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arXiv:1807.04674 · Physical Review A

Quantifying the randomness of copies of noisy Popescu-Rohrlich correlations

Boris Bourdoncle, Stefano Pironio, Antonio Acín

Published Oct 25, 2018• 2 citations
arXivHTMLPDF

Abstract

In a no-signaling world, the outputs of a nonlocal box cannot be completely predetermined, a feature that is exploited in many quantum information protocols exploiting nonlocality, such as device-independent randomness generation and quantum key distribution. This relation between nonlocality and randomness can be formally quantified through the min-entropy, a measure of the unpredictability of the outputs that holds conditioned on the knowledge of any adversary that is limited only by the no-signaling principle. This quantity can easily be computed for the noisy Popescu-Rohrlich (PR) box, the paradigmatic example of nonlocality. In this paper, we consider the min-entropy associated to several copies of noisy PR boxes. In the case where $n$ noisy PR boxes are implemented using $n$ noncommunicating pairs of devices, it is known that each PR box behaves as an independent biased coin: the min-entropy per PR box is constant with the number of copies. We show that this does not hold in more general scenarios where several noisy PR boxes are implemented from a single pair of devices, either used sequentially $n$ times or producing $n$ outcome bits in a single run. In this case, the min-entropy per PR box is smaller than the min-entropy of a single PR box, and it decreases as the number of copies increases.

DOI
10.1103/physreva.98.042130

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In a no-signaling world, the outputs of a nonlocal box cannot be completely predetermined, a feature that is exploited in many quantum information protocols exploiting nonlocality, such as device-independent randomness generation and quantum key distribution. This relation between nonlocality and randomness can be formally quantified through the min-entropy, a measure of the unpredictability of the outputs that holds conditioned on the knowledge of any adversary that is limited only by the no-signaling principle. This quantity can easily be computed for the noisy Popescu-Rohrlich (PR) box, the paradigmatic example of nonlocality. In this paper, we consider the min-entropy associated to several copies of noisy PR boxes. In the case where nnn noisy PR boxes are implemented using nnn noncommunicating pairs of devices, it is known that each PR box behaves as an independent biased coin: the min-entropy per PR box is constant with the number of copies. We show that this does not hold in more general scenarios where several noisy PR boxes are implemented from a single pair of devices, either used sequentially nnn times or producing nnn outcome bits in a single run. In this case, the min-entropy per PR box is smaller than the min-entropy of a single PR box, and it decreases as the number of copies increases.
Imagine a task in which a group of separated players aim to simulate a statistic that violates a Bell inequality. Given measurement choices the players shall announce an output based solely on the results of local operations---which they can discuss before the separation---on shared random data and shared copies of a so-called unit correlation. In the first part of this paper we show that in such a setting the simulation of any bipartite correlation, not containing the possibility of signaling, can be made arbitrarily accurate by increasing the number of shared Popescu-Rohrlich (PR) boxes. This establishes the PR box as a simple asymptotic unit of bipartite nonlocality. In the second part we study whether this property extends to the multipartite case. More generally, we ask if it is possible for separated players to asymptotically reproduce any nonsignaling statistic by local operations on bipartite unit correlations. We find that nonadaptive strategies are limited by a constant accuracy and that arbitrary strategies on nnn resource correlations make a mistake with a probability greater or equal to c/nc/nc/n, for some constant ccc.
Quantum mechanics postulates random outcomes. However, a model making the same output predictions but in a deterministic manner would be, in principle, experimentally indistinguishable from quantum theory. In this work we consider such models in the context of nonlocality on a device-independent scenario. That is, we study pairs of nonlocal boxes that produce their outputs deterministically. It is known that, for these boxes to be nonlocal, at least one of the boxes' outputs has to depend on the other party's input via some kind of hidden signaling. We prove that, if the deterministic mechanism is also algorithmic, there is a protocol that, with the sole knowledge of any upper bound on the time complexity of such an algorithm, extracts that hidden signaling and uses it for the communication of information.
One of the striking properties of quantum mechanics is the occurrence of the Bell-type non-locality. They are a fundamental feature of the theory that allows two parties that share an entangled quantum system to observe correlations stronger than possible in classical physics. In addition to their theoretical significance, non-local correlations have practical applications, such as device-independent randomness generation, providing private unpredictable numbers even when obtained using devices delivered by an untrusted vendor. Thus, determining the quantity of certifiable randomness that can be produced using a specific set of non-local correlations is of significant interest. First, we present an experimental realization of recent Bell-type operators designed to provide private random numbers that are secure against adversaries with quantum resources. We use semi-definite programming to give lower bounds on the generated randomness in terms of both min-entropy and von Neumann entropy in a device-independent scenario. Our results demonstrate the first experiment that certifies nearly two bits of randomness from binary measurements of two parties [1], [2]. Apart from single-round certification, we provide an analysis of finite-key protocol for quantum randomness expansion using the Entropy Accumulation Theorem. Second, we present an efficient and practical method for certifying quantum randomness using generalized measurements. Indeed, we have derived a method for self-testing the presence of POVMs. We present the certification of more than one bit of min-entropy from a single POVM on one of the qubits of an entangled state using a variant of the elegant Bell inequality. We also obtain more than one bit of randomness in a prepare and measure scenario of a similar structure with a POVM on a single qubit [3]. We provide numerical simulations to demonstrate the effectiveness of our randomness certification and expansion.
The question of security of practical device-independent protocols against no-signalling adversaries, the ultimate form of cryptographic security, has remained open. A key ingredient is to identify how the entropy in the raw outputs of a Bell test accumulates over nnn sequential runs (termed time-ordered no-signalling) against a no-signalling adversary. Previous numerical and analytical investigations for small nnn (≤5\leq 5≤5) had suggested that the min-entropy might not accumulate linearly in contrast to the case of quantum adversaries. Here we point out that despite the findings for small nnn, the min-entropy does in fact accumulate linearly for large nnn. We illustrate the difference in randomness accumulation against quantum and no-signalling adversaries with the paradigmatic example of the Chained Bell test for which we analytically derive the min-entropy. Finally, we illustrate the power of the no-signalling adversary by providing a class of attacks that allow an eavesdropper to perfectly guess the outputs of one player in general bipartite Pseudotelepathy games.
The key feature in correlations established by multi-party quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with no-signaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a Clauser-Horne-Shimony-Holt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest Popescu-Rohrlich set of causal stochastic processes (aka a PR box) in the no-signaling polytope, related by polyhedral duality to Bell's correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction [Formula: see text] of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
Given a pair of isolated devices that accept random binary inputs and return binary outputs, a user can deduce from the observed data alone if the underlying mechanism can be explained classically. Bell's theorem further states that a classical explanation can be ruled out if the devices perform certain measurements on an entangled quantum state, underpinning the security of cryptographic protocols that are device-independent (DI). For certain protocols, such as those performed in a tight space, it might be difficult to perfectly enforce the non-signaling assumption required in Bell's theorem. This prompts the question: is quantum nonlocality robust to such imperfections? We show that if a binary channel sends a noisy copy of one party's input to the other before any measurements take place, the answer is yes. We completely characterize the vertices and facets of the local polytope in this scenario, and identify Bell inequalities that certify non-signaling quantum correlations. This is possible even when a near perfect copy of the input is sent. We go on to show that the identified inequalities are more robust to depolarizing noise than the Clauser-Horne-Shimony-Holt inequality when certifying DI randomness in this scenario. In addition, we characterize the local polytope when both parties receive a noisy copy of each other's input and make similar conclusions, leaving many new potential Bell inequalities to be explored.