Quantum Correlations in the Minimal Scenario

• URL: http://arxiv.org/abs/2111.06270v2
• Date: Mon, 2 May 2022 09:17:39 GMT
• Title: Quantum Correlations in the Minimal Scenario
• Authors: Thinh P. Le, Chiara Meroni, Bernd Sturmfels, Reinhard F. Werner, and Timo Ziegler
• Abstract summary: In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. The resulting four-dimensional convex body of correlations, denoted $mathcalQ$, is fundamental for quantum information theory.
• Score: 0.6116681488656471
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\mathcal{Q}$, is fundamental for quantum information theory. It is here studied through the lens of convex algebraic geometry. We review and systematize what is known and add many details, visualizations, and complete proofs. A new result is that $\mathcal{Q}$ is isomorphic to its polar dual. The boundary of $\mathcal{Q}$ consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These share all basic properties with the usual maximally CHSH-violating correlations. 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, the application of the sine function to each coordinate. This transforms the classical polytope exactly into the correlation body $\mathcal{Q}$, also identifying the boundary structures. The second principle, self-duality, reveals the polar dual, i.e., the set of all Tsirelson inequalities satisfied by all quantum correlations. The convex body $\mathcal{Q}$ includes the classical correlations, a cross polytope, and is contained in the no-signaling body, a 4-cube. These polytopes are dual to each other, and the linear transformation realizing this duality also identifies $\mathcal{Q}$ with its dual.

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