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arXiv:1410.0947 · Physical Review Letters

No Quantum Realization of Extremal No-Signaling Boxes

Ravishankar Ramanathan, Jan Tuziemski, Michał Horodecki, Paweł Horodecki

Published Jul 25, 2016• 22 citations
arXivHTMLPDF

Abstract

The study of quantum correlations is important for fundamental reasons as well as for quantum communication and information processing tasks. On the one hand, it is of tremendous interest to derive the correlations produced by measurements on separated composite quantum systems from within the set of all correlations obeying the no-signaling principle of relativity, by means of information-theoretic principles. On the other hand, an important ongoing research program concerns the formulation of device-independent cryptographic protocols based on quantum nonlocal correlations for the generation of secure keys, and the amplification and expansion of random bits against general no-signaling adversaries. In both these research programs, a fundamental question arises: Can any measurements on quantum states realize the correlations present in pure extremal no-signaling boxes? Here, we answer this question in full generality showing that no nontrivial (not local realistic) extremal boxes of general no-signaling theories can be realized in quantum theory. We then explore some important consequences of this fact.

DOI
10.1103/physrevlett.117.050401

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The study of quantum correlations is important for fundamental reasons as well as for quantum communication and information processing tasks. On the one hand, it is of tremendous interest to derive the correlations produced by measurements on separated composite quantum systems from within the set of all correlations obeying the no-signaling principle of relativity, by means of information-theoretic principles. On the other hand, an important ongoing research program concerns the formulation of device-independent cryptographic protocols based on quantum nonlocal correlations for the generation of secure keys, and the amplification and expansion of random bits against general no-signaling adversaries. In both these research programs, a fundamental question arises: Can any measurements on quantum states realize the correlations present in pure extremal no-signaling boxes? Here, we answer this question in full generality showing that no nontrivial (not local realistic) extremal boxes of general no-signaling theories can be realized in quantum theory. We then explore some important consequences of this fact.
Abstract The problem of achieving security of device-independent (or semi-device-independent) cryptography (for quantum key distribution and randomness generation) against the most general no-signaling adversaries has remained open. It has been recognized that the realization of extremal no-signaling non-local boxes (or extremal no-signaling non-local assemblages) could provide a route toward devising such highly secure protocols. We first prove a general no-go result that in the Bell non-locality scenario, quantum theory does not allow us to realize any extremal no-signaling non-local box, even if scenarios of arbitrary sequential measurements are considered. On the other hand, we secondly prove a positive result showing that a one-sided device-independent scenario where a single party trusts their qubit system is already sufficient for quantum theory to realize a self-testing extremal non-local point within the set of no-signaling assemblages.
Extremal non-signaling (ENS) boxes are correlations that correspond to vertices of the non-signaling polytope of a Bell scenario. Neither quantum theory nor any theory for ideal measurements allows for ENS boxes. That is, according to quantum theory, ENS boxes are nonphysical. Still, ENS boxes are crucial for addressing a number of problems in Bell nonlocality. Here, we obtain ENS boxes in arbitrary bipartite Bell scenarios and present the complete list of ENS boxes for several unexplored scenarios. Equipped with the boxes, we revisit several foundational questions. We find that already two copies of any ENS box violate the exclusivity (or local orthogonality) and Specker's principles. We provide the minimal decomposition of the magic square correlation - the simplest known perfect correlation in nature - in terms of ENS boxes. We identify the minimal scenario in which a dit of communication (with d < 6) is insufficient to simulate ENS boxes. Our results show that the ENS boxes approach leads to new results and opens new avenues for research.
Nonlocality is arguably one of the most fundamental and counterintuitive aspects of quantum theory. Nonlocal correlations could, however, be even more nonlocal than quantum theory allows, while still complying with basic physical principles such as no-signaling. So why is quantum mechanics not as nonlocal as it could be? Are there other physical or information-theoretic principles which prohibit this? So far, the proposed answers to this question have been only partially successful, partly because they are lacking genuinely multipartite formulations. In Fritz et al. [Nat. Commun. 4, 2263 (2013)], we introduced the principle of local orthogonality (LO), an intrinsically multipartite principle which is satisfied by quantum mechanics but is violated by nonphysical correlations. Here we further explore the LO principle, presenting additional results and explaining some of its subtleties. In particular, we show that the set of no-signaling boxes satisfying LO is closed under wirings, present a classification of all LO inequalities in certain scenarios, show that all extremal tripartite boxes with two binary measurements per party violate LO, and explain the connection between LO inequalities and unextendible product bases.
A key aspect in quantum information is to understand the advantage offered by quantum systems over classical ones in communication tasks. In recent years, a fundamental approach to this problem has been developed, focusing on quantum correlations in prepare-and-measure scenarios. Inspired by the developments in Bell nonlocality and device-independent information processing, this line of research aims to characterize the possibilities and limits of quantum systems for communication, in particular to precisely capture the advantage they offer over classical systems. In addition to fundamental insights, these ideas also underpin the concept of semi-device-independent quantum information processing. Exploring trade-offs between security, performance and ease-of-implementation, this approach opens promising directions for novel quantum information processing technologies and devices. A number of protocols and proof-of-principle demonstrations have been reported in recent years, in particular for quantum randomness certification and key distribution. Here, we provide a comprehensive introduction to quantum prepare-and-measure correlations and semi-device independent applications.
One of the most fascinating consequences of quantum theory is non‐locality, i.e., the fact that the measurement outcomes of (spatially separated) parts of a system can have a correlation unexplainable by shared classical information. Note that at the same time, these correlations are non‐signaling and do not allow for message transmission. Popescu and Rohrlich have defined a non‐local box as a “basic building block of non‐locality” and initiated a systematic study of non‐local correlations and their applications. However, they left open the question of whether any non‐signaling correlation can be simulated by such non‐local boxes. We show that the answer is yes with respect to arbitrarily accurate approximations.
In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k) -parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.