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arXiv:1304.6425 · Physical Review A

Lack of measurement independence can simulate quantum correlations even when signaling can not

Manik Banik

Published Sep 27, 2013• 12 citations
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Abstract

In the Bell scenario, any nonlocal correlation shared between two spatially separated parties can be modeled deterministically either by allowing communications between the two parties or by restricting their free will in choosing the measurement settings. Recently, the Bell scenario has been generalized into a ``semiquantum'' scenario where external quantum inputs are provided to the parties. We show that in the semiquantum scenario, entangled states produce correlations whose deterministic explanation is possible only if measurement independence is reduced. Thus in simulating quantum correlation the semiquantum scenario reveals a qualitative distinction between signaling and measurement dependence which is absent in the Bell scenario. We further show that such distinction is not observed in the ``steering-game'' scenario, a special case of the semiquantum scenario.

DOI
10.1103/physreva.88.032118

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In the Bell scenario, any nonlocal correlation shared between two spatially separated parties can be modeled deterministically either by allowing communications between the two parties or by restricting their free will in choosing the measurement settings. Recently, the Bell scenario has been generalized into a semiquantum'' scenario where external quantum inputs are provided to the parties. We show that in the semiquantum scenario, entangled states produce correlations whose deterministic explanation is possible only if measurement independence is reduced. Thus in simulating quantum correlation the semiquantum scenario reveals a qualitative distinction between signaling and measurement dependence which is absent in the Bell scenario. We further show that such distinction is not observed in the steering-game'' scenario, a special case of the semiquantum scenario.
In a world described by quantum mechanics, one cannot have determinism, no-signaling and free will all together. Under validation of all these three assumptions, together, the set of statistical correlations are bounded to the set of Bell local correlations. In standard Bell scenario any non-local correlation has a deterministic explanation if one either sacrifices free will or allows signaling. Recently in [Phys. Rev. Lett. 108, 200401 (2012)] F. Buscemi generalizes the standard Bell scenario into `semi-quantum' Bell scenario. Interestingly, in this article, we show that in `semi-quantum' Bell game one can get correlations that cannot be obtained by giving up no signaling, alone; whereas such correlations can be obtained by sacrificing free will.
A central aspect of quantum information is that correlations between spacelike separated observers sharing entangled states cannot be reproduced by local hidden variable (LHV) models, a phenomenon known as Bell nonlocality. If one wishes to explain such correlations by classical means, a natural possibility is to allow communication between the parties. In particular, LHV models augmented with two bits of classical communication can explain the correlations of any two-qubit state. Would this still hold if communication is restricted to measurement outcomes? While in certain scenarios with a finite number of inputs the answer is yes, we prove that if a model must reproduce all projective measurements, then for any qubit-qudit state the answer is no. In fact, a qubit-qudit under projective measurements admits an LHV model with outcome communication if and only if it already admits an LHV model without communication. On the other hand, we also show that when restricted sets of measurements are considered (for instance, when the qubit measurements are in the upper hemisphere of the Bloch ball), outcome communication does offer an advantage. This exemplifies that trivial properties in standard LHV scenarios, such as deterministic measurements and outcome-relabelling, play a crucial role in the outcome communication scenario.
Bell nonlocality is a cornerstone of quantum theory with applications in information processing ranging from cryptography to distributed computing and game theory. Indeed, it is known that Bell's theorem can be formally linked to Bayesian games, allowing the use of nonlocal correlations to advise players and thereby achieve new points of equilibrium that are unavailable classically. Here we generalize this link, proving the connection between multistage games of incomplete information with Bell scenarios involving the communication of measurement outcomes between the parties. We apply the general framework for a few cases of interest and analyze the equilibrium reached by quantum nonlocal correlations.
Simulation tasks are insightful tools to compare information-theoretic resources. Considering a generalization of usual Bell scenarios where external quantum inputs are provided to the parties, we show that any entangled quantum state exhibits correlations that cannot be simulated using only shared randomness and classical communication, even when the amount and rounds of classical communication involved are unrestricted. We indeed construct explicit Bell-like inequalities that are necessarily satisfied by such classical resources but nevertheless violated by correlations obtainable from entangled quantum states, when measured a single copy at a time.
It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are nonlocal, in the sense that a Bell inequality is violated. They cannot, however, be used for superluminal signaling. It is also known that it is possible to write down sets of ``superquantum'' correlations that are more nonlocal than is allowed by quantum mechanics, yet are still nonsignaling. Viewed as an information-theoretic resource, superquantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signaling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signaling polytope in the case that two observers each choose from two possible measurements with ddd outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.
Abstract Classical and quantum physics provide fundamentally different predictions about experiments with separate observers that do not communicate, a phenomenon known as quantum nonlocality. This insight is a key element of our present understanding of quantum physics, and also enables a number of information processing protocols with security beyond what is classically attainable. Relaxing the pivotal assumption of no communication leads to new insights into the nature quantum correlations, and may enable new applications where security can be established under less strict assumptions. Here, we study such relaxations where different forms of communication are allowed. We consider communication of inputs, outputs, and of a message between the parties. Using several measures, we study how much communication is required for classical models to reproduce quantum or general no-signalling correlations, as well as how quantum models can be augmented with classical communication to reproduce no-signalling correlations.