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arXiv:1706.03528

All entangled states can generate certified randomness

Xing Chen

Published Jun 12, 2017• 0 citations
arXivHTMLPDF

Abstract

Random number has many applications, it plays an important role in quantum information processing. It's not difficult to generate true random numbers, the main difficulty is how to certify the random numbers generated by untrusted devices. In [Nature(London) 464, 1021 (2010)], the authors provided us a way to generate certified random number by Bell's theorem. In their scheme, we can use the nonlocal behavior of entangled states to generate certified randomness. But there are entangled states, which admit a local hidden variable model, could not be used in their scheme. We show in our paper that the nonlocal correlations in every entangled state can be used to generate certified randomness, and we use Werner states as an example to show how to quantify the output randomness.

DOI
10.48550/arxiv.1706.03528

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Random number has many applications, it plays an important role in quantum information processing. It's not difficult to generate true random numbers, the main difficulty is how to certify the random numbers generated by untrusted devices. In [Nature(London) 464, 1021 (2010)], the authors provided us a way to generate certified random number by Bell's theorem. In their scheme, we can use the nonlocal behavior of entangled states to generate certified randomness. But there are entangled states, which admit a local hidden variable model, could not be used in their scheme. We show in our paper that the nonlocal correlations in every entangled state can be used to generate certified randomness, and we use Werner states as an example to show how to quantify the output randomness.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities -- entanglement, non-locality and randomness -- relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: For systems in a given entangled state, how many nonlocal correlations in a sequence can we obtain by measuring them repeatedly? And from this generated sequence of non-local correlations, how many random numbers is it possible to certify? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from an entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities – entanglement, non-locality and randomness – relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: Given a single entangled system, how many nonlocal correlations in a sequence can we obtain? And from this single entangled system, how many certified random numbers is it possible to generate? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from a single entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.
Nonlocal correlations obtained from entangled quantum states are useful for device independent (DI) randomness certification [Nature (London) 464, 1021 (2010)]. The advantage of this DI protocol over the conventional quantum protocol is that even with untrusted experimental apparatus randomness can be certified. Although quantum entanglement is the necessary physical source for nonlocal correlation, these two are different concepts. There exist entangled states which produce no nonlocal correlation and hence are not useful for the DI randomness certification task. In this work we introduce measurement-device-independent randomness certification task and show that entangled states, having local description, can be useful resource in such task which otherwise are useless in corresponding DI scenario.
Measurements on entangled quantum systems necessarily yield outcomes that are intrinsically unpredictable if they violate a Bell inequality. This property can be used to generate certified randomness in a device-independent way, i.e., without making detailed assumptions about the internal working of the quantum devices used to generate the random numbers. Furthermore these numbers are also private; i.e., they appear random not only to the user but also to any adversary that might possess a perfect description of the devices. Since this process requires a small initial random seed to sample the behavior of the quantum devices and to extract uniform randomness from the raw outputs of the devices, one usually speaks of device-independent randomness expansion. The purpose of this paper is twofold. First, we point out that in most real, practical situations, where the concept of device independence is used as a protection against unintentional flaws or failures of the quantum apparatuses, it is sufficient to show that the generated string is random with respect to an adversary that holds only classical side information; i.e., proving randomness against quantum side information is not necessary. Furthermore, the initial random seed does not need to be private with respect to the adversary, provided that it is generated in a way that is independent from the measured systems. The devices, however, will generate cryptographically secure randomness that cannot be predicted by the adversary, and thus one can, given access to free public randomness, talk about private randomness generation. The theoretical tools to quantify the generated randomness according to these criteria were already introduced in S. Pironio et al. [Nature (London) 464, 1021 (2010)], but the final results were improperly formulated. The second aim of this paper is to correct this inaccurate formulation and therefore lay out a precise theoretical framework for practical device-independent randomness generation.
Random number generation is vital for a wide range of applications, including numerical simulation, gambling, and cryptography. However, verifying the randomness of a bit stream generated by a device is exceedingly difficult. Even if a sequence of numbers passes all of the standard statistical tests, it is often impossible to certify the authenticity of the device and rule out the possibility that an adversary has loaded a predetermined sequence into an internal memory. Recently, it has been shown that the non-local correlations between entangled quantum systems can be used to verify the generation of true random numbers [1, 2]. This insight enables the construction of a private random number generator whose output can be verified as random through the violation of a Bell inequality, without requiring any assumptions about the internal mechanisms of the device.
Abstract Certification of quantum nonlocality plays a central role in practical applications like device-independent quantum cryptography and random number generation protocols. These applications entail the challenging problem of certifying quantum nonlocality, something that is hard to achieve when the target quantum state is only weakly entangled, or when the source of errors is high, e.g. when photons propagate through the atmosphere or a long optical fiber. Here we introduce a technique to find a Bell inequality with the largest possible gap between the quantum prediction and the classical local hidden variable limit for a given set of measurement frequencies. Our method represent an efficient strategy to certify quantum nonlocal correlations from experimental data without requiring extra measurements, in the sense that there is no Bell inequality with a larger gap than the one provided. Furthermore, we also reduce the photodetector efficiency required to close the detection loophole. We illustrate our technique by improving the detection of quantum nonlocality from experimental data obtained with weakly entangled photons.