Classical simulation of traceless binary observables on any bipartite quantum state Julien Degorre, 1,2 Sophie Laplante, 1 and Jérémie Roland 1,3 1 Laboratoire de Recherche en Informatique, UMR 8263, Université Paris-Sud, 91405 Orsay, France 2 Laboratoire d’Informatique Théorique et Quantique, Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, Canada 3 Quantum Information and Communication, Ecole Polytechnique, CP 165/59, Université Libre de Bruxelles, 1050 Bruxelles, Belgium Received 14 July 2006; published 10 January 2007 We present a protocol to simulate the quantum correlations of an arbitrary bipartite state, when the parties perform a measurement according to two traceless binary observables. We show that log 2 d bits of classical communication is enough on average, where d is the dimension of both systems. To obtain this result, we use the sampling approach for simulating the quantum correlations. DOI: 10.1103/PhysRevA.75.012309 PACS numbers: 03.67.Hk, 03.65.Ud I. INTRODUCTION In 1964 John Bell showed that the correlations exhibited by the EPR gendanken experiment 1,2 could not be repro- duced by a so-called local hidden variable model, that is, a model where the parties share an infinite amount of locally created hidden variables. This nonlocal aspect is one of the strangest properties of quantum physics, and understanding this notion remains an important problem. Recently, quantum information processing has provided a new point of view to understand quantum nonlocality. In particular, the framework of communication complexity has provided tools to study nonlocality. For example, two parties, whom we call Alice and Bob, cannot reproduce quantum correlations if they share only hidden random variables shared randomness, but in some cases, if they are allowed to use some additional resources, it becomes possible for them to reproduce the quantum correlations. It is precisely this amount of addi- tional resources which we consider here; they allow us to quantify quantum nonlocality. The problem of reproducing the statistics of projective measurements on the singlet has been widely studied, with communication as the additional resource. Maudlin 3 pre- sented a protocol in the case of measurements in the real plane and proved an average-case communication upper bound of 1.17 bits, and Brassard, Cleve, and Tapp gave a protocol with a worst-case communication upper bound of 8 bits, for arbitrary projective measurements 4. Steiner, in- dependently of Maudlin, gave a protocol for projective mea- surements in the real plane with an average-case upper bound on communication of 1.48 bits, and Cerf, Gisin, and Massar 5 proved that for an arbitrary projective measure- ments, 1.19 bits of communication sufficed on average. Re- cently, Toner and Bacon have shown that one bit of commu- nication is always enough to reproduce the quantum correlations for arbitrary projective measurements on the sin- glet state 6. Some other resources have been used to simulate quantum correlations resulting from projective measurements on the singlet state. These include post-selection 7 and nonlocal boxes 8. In 2005, we have shown that simulating these quantum correlations could be reduced to a sampling prob- lem, from which we derived many of the above mentioned protocols, in a unified framework 9. Nevertheless, these results address the simplest scenario, that is, simulating the correlations resulting from measure- ments on the singlet state mostly for projective measure- ments, with a few extensions to POVMs. There are few results about nonmaximally entangled pairs, multiparty states, higher dimensional states, or more general measure- ments. One significant result in this direction is a protocol from Massar et al. able to reproduce the correlations of ar- bitrary measurements on any entangled pair of d-dimensional states qudits using Od log 2 d bits of communication but no local hidden variables 10. In this paper, we use the sampling approach developed in Ref. 9 and generalize it to the case of a bipartite pair of arbitrary-dimension states qudits. We study the case where the parties make a restricted type of measurement with only two opposite outcomes 1,-1, that we call traceless binary observable, or TBO. Furthermore we impose no constraint on the bipartite pure state whose correlations the parties wish to simulate; it could be maximally entangled or non-maximally entangled. For an arbitrary bipartite qudit pair, we show that log 2 d bits of communication on average are enough to simulate the joint correlations of the outcomes where the joint correla- tion is defined as the expectation value of the product of Alice’s and Bob’s outcome. In the special case of maximally entangled qudit pairs, our protocol also reproduces the mar- ginal probabilities, and therefore the full probability distribu- tion. We will begin by describing the quantum correlations in arbitrary dimensions that we want to simulate classically. Then, using the sampling approach, we will present a gener- alization of the local biased random variable model for arbi- trary dimensions, and present a classical protocol which uses log 2 d bits of communication to simulate the joint quantum correlations of an arbitrary bipartite qudit pair. II. THE QUANTUM CORRELATIONS In this section we describe the system that we want to simulate classically using some communication. Two parties, Alice and Bob, share an arbitrary bipartite qudit pair. They each perform a measurement on their part, where the mea- PHYSICAL REVIEW A 75, 012309 2007 1050-2947/2007/751/0123095 ©2007 The American Physical Society 012309-1