet, 581 83 Linköping, Sweden

Abstract

In any Bell test, loopholes can cause issues in the interpretation of the results, since an apparent violation of the inequality may not correspond to a violation of local realism. An important example is the coincidence-time loophole that arises when detector settings might influence the time when detection will occur. This effect can be observed in many experiments where measurement outcomes are to be compared between remote stations because the interpretation of an ostensible Bell violation strongly depends on the method used to decide coincidence. The coincidence-time loophole has previously been studied for the Clauser-Horne-Shimony-Holt (CHSH) and Clauser-Horne (CH) inequalities, but recent experiments have shown the need for a generalization. Here, we study the generalized “chained” inequality by Pearle-Braunstein-Caves (PBC) with $N\geq 2$ settings per observer. This inequality has applications in, for instance, Quantum Key Distribution where it has been used to re-establish security. In this paper we give the minimum coincidence probability for the PBC inequality for all $N\geq 2$ and show that this bound is tight for a violation free of the fair-coincidence assumption. Thus, if an experiment has a coincidence probability exceeding the critical value derived here, the coincidence-time loophole is eliminated.

I Introduction

In recent years there has been an increased interest in the “chained” generalization by Pearle, Braunstein and Caves (PBC) Pearle1970Hidden (1, 2) of the CHSH Clauser1969Proposed (3, 4) inequality due to its applications in re-establishing a full Bell violation. An important application is Quantum Key Distribution (QKD) based on the Franson interferometer Franson1989Bell (5) where it is known Aerts1999Twophoton (6, 7, 8) that the CHSH inequality is insufficient as a security test. If the switch to the full PBC is made, full security can be re-established Jogenfors2014Energy (7, 9).

Where the standard CHSH inequality is limited to two possible measurement settings per observer, the PBC inequality generalizes this to $N\geq 2$ settings. In order for Franson-based systems to function, $N\geq 3$ is required at the cost of significantly higher experimental requirements. Specifically, such an experiment requires a very high visibility, and until recently it was believed Jogenfors2014Energy (7) that these requirements were too impractical to achieve. Recent works Tomasin2017Highvisibility (9), however, showed it possible to meet these requirements by reaching a full violation of the PBC inequality for $N=3$ , $4$ , and $5$ with visibility in excess of $94.63\text{\,}\mathrm{\char 37\relax}$ .

Compared to other types of QKD such as BB84 Bennett1984Quantum (10) and E91 Ekert1991Quantum (11), the Franson design promises a simpler approach with fewer moving parts. This advantage could allow the Franson system to pave the way for commercial applications and widespread QKD adoption by reducing end-user complexity Jogenfors2015Hacking (8). Therefore, the possibility of re-establishing full security in the Franson interferometer is a strong motivation of further study of the PBC inequality.

Previous works Larsson2004Bell (12, 13) have shown that the CHSH and CH inequalities are vulnerable to the coincidence-time loophole which relates to the problem of attributing detector clicks to the correct pair of events. Bipartite Bell experiments measure correlations of outcomes between remote stations, and as this is done for each pair of detections, one must reliably decide which detector clicks correspond to which pair. This is more difficult than it might first appear due to high levels of non-detections, jitter in detection times, and dark counts. If coincidences are lost, one needs to apply the “fair-coincidence” assumption Larsson2014Bell (13), i.e. that the outcome statistics is not skewed from these losses. According to Larsson2014Bell (13), this fair-coincidence assumption appears to have been implicitly made in at least every experiment before 2015.

This paper formally derives bounds for the coincidence probability so that a violation of the PBC inequality can be performed without the fair-coincidence assumption. Therefore, if the coincidence probability is high enough we can eliminate the coincidence-time loophole. It should be noted that switching to the generalized PBC inequality comes at a cost. As shown by Cabello2009Minimum (14), the minimum required detection efficiency is strictly increasing with $N$ . Similarly, the PBC inequality in general has higher requirements for the coincidence probability than the CHSH inequality.

We begin by formally defining the coincidence probability for PBC-based experiments, followed by a sufficient condition for eliminating the coincidence-time loophole. Then, we show that our bound is tight by constructing a classical model that precisely reproduces the output statistics whenever the losses exceed the bound. Finally, we conclude that our results reduce to the special case of CHSH Larsson2004Bell (12) by choosing $N=2$ and compare with the corresponding limits on detection efficiency Cabello2009Minimum (14).

II The coincidence-time loophole

We use the symbol $\lambda$ for the hidden variable, which can take values in a sample space $\Lambda$ , that in turn is the domain of random variables $A(\lambda)$ and $B(\lambda)$ denoting the measurement outcomes at Alice’s and Bob’s measurement stations, respectively. We further assume that the space $\Lambda$ has a probability measure $P$ which induces an expectation value $E$ in the standard way. We now give the formal definition of the PBC inequality Pearle1970Hidden (1, 2):

Theorem 1 (Pearle-Braunstein-Caves).

Let $N$ be an integer $\geq 2$ and $i$ , $j$ , and $k$ be integers between $1$ and $2N$ , and assume the following three prerequisites to hold almost everywhere:

(i)

Realism: Measurement results can be described by probability theory, using two families of random variables $A_{i,j}$ , $B_{i,j}$ , e.g.,

[TABLE] 2. (ii)

Locality: A measurement result should be independent of the remote setting, e.g., for $k\neq i$ , $l\neq j$ we have

[TABLE] 3. (iii)

Measurement result restriction: The results may only range from $-1$ to $+1$ ,

[TABLE]

Then, by defining

[TABLE]

we get

[TABLE]

The proof consists of simple algebraic manipulations, adding of integrals and an application of the triangle inequality Pearle1970Hidden (1, 2).

The right-hand value of Equation 5 is the highest value $S_{N}$ can attain with a local realist model. Compare this with the prediction of quantum mechanics Pearle1970Hidden (1, 2):

[TABLE]

Note that $2N\cos\left(\pi/2N\right)>2N-2$ which, in the spirit of Bell1964Einstein (4), shows that the outcomes of a quantum-mechanical experiment cannot be explained in local realist terms.

Computing the Bell value requires computing the correlation between outcomes at remote stations. Importantly, data must be gathered in pairs, so that products such as $A_{1}B_{1}$ can be computed (see Equation 4). Experimentally, this is done by letting a source device generate pairs of (possibly entangled) particles that are sent to Alice and Bob for measurement. Detectors at either end record the measurement outcomes, and as previously mentioned there will is always be variations on the detection times due to experimental effects. As a consequence, it is not always obvious which detector clicks correspond to which pairs of particles.

After a number of trials, Alice and Bob must determine in which trial if they have coincidence (simultaneous clicks at Alice and Bob), a single event (only one party gets a detection) or no detection at all. This is especially pronounced if the experimental setup uses down-conversion where a continuous-wave laser pumps a nonlinear crystal in order to spontaneously create pairs of entangled photons. In that case the emission time is uniformly distributed over the duration of the experiment so it becomes a probabilistic process that further complicates pair detection.

A typical strategy used in quantum optics experiments to reduce the influence of noise in a Bell experiment is to have a time window of size $\Delta T$ around, for example, Alice’s detection event Giustina2013Bell (15, 16). If a detection event has occurred at Bob’s side within this window it is counted as a coincident pair. This is a non-local strategy as it involves comparing data between remote stations and is used in many experiments.

For the experimenter it is tempting to choose a small $\Delta T$ since it filters out noise and therefore increases the measured Bell value. At this point, there is apparently no immediately obvious drawback of picking a very small $\Delta T$ . However, rejecting experimental data in a Bell experiment modifies the underlying statistical ensemble and it is known Larsson1998Bell (17) to lead to inflated Bell values and a false violation of the Bell inequality. This is a so-called loophole that can arise in Bell testing, and many such loopholes have been studied in recent years (see Larsson2014Loopholes (18) for a review).

A coincidence window that is too small discards some truly coincident events as noise. Therefore, the Bell value measurement only occurs on a subset of the statistical ensemble which means a number of events are not accounted for when the Bell value is computed. While the Bell value $S_{N}$ from Theorem 1 is in violation of a Bell inequality, this violation might be a mirage. Specifically, the loophole that arises from choosing a small $\Delta T$ is called the coincidence-time loophole and has previously been studied Larsson2004Bell (12) for the special CHSH case $N=2$ . In addition, more recent works Larsson2014Bell (13) derive similar bounds for the CH Clauser1974Experimental (19) inequality.

We generalize the results previously obtained for the special case of CHSH by deriving tight bounds for the coincidence-time loophole in the PBC inequality (Theorem 1) for all $N\geq 2$ . This contribution will be useful for future experiments investigating, among others, Franson-based QKD. Other works Cabello2009Minimum (14) have studied the effects of reduced detector efficiency for the full PBC inequality, which in turn is a generalization of an older result Larsson1998Bell (17) that only discussed the special CHSH case.

For the rest of this paper, Alice and Bob perform measurements on some underlying, possibly quantum, system. Their measurements are chosen from $\{A_{i}\}$ and $\{B_{j}\}$ , respectively, i.e. sets of $N$ measurement settings each. As discussed by Larsson2004Bell (12), Alice’s and Bob’s choice of measurement settings might influence whether an event is coincident or not. Following the formalism in Larsson1998Bell (17) we will therefore model non-coincident settings $\lambda$ as subsets of $\Lambda$ where the random variables $A_{i}(\lambda)$ and $B_{i}(\lambda)$ are undefined. We must therefore modify the expectation values in Equation 4 to be conditioned on coincidence in order for $S_{N}$ to be well-defined (see Equation 10). The time of arrival at Alice’s and Bob’s measurement stations is defined as

[TABLE]

respectively. Since this notation will become cumbersome, we will introduce a simplification. Let ${{\{b_{i}\}}_{1}}^{2N}=\{1,1,2,2,\ldots,N,N\}$ and $a_{i}$ rotated one step so that ${{\{a_{i}\}}_{1}}^{2N}=\{1,2,2,\ldots,N,N,1\}$ . Then ${{\{(a_{i},b_{i})\}}_{1}}^{2N}=\{(1,1)$ , $(2,1)$ , $(2,2)$ , $(3,2)$ , $\ldots$ , $(N,N)$ , $(1,N)\}$ . This allows us to define subsets of $\Lambda$ as the sets on which Alice’s and Bob’s measurement settings give coincident outcomes. For $1\leq i\leq 2N$ we have

[TABLE]

We can now calculate the probability of coincidence as

[TABLE]

Finally, for $1\leq i\leq 2N$ we have the conditional expectation defined as

[TABLE]

where we use the convenient shorthand

[TABLE]

for the product of the outcomes of Alice and Bob.

III The PBC inequality with coincidence

probability

We can now re-state Theorem 1 in terms of coincidence probability.

Theorem 2 (PBC with coincidence probability).

Let $N$ be an integer $\geq 2$ and $i$ , $j$ , and $k$ be integers between $1$ and $2N$ , and assume the prerequisites $(i)-(iii)$ of Theorem 1 hold almost everywhere together with

(iv)

Coincident events: Correlations are obtained on $\Lambda_{i}\subset\Lambda$ .

Then by defining

[TABLE]

we get

[TABLE]

The remainder of this section is dedicated to proving this result. Note that while the proof of Theorem 1 consists of adding expectation values, this cannot be done for Theorem 2 since $\Lambda_{i}\neq\Lambda_{j}$ in general. Again, the ensemble changes with Alice’s and Bob’s measurement settings, so the ensemble that Theorem 1 implicitly acts upon is really

[TABLE]

i.e. the intersection of all coincident subspaces of $\Lambda$ . In other words, prerequisites $(i)-(iii)$ yield

[TABLE]

which again is a more precise re-statement of Theorem 1 where we stress the conditional part. An experiment, however, will give us results on the form $E(X_{i}|\Lambda_{i})$ , i.e. $E(X_{i}|\Lambda_{I})$ is unavailable to the experimenter. We therefore need to bridge the gap between experimental data and Theorem 2, so we define following quantity which will act as a stepping stone:

[TABLE]

Note that it is possible for the ensemble $\Lambda_{I}$ to be empty, but only when $\delta=0$ and then Equation 13 is trivial. We can therefore assume $\delta>0$ for the rest of the proof and our goal now is to give a lower bound to $\delta$ in terms of the coincidence probability $\gamma_{N}$ . We fix $i$ and apply Boole’s inequality:

[TABLE]

and rewrite the summation terms:

[TABLE]

Inserting Equation 18 into Equation 17 we get

[TABLE]

and as Equation 18 is independent of $i$ , inserting into Equation 16 gives

[TABLE]

and this is the desired lower bound. We now bound $S_{C,N}$ from above by adding and subtracting $\delta E(X_{i}|\Lambda_{I})$ in every term before applying the triangle inequality and use Equation 15:

[TABLE]

To give an upper bound to the last sum, we need the following lemma:

Lemma 1.

For $1\leq i\leq 2N$ and $0\leq\delta\leq 1$ we have the following inequality:

[TABLE]

Proof 1.

It is clear that $\Lambda_{I}\subset\Lambda_{i}$ . We can therefore split $\Lambda_{i}$ in two disjoint sets: $\Lambda_{*}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\Lambda_{i}\setminus\Lambda_{I}$ and $\Lambda_{I}$ . It follows that $\Lambda_{I}\cup\Lambda_{*}=\Lambda_{i}$ and we have

[TABLE]

Lemma 1 gives us

[TABLE]

The final step is to use Equations 24, 5 and 20 on Equation 21 which proves the desired result.

IV Minimum coincidence probability

The right-hand-side of Equation 13 increases as $\gamma_{N}$ goes down so there exists a unique $\gamma_{N}$ so the bound on $S_{C,N}$ coincides with the quantum-mechanical prediction in Equation 6. We define this critical coincidence probability as $\gamma_{\text{crit},N}$ and find it by solving the following equation:

[TABLE]

and get

[TABLE]

What remains to show is that for all $\gamma_{N}\leq\gamma_{\text{crit},N}$ there exists a local hidden variable (LHV) model that produces a $S_{C,N}$ that mimics the predictions of quantum theory. Formally, we have the following theorem:

Theorem 3.

Let $N$ be an integer $\geq 2$ . For every $\gamma_{N}\leq\gamma_{\text{crit},N}$ is is possible to construct an LHV model fulfilling the prerequisites (i) — (iv) of Equation 13 so that

[TABLE]

We explicitly prove Theorem 3 by constructing the LHV model depicted in Figure 1. Here, the hidden variable is on the form $(r,\theta)$ and uniformly distributed over $0\leq r\leq 1$ and $0\leq\theta\leq 2\pi$ . The LHV model defines the random variables $A_{i}$ and $B_{i}$ and arrival times $T_{i}$ and $T^{\prime}_{i}$ , where we adapt the shorthand from Equation 11 to the definition in Equation 7. We choose $\phi$ to be a function of $i$ in the following way for Alice’s detector:

[TABLE]

and the following way for Bob’s detector:

[TABLE]

In Figure 1, $\phi$ acts as a shift in the $\theta$ direction (with wraparound when neccessary). The case $i=1$ is depicted in Figure 2, and by choosing $\Delta T=3/2$ we get coincidence for a time difference of 0 and 1 units (solid background), and non-coincidence for a time difference of two units (cross-hatched background).

We compute the probability of coincidence in Figure 2, $P(\Lambda_{i})=(2N-1+p)/2N$ and find that it is independent of $i$ . Therefore,

[TABLE]

In addition, for $1\leq i\leq 2N-1$ ,

[TABLE]

and

[TABLE]

which gives

[TABLE]

for $1\leq i\leq 2N-1$ . A similar calculation yields

[TABLE]

We now insert the predictions of the LHV model into Equation 12 to get

[TABLE]

As we want the LHV model to mimic the predictions of quantum mechanics (from Equation 6) we put

[TABLE]

which gives

[TABLE]

Solving for $p$ we get

[TABLE]

and Equation 30 then gives us

[TABLE]

which coincides with $\gamma_{\text{crit},N}$ . The model in Figure 1 is a constructive proof of Theorem 3 as it produces the same output statistics as quantum mechanics with coincidence probability $\gamma_{\text{crit},N}$ . We finally note that it is trivial to modify the LHV model to give any $\gamma\leq\gamma_{\text{crit},N}$ which finishes the proof.

The LHV model in Figure 1 mimics almost every statistical property of a truly quantum-mechanical experiment (see Larsson2004Bell (12)) and shows it is possible to fake a violation of the PBC inequality if the coincidence probability is lower than the critical value. It is therefore important that any experiment relying on a PBC inequality violation takes the coincidence probability into account before ruling out a classical model.

As the number of measurement settings $N$ goes to infinity the critical coincidence probability $\gamma_{\text{crit},N}$ goes to 1. Therefore, achieving the required coincidence becomes harder as more measurement settings are used. If we define $\eta_{\text{crit},N}$ as the minimum required detection efficiency for a violation of the PBC inequality free of the detection loophole (see Cabello2009Minimum (14) for full details) we get

[TABLE]

and note that $\gamma_{\text{crit},N}>\eta_{\text{crit},N}$ for all $N\geq 2$ . In addition, the critical coincidence probability for the special CHSH case $N=2$ is $87.87\text{\,}\mathrm{\char 37\relax}$ which agrees with previous works Larsson2004Bell (12). See Table 1 for critical probabilities for the cases $N=2,3,4,5$ and note that both $\gamma_{\text{crit},N}$ and $\eta_{\text{crit},N}$ are strictly increasing in $N$ . Note that a loophole-free experiment requires both the coincidence probability and detection efficiency be in excess of their respective thresholds.

While reaching $\gamma_{\text{crit},N}$ is less challenging for small $N$ , some applications do requires a PBC inequality with a higher number of settings. An example is the Franson interferometer Franson1989Bell (5), where postselection leads to a loophole for $N=2$ but not for $N\geq 3$ Jogenfors2014Energy (7). In fact, $N=5$ is optimal for that setup in terms of violation, however Table 1 shows that the corresponding minimal coincidence probability is as high as $92.26\text{\,}\mathrm{\char 37\relax}$ , which is a considerable challenge.

V Conclusion

The PBC inequality is a powerful tool for testing local realism in applications where the CHSH test is insufficient. We have found the minimum required coincidence probability for a violation of the PBC inequality without the fair-coincidence assumption. This bound is tight, so any application of the PBC inequality that relies on a violation of local realism must have at least this coincidence probability, unless the perilous fair-coincidence assumption is to be made. If not, and if the coincidence probability is below the critical threshold, an attacker can construct a local realist model from which all measurements can be predicted.

Bibliography19

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1(1) P. Pearle “Hidden-Variable Example Based upon Data Rejection” In Phys. Rev. D 2 , 1970, pp. 1418–1425 DOI: 10.1103/Phys Rev D.2.1418 · doi ↗
2(2) S.L. Braunstein and C.M. Caves “Wringing out Better Bell Inequalities” In Annals of Physics 202.1 , 1990, pp. 22–56 DOI: 10.1016/0003-4916(90)90339-P · doi ↗
3(3) John F. Clauser, Michael A. Horne, Abner Shimony and Richard A. Holt “Proposed Experiment to Test Local Hidden-Variable Theories” In Phys. Rev. Lett. 23.15 , 1969, pp. 880–884 DOI: 10.1103/Phys Rev Lett.23.880 · doi ↗
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7(7) Jonathan Jogenfors and Jan-Åke Larsson “Energy-Time Entanglement, Elements of Reality, and Local Realism” In Journal of Physics A: Mathematical and Theoretical 47.42 , 2014, pp. 424032 DOI: 10.1088/1751-8113/47/42/424032 · doi ↗
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TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Tight Bounds for the Pearle-Braunstein-Caves Chained Inequality Without

Abstract

I Introduction

II The coincidence-time loophole

Theorem 1** (Pearle-Braunstein-Caves).**

III The PBC inequality with coincidence

Theorem 2** (PBC with coincidence probability).**

Lemma 1**.**

Proof 1**.**

IV Minimum coincidence probability

Theorem 3**.**

V Conclusion

Theorem 1 (Pearle-Braunstein-Caves).

Theorem 2 (PBC with coincidence probability).

Lemma 1.

Proof 1.

Theorem 3.