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arXiv:1504.05306 · IEEE Transactions on Information Theory

Tight Bound on Randomness for Violating the Clauser–Horne–Shimony–Holt Inequality

Yifeng Teng, Shenghao Yang, Siwei Wang, Mingfei Zhao

Published Feb 26, 2016• 2 citations
arXivHTMLPDF

Abstract

Free will (or randomness) has been studied to achieve loophole-free Bell's inequality test and to provide device-independent quantum key distribution security proofs. The required randomness such that a local hidden variable model (LHVM) can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality has been studied, but a tight bound has not been proved for a practical case that: 1) the device settings of the two parties in the Bell test are independent and 2) the device settings of each party can be correlated or biased across different runs. Using some information theoretic techniques, we prove, in this paper, a tight bound on the required randomness for this case, such that the CHSH inequality can be violated by certain LHVM. Our proof has a clear achievability and converse style. The achievability part is proved using type counting. To prove the converse part, we introduce a concept called profile for a set of binary sequences and study the properties of profiles. Our profile-based converse technique is also of independent interest.

DOI
10.1109/tit.2016.2535179

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Free will (or randomness) has been studied to achieve loophole-free Bell's inequality test and to provide device-independent quantum key distribution security proofs. The required randomness such that a local hidden variable model (LHVM) can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality has been studied, but a tight bound has not been proved for a practical case that: 1) the device settings of the two parties in the Bell test are independent and 2) the device settings of each party can be correlated or biased across different runs. Using some information theoretic techniques, we prove, in this paper, a tight bound on the required randomness for this case, such that the CHSH inequality can be violated by certain LHVM. Our proof has a clear achievability and converse style. The achievability part is proved using type counting. To prove the converse part, we introduce a concept called profile for a set of binary sequences and study the properties of profiles. Our profile-based converse technique is also of independent interest.
The Clauser-Horne-Shimony-Holt inequality test is widely used as a means of invalidating the local deterministic theories and a tool of device-independent quantum cryptographic tasks. There exists a randomness (free will) loophole in the test, which is widely believed impossible to be closed perfectly, that is, certain random inputs are required for the test. Following a randomness quantification method used in literature, we investigate the randomness required in the test under various assumptions. By comparing the results, we conclude that, in order to make the test result reliable, it is more important to rule out the correlation between multiple runs than the correlation between two parties.
A device-independent randomness expansion protocol aims to take an initial random string and generate a longer one, where the security of the protocol does not rely on knowing the inner workings of the devices used to run it. In order to do so, the protocol tests that the devices violate a Bell inequality and one then needs to bound the amount of extractable randomness in terms of the observed violation. The entropy accumulation theorem gives a bound in terms of the single-round von Neumann entropy of any strategy achieving the observed score. Tight bounds on this are known for the one-sided randomness when using the Clauser-Horne-Shimony-Holt (CHSH) game. Here we find the minimum von Neumann entropies for a given CHSH score relevant for one and two sided randomness that can be applied to various protocols. In particular, we show the gain that can be made by using the two-sided randomness and by using a protocol without spot-checking where the input randomness is recycled. We also discuss protocols that fully close the locality loophole while expanding randomness. Although our bounds are mostly numerical, we conjecture analytic formulae for the curves in two cases.
A device-independent randomness expansion protocol aims to take an initial random string and generate a longer one, where the security of the protocol does not rely on knowing the inner workings of the devices used to run it. In order to do so, the protocol tests that the devices violate a Bell inequality and one then needs to bound the amount of extractable randomness in terms of the observed violation. The entropy accumulation theorem gives a bound in terms of the single-round von Neumann entropy of any strategy achieving the observed score. Tight bounds on this are known for the one-sided randomness when using the Clauser-Horne-Shimony-Holt (CHSH) game. Here we find the minimum von Neumann entropies for a given CHSH score relevant for one and two sided randomness that can be applied to various protocols. In particular, we show the gain that can be made by using the two-sided randomness and by using a protocol without spot-checking where the input randomness is recycled. We also discuss protocols that fully close the locality loophole while expanding randomness. Although our bounds are mostly numerical, we conjecture analytic formulae for the curves in two cases.
Conference key agreement allows multiple remote parties to establish a shared secret key with information-theoretical security. In device-independent conference key agreement, security can be guaranteed with minimal assumptions on the devices used, provided that a violation of a Bell inequality is observed. However, implementations are extremely challenging because high detection efficiency is required to observe loophole-free Bell violations. Here, we enhance the robustness of device-independent conference key agreement by introducing a new family of multipartite Bell inequalities called the asymmetric parity-Clauser-Horne-Shimony-Holt (CHSH) inequalities. We derive a tight analytical lower bound on the conditional von Neumann entropy of the outcomes of one of the parties in a protocol based on this inequality, including noisy preprocessing. Using this bound, we analyze robustness to detection inefficiencies as well as local and global depolarizing noise. We show that the combination of the asymmetric parity-CHSH inequality and noisy preprocessing can significantly improve the robustness to imperfections.
Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given nonlocal correlations is a key quantity of interest. Here, we derive tight analytic bounds on the maximum certifiable randomness as a function of the nonlocality as expressed using the Clauser-Horne-Shimony-Holt (CHSH) value. We show that for every CHSH value greater than the local value (2) and up to 3sqrt[3]/2≈2.598 there exist quantum correlations with that CHSH value that certify a maximal two bits of global randomness. Beyond this CHSH value the maximum certifiable randomness drops. We give a second family of Bell inequalities for CHSH values above 3sqrt[3]/2, and show that they certify the maximum possible randomness for the given CHSH value. Our work hence provides an achievable upper bound on the amount of randomness that can be certified for any CHSH value. We illustrate the robustness of our results, and how they could be used to improve randomness generation rates in practice, using a Werner state noise model.
In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nthn\mathrm{th}nth particle pair would be independent of measurement choices and outcomes for the first $(n\ensuremath{-}1)$ pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the two-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the two-sided memory loophole allows a systematic violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality when the data are analyzed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in the existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearized CHSH inequality as well as other Bell-type inequalities.