Quantum Correlations in the Minimal Scenario

Abstract

In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted Q, is fundamental for quantum information theory. We review and systematize what is known about Q, and add many details, visualizations, and complete proofs. In particular, we provide a detailed description of the boundary, which consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These patches are separated by cubic surfaces of non-exposed extreme points. We provide a trigonometric parametrization of all extreme points, along with their exposing Tsirelson inequalities and quantum models. All non-classical extreme points (exposed or not) are self-testing, i.e., realized by an essentially unique quantum model.Two principles, which are specific to the minimal scenario, allow a quick and complete overview: The first is the pushout transformation, i.e., the application of the sine function to each coordinate. This transforms the classical correlation polytope exactly into the correlation body Q, also identifying the boundary structures. The second principle, self-duality, is an isomorphism between Q and its polar dual, i.e., the set of affine inequalities satisfied by all quantum correlations (“Tsirelson inequalities''). The same isomorphism links the polytope of classical correlations contained in Q to the polytope of no-signalling correlations, which contains Q.We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observables, and establish a new non-linear inequality for Q involving the determinant of the correlation matrix.

Quantum circuits for measuring weak values, Kirkwood–Dirac quasiprobability distributions, and state spectra

“…This was observed numerically in [85]. Moreover, its bounds were analytically found to correspond to the boundaries of an elliptope, discussed in detail in [88]. 21 More formally, we establish convexity by showing that if a triple of overlaps (∆ 12 , ∆ 13 , ∆ 23 ) is pure-state realizable, then so is any triple in the line segment connecting it to (0, 0, 0), i.e.…”

“…This set was shown numerically to be convex in [85]. It is sometimes referred to as the body of quantum correlations, and parts of it correspond to the elliptope of [88]. In [85,88] it was shown that points in Q • 3 satisfy the inequality…”

“…There is a dual relationship between the scenarios investigated in[88] and those in[85]. For more information about these aspects, we refer the reader to the discussion in[86].…”

Broken Arrows: Hardy–Unruh Chains and Quantum Contextuality

“…If, for instance, we want to simulate the correlation array for a Hardy-Unruh setup in Figure 9 or Figure 11 with one of our raffles, the problem is not to obtain a non-zero probability for one particular outcome, but to obtain it while at the same time obtaining zero probabilities for several other outcomes. 13 In other words, rather than focusing on individual entries, we need to consider a correlation array as a whole. 14 Despite being taken down a notch, Hardy-Unruh chains remain valuable.…”

“…The extra minus sign affects Table 1, Figure 7 and Equations ( 13) and ( 14) below. 5 See Goh et al [11], Chen et al [12] and Le et al [13] for interesting recent work in this tradition, to which Tsirelson made important early contributions (see, for instance, [14], p. 3, Figure 2). In a paper from 1987 [15], Tsirelson already noted: "As one can easily see, the set Cor(m, n) of all quantum realized m×n correlation matrices .…”