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Bell Inequalities with Communication

Dave Bacon, Ben Toner

Published Aug 9, 2002• 0 citations
OpenAlex

Abstract

What is the communication cost of the correlations produced by quantum theory? We generalize Bell inequalities to the setting of local realistic theories augmented by a fixed amount of classical communication. Suppose two parties choose one of M two-outcome measurements and exchange one bit of information. We present the complete set of inequalities for M=2, and the complete set of inequalities for the joint correlation observable for M=3. Remarkably, we find that correlations produced by quantum theory satisfy both of these sets of inequalities, irrespective of the particular quantum state or the specific measurements. One bit of communication is therefore sufficient to simulate quantum theory in both of these scenarios.

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What is the communication cost of the correlations produced by quantum theory? We generalize Bell inequalities to the setting of local realistic theories augmented by a fixed amount of classical communication. Suppose two parties choose one of M two-outcome measurements and exchange one bit of information. We present the complete set of inequalities for M=2, and the complete set of inequalities for the joint correlation observable for M=3. Remarkably, we find that correlations produced by quantum theory satisfy both of these sets of inequalities, irrespective of the particular quantum state or the specific measurements. One bit of communication is therefore sufficient to simulate quantum theory in both of these scenarios.
What is the communication cost of simulating the correlations produced by quantum theory? We generalize Bell inequalities to the setting of local realistic theories augmented by a fixed amount of classical communication. Suppose two parties choose one of M two-outcome measurements and exchange 1 bit of information. We present the complete set of inequalities for M=2, and the complete set of inequalities for the joint correlation observable for M=3. We find that correlations produced by quantum theory satisfy both of these sets of inequalities. One bit of communication is therefore sufficient to simulate quantum correlations in both of these scenarios.
In this paper, we consider the possible correlations between two parties using local machines and shared randomness with an additional amount of classical communication. This is a continuation of the work initiated by Bacon and Toner [Phys. Rev. Lett. 90, 157904 (2003)] who characterized the correlation polytope for $2\ifmmode\times\else\texttimes\fi{}2$ measurement settings with binary outcomes plus one bit of communication. Here, we derive a complete set of Bell inequalities for $3\ifmmode\times\else\texttimes\fi{}2$ measurement settings and a shared bit of communication. When the communication direction is fixed, nine Bell inequalities characterize the correlation polytope, whereas when the communication direction is bidirectional, 143 inequalities describe the correlations. We then prove a tight lower bound on the amount of communication needed to simulate all no-signaling correlations for a given number of measurement settings.
To reproduce in a local hidden variables theory correlations that violate Bell inequalities, communication must occur between the parties. We show that the amount of violation of a Bell inequality imposes a lower bound on the average communication needed to produce these correlations. Moreover, for every probability distribution there exists an optimal inequality for which the degree of violation gives the minimal average communication. As an example, to produce using classical resources the correlations that maximally violate the Clauser-Horne-Shimony-Holt inequality, $\sqrt{2}\ensuremath{-}1\ensuremath{\simeq}0.4142$ bits of communication are necessary and sufficient. For Bell tests performed on two entangled states of dimension d> 3d>~3d> 3 where each party has the choice between two measurements, our results suggest that more communication is needed to simulate outcomes obtained from certain nonmaximally entangled states than maximally entangled ones.
We investigate the amount of communication that must augment classical local hidden variable models in order to simulate the behavior of entangled quantum systems. We consider the scenario where a bipartite measurement is given from a set of possibilities and the goal is to obtain exactly the same correlations that arise when the actual quantum system is measured. We show that, in the case of a single pair of qubits in a Bell state, a constant number of bits of communication is always sufficient---regardless of the number of measurements under consideration. We also show that, in the case of a system of nnn Bell states, a constant times 2n{2}^{n}2n bits of communication is necessary.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
A quantum system of any fixed dimension can be prepared in a continuum of states, yet it cannot be used to transmit an unlimited amount of classical information. Similarly, the correlations observed between measurement outcomes on separate parts of a shared quantum system can be stronger than classical correlations, but they cannot transmit information. These fundamental limitations suggest that the statistics observed from quantum communication and quantum correlations may admit a simulation using a finite amount of classical communication. This expectation is confirmed in the smallest nontrivial quantum dimension, with two classical bits being necessary and sufficient to exactly simulate qubit communication and all correlations between qubits. Despite significant efforts during the previous decades, this remained the only solved case. Here we resolve both problems for every quantum dimension. The solution reveals an unexpected qualitative transition starting at dimension four: no finite amount of classical communication can exactly simulate ququart communication nor all quantum correlations of two entangled ququarts, even with unlimited shared randomness. One might have expected this transition, if it existed, to appear already for qutrits. Instead, we construct an explicit protocol that exactly simulates qutrit communication using 357357357 classical bits, and consequently, all correlations of two entangled qutrits.