Maximally Nonlocal and Monogamous Quantum Correlations

Abstract

We introduce a version of the chained Bell inequality for an arbitrary number of measurement outcomes and use it to give a simple proof that the maximally entangled state of two d-dimensional quantum systems has no local component. That is, if we write its quantum correlations as a mixture of local correlations and general (not necessarily quantum) correlations, the coefficient of the local correlations must be zero. This suggests an experimental program to obtain as good an upper bound as possible on the fraction of local states and provides a lower bound on the amount of classical communication needed to simulate a maximally entangled state in d d dimensions. We also prove that the quantum correlations violating the inequality are monogamous among nonsignaling correlations and, hence, can be used for quantum key distribution secure against postquantum (but nonsignaling) eavesdroppers. Quantum theory predicts that measurements on separated entangled systems will produce outcome correlations that, in Bell's terminology [1], are not locally causal or, in what has become standard terminology, are nonlocal. In particular, if a Bell inequality is violated, then one cannot consistently assume that the outcomes of measurements on each system are predetermined and independent of the measurements carried out on the other system(s). Violation of Bell inequalities has been confirmed in numerous experiments [2].Violation of Bell inequalities not only tells us something fundamental about nature but also has practical applications. For example, the nonlocality of quantum correlations allows communication complexity problems to be solved using an amount of communication that is smaller than is possible classically [3]. Barrett-Hardy-Kent (BHK) recently showed that testing particular nonlocal quantum correlations allows two parties to distribute a secret key securely, in such a way that the security is guaranteed by the no-signaling principle alone [4] (i.e., without relying on the validity of quantum theory).In this Letter, we extend the chained Bell inequality to an inequality with an arbitrary number of measurement outcomes, which can thus be applied to states in arbitrary dimensions. In the limit of a large number of measurement settings, quantum mechanics predicts correlations for a maximally entangled bipartite state that resemble those of the tripartite Greenberger-Horne-Zeilinger (GHZ) state in that, with probability tending to one, the predictions of any local hidden variable model contradict those of quantum mechanics for at least one pair of measurements. We use this to give a constructive proof of a result originally due to Elitzur, Popescu, and Rohrlich (EPR) [5]: If the quantum correlations of a maximally entangled state of two qubits are written as a convex combination of local and nonlocal correlations, then the local fraction must be zero. Our proof extends EPR's result to maximally entangled states in any dimension and also removes the need for a technical assumption required for EPR's original proof [6]. Mor...

Local content of bipartite qubit correlations

“…no EPR2 decomposition with p L > 0 exists for this state. This is in fact a much more general result, as shown later by Barrett et al [4]: the maximally entangled state of two d-dimensional quantum systems, for any dimension d, has no local component.…”

“…On the other hand, an upper bound onp L (θ) can be obtained with the help of Bell inequalities [4]. Let I ≤ I L be a Bell inequality (defined by a linear combination of conditional probabilities), I Q the quantum value obtainable with the probability distribution P Q , and I N S (> I L ) the maximum value obtainable with non-signaling distributions.…”

“…Inspired also by models that Bell devised to reproduce the measurement statistics on a single qubit in the state |0 [12] (which gives precisely the marginals we want), or to approximate the statistics of the singlet state Marginals. Alice's and Bob's marginals corresponding to our local probability distribution P L are, as required in (14): 4 However, one hint that the argument is questionable is the fact that the no-signaling polytopes for arbitrarily many measurements but binary outcomes contain extremal points with non-random marginals [10,11].…”

“…The intuition is that the marginals are local properties, which should be concentrated on the local component only 4 .…”

“…The goal is to find, for a given state, a decomposition with the largest possible value for p L , denotedp L (θ), which characterizes the locality of the probability distribution P Q as defined by (3)(4)(5).…”