Quantum Correlations in the Minimal Scenario

Thinh P. Le1, Chiara Meroni2, Bernd Sturmfels3,4, Reinhard F. Werner5, and Timo Ziegler5

1Institute for Quantum Optics and Quantum Information Vienna, Boltzmanngasse 3 1090 Vienna, Austria
2Institute for Computational and Experimental Research in Mathematics, 121 South Main Street Providence RI 02903, USA
3Max Planck Institute for Mathematics in the Sciences Leipzig, Inselstrasse 22 04103 Leipzig, Germany
4Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840 Berkeley CA 94720-3840, USA
5Insitute für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2 30167 Hannover, Germany

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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. We review and systematize what is known about $\mathcal{Q}$, 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 $\mathcal{Q}$, also identifying the boundary structures. The second principle, self-duality, is an isomorphism between $\mathcal{Q}$ 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 $\mathcal{Q}$ to the polytope of no-signalling correlations, which contains $\mathcal{Q}$.
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 $\mathcal{Q}$ involving the determinant of the correlation matrix.

Characterizing and understanding the set of allowed quantum correlations has been an important goal since the birth of quantum theory. In this work, we deliver the most comprehensive understanding of the set of quantum correlation in the smallest nontrivial scenario from several perspectives: geometry and applications. We supplement our theoretical understanding with lots of exact visualizations in three dimensions.

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The above citations are from Crossref's cited-by service (last updated successfully 2024-06-19 18:09:41) and SAO/NASA ADS (last updated successfully 2024-06-19 18:09:42). The list may be incomplete as not all publishers provide suitable and complete citation data.