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arXiv:1508.01494 · Physical Review A

Bell inequalities violated using detectors of low efficiency

Károly F. Pál, Tamás Vértesi

Published Nov 9, 2015• 30 citations
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Abstract

We define a family of binary-outcome $n$-party $m\ensuremath{\le}n$ settings per party Bell inequalities whose members require the least detection efficiency for their violation among all known inequalities of the same type. This gives upper bounds for the minimum value of the critical efficiency---below which no violation is possible---achievable for such inequalities. For $m=2$, our family reduces to the one given by Larsson and Semitecolos in 2001 [Phys. Rev. A 63, 022117 (2001)]. For $m>2$, a gap remains between these bounds and the best lower bounds. The violating state near the threshold efficiency always approaches a product state of $n$ qubits.

DOI
10.1103/physreva.92.052104

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We define a family of binary-outcome nnn-party m\ensuremath≤nm\ensuremath{\le}nm\ensuremath≤n settings per party Bell inequalities whose members require the least detection efficiency for their violation among all known inequalities of the same type. This gives upper bounds for the minimum value of the critical efficiency---below which no violation is possible---achievable for such inequalities. For m=2m=2m=2, our family reduces to the one given by Larsson and Semitecolos in 2001 [Phys. Rev. A 63, 022117 (2001)]. For m>2m>2m>2, a gap remains between these bounds and the best lower bounds. The violating state near the threshold efficiency always approaches a product state of nnn qubits.
Abstract Imperfect detection efficiency remains one of the major obstacles in achieving loophole-free Bell tests over long distances. At the same time, the challenge of establishing a common reference frame for measurements becomes more pronounced as the separation between parties increases. In this work, we tackle both of these issues by examining the impact of limited detection efficiency on the probability of Bell inequality violation with Haar random measurement settings. We derive analytical lower bounds on the violation probability for a two-qubit maximally entangled state, which is tight for correlation inequalities and perfect detection efficiencies. We further investigate it numerically for more qubits and settings using two detection efficiency models and an original method based on the linear programming. Beyond that, we show that the so-called typicality of Bell inequality violation, i.e. almost certain violation of local realism with sufficiently many particles or random measurement directions, holds even if the detection efficiency is limited. Our findings reveal that increasing the number of measurement settings can compensate for efficiency losses above the critical threshold. Additionally, we determine critical detection efficiencies for both two-party and three-party scenarios. For two parties, we recover previously established results, while for three parties, we derive a symmetric critical efficiency of η crit = 2 / 3 for the W and GHZ states within the binning model. In cases involving the no-detection outcome, we present a modified inequality using a pair of orthogonal observables for each party with η crit ≈ 0.7208 , which is notably less than 0.75 for the GHZ state. These results offer deeper insights into the limitations and possibilities for certifying non-locality in the presence of limited detection efficiency, shedding light on its robustness in practical Bell tests.
The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.
Abstract The detection loophole problem arises when quantum devices fail to provide an output for some runs. If treating these devices in a device-independent manner, failure to include the unsuccessful runs in the output statistics can lead to an adversary falsifying security i.e. Bell inequality violation. If the devices fail with too high frequency, known as the detection threshold, then no security is possible, as the full statistics cannot violate a Bell inequality. In this work we provide an intuitive local hidden-variable strategy that the devices may use to falsify any two-party, binary-outcome no-signalling distribution up to a threshold of 2(m A + m B − 8)/(m A m B − 16), where m A, m B refer to the number of available inputs choices to the two parties. This value is the largest analytically predicted lower bound for no-signalling distributions. We strongly conjecture it gives the true detection threshold for m A = m B, and for computationally tractable scenarios we provide the Bell inequality which verifies this. We also prove that a non-trivial detection threshold remains, even when allowing one party an arbitrary number of input choices.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
In many quantum information applications, a minimum detection efficiency must be exceeded to ensure success. Protocols depending on the violation of a Bell inequality, for instance, may be subject to the so-called detection loophole: Imperfect detectors may yield spurious violations, which consequently cannot be used to ensure, say, quantum cryptographic security. Hence, we investigate the possibility of giving lower bounds on detector efficiency even if an adversary has full control over both the source and the detectors. To this end, we present a technique to systematically derive Bell inequalities free from the detection loophole using only the observed measurements statistics. The violation of these inequalities certifies that the detectors used exceed a certain minimal efficiency.