On the relationship between convex bodies related to correlation experiments with dichotomic observables

Abstract

In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. Phys. A: Math. Gen. 38 10971-87) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329-45) to represent hidden deterministic behaviors, quantum behaviors, and no-signalling behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the no-signalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m, n) dichotomic observables when m = 4, n = 4 and when min{m, n} ≤ 3, and give a new general family of correlation inequalities. PACS classificationnumbers: 03.65.Ud, 02.40.Ft, 02.10.Ud Keywords: Bell inequalities, quantum behaviors, the cut polytope, the no-signaling polytope, semidefinite programming 1 The convex sets arising from the classical, quantum and nosignaling correlation experiments Our terminology for bodies related to quantum correlations follows closely that of Tsirelson [33] whilst for cut polyhedra we follow that of Deza and Laurent [12]. We consider the following (quantum) correlation experiment. Suppose that two parties, say Alice and Bob, share a quantum mixed state ρ, or a nonnegative Hermitian operator ρ on a Hilbert space H A ⊗ H B with tr ρ = 1. Here H A and H B are Hilbert spaces representing the subsystems owned by Alice and Bob, respectively. Alice has m ±1-valued observables A 1 , . . . , A m in the space H A , i.e. Hermitian operators A i on H A whose eigenvalues are within [−1, 1]. Similarly Bob has n ±1-valued observables B 1 , . . . , B n in H B . Alice

Bounds on quantum correlations in Bell-inequality experiments

“…If so this could complement the methods of Refs. [22,24,25] Here, we provide an explicit form for the matrices F m and constants c m that define the semidefinite program (14). By setting…”

“…In particular, Tsirelson [19] has demonstrated, using what is now known as Tsirelson's vector construction, that in a Bell-CHSH setup, bipartite quantum systems of arbitrary dimensions cannot exhibit correlations stronger than 2 √ 2 -a value now known as Tsirelson's bound. Recently, analogous bounds for more complicated Bell inequalities have also been investigated by Filipp and Svozil [20], Buhrman and Massar [21], Wehner [22], Toner [23], Avis et al [24] and Navascués et al [25]. On a related note, bounds on quantum correlations for given local measurements, rather than given quantum state, have also been investigated by Cabello [26] and Bovino et al [27].…”

More efficient Bell inequalities for Werner states

“…For m > 2 one needs to resort to numerical programs for computing the inequalities corresponding to the inequivalent facets of the correlation polytope. Up to m = 4 all the correlation inequalities have been computed [26], and the two inequivalent inequalities obtained are in fact less efficient than the CHSH one for Werner states. However, the complexity of the computation exponentially grows with m (in fact, this is an NP-complete problem [24]), therefore there is no hope to completely characterize all the facets of the correlation polytope for any given m. Thus in general one needs to look for alternative methods.…”

Leggett-Garg inequalities and the geometry of the cut polytope

“…One could also consider the maximal violations of LGIs possible in quantum mechanics. For Bell inequalities, the maximal violations are closely related to the elliptope, a semidefinite relaxation of the cut polytope defined by the set of negative type inequalities [12]. In the case of LGIs, however, an experimentalist is free to perform a measurement and ignore the outcomes: by exploiting the quantum Zeno effect [17] it then becomes possible in principle to avoid any meaningful constraint on the correlation functions.…”

“…[10]. Avis et al described a relationship between the Bell polytope and a projecion of the cut polytope [11,12], a polytope which is isomorphic to the correlation polytope, and studied in depth in Ref.[10]. They were then able to offer 44,368,793 inequivalent tight Bell inequalities other than those of the CHSH form for the bipartite setting where each party measures ten dichotomic observables [11].…”