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arXiv:0802.3632 · Physical Review A

Geometry of quantum correlations

Itamar Pitowsky

Published Jun 18, 2008• 23 citations
arXivHTMLPDF

Abstract

We consider the set $\mathcal{Q}$ of quantum correlation vectors for two observers, each with two possible binary measurements. Quadric (hyperbolic) inequalities which are satisfied by every $q∊\mathcal{Q}$ are proved, and equality holds on a two-dimensional manifold consisting of the local boxes and all quantum correlation vectors that maximally violate the Clauser-Horne-Shimony-Holt (CHSH) inequality. The quadric inequalities are tightly related to the CHSH inequality; they are their iterated versions. Consequently, it is proved that $\mathcal{Q}$ is contained in a hyperbolic cube whose axes lie along the nonlocal (Popescu-Rohrlich) boxes. As an application, a tight constraint on the rate of local boxes that must be present in every quantum correlation is derived. The inequalities allow one to test the validity of quantum mechanics on the basis of data available from experiments which test the violation of the CHSH inequality. It is noted how these results can be generalized to the case of $n$ sites, each with two possible binary measurements.

DOI
10.1103/physreva.77.062109

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We consider the set Q\mathcal{Q}Q of quantum correlation vectors for two observers, each with two possible binary measurements. Quadric (hyperbolic) inequalities which are satisfied by every q∊Qq∊\mathcal{Q}q∊Q are proved, and equality holds on a two-dimensional manifold consisting of the local boxes and all quantum correlation vectors that maximally violate the Clauser-Horne-Shimony-Holt (CHSH) inequality. The quadric inequalities are tightly related to the CHSH inequality; they are their iterated versions. Consequently, it is proved that Q\mathcal{Q}Q is contained in a hyperbolic cube whose axes lie along the nonlocal (Popescu-Rohrlich) boxes. As an application, a tight constraint on the rate of local boxes that must be present in every quantum correlation is derived. The inequalities allow one to test the validity of quantum mechanics on the basis of data available from experiments which test the violation of the CHSH inequality. It is noted how these results can be generalized to the case of nnn sites, each with two possible binary measurements.
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A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of 2\sqrt 22​, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.
Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value 222\sqrt{2}22​, known as Tsirelson's bound. In this paper, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X1,…,Xn{X}_{1},\dots{},{X}_{n}X1​,…,Xn​ and Y1,…,Yn{Y}_{1},\dots{},{Y}_{n}Y1​,…,Yn​ with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$ we have $\ensuremath{\mid}⟨{X}_{1}{Y}_{1}⟩+⟨{X}_{2}{Y}_{1}⟩+⟨{X}_{2}{Y}_{2}⟩+⟨{X}_{3}{Y}_{2}⟩+\ensuremath{\cdots}+⟨{X}_{n}{Y}_{n}⟩\ensuremath{-}⟨{X}_{1}{Y}_{n}⟩\ensuremath{\mid}\ensuremath{\leqslant}2n\phantom{\rule{0.2em}{0ex}}\mathrm{cos}[\ensuremath{\pi}∕(2n)]$. It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.
The key feature in correlations established by multi-party quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with no-signaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a Clauser-Horne-Shimony-Holt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest Popescu-Rohrlich set of causal stochastic processes (aka a PR box) in the no-signaling polytope, related by polyhedral duality to Bell's correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction [Formula: see text] of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two dimensions. This inequality is optimal in the same sense as the CHSH inequality for two-dimensional systems, namely, the maximal amount by which the inequality is violated consists of the maximal resistance to noise. We also discuss the physical meaning and general definition of the correlation functions. Furthermore, by giving another specific set of the correlation functions with the same physical meaning, we realize the inequality presented by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]].
The best upper bound for the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of ±1 valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendieck's constant as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four ±1 valued observables on both sites of a correlation experiment and show that the quantum violation approaches 1.58. Moreover, we estimate the maximal quantum violation of a correlation experiment for large and equal number of ±1 valued observables on both sites. In this case, the maximal quantum violation converges to for very large n , which coincides with the approximate value of Grothendieck's constant.