Geometry of the set of quantum correlations

Abstract

It is well known that correlations predicted by quantum mechanics cannot be explained by any classical (local-realistic) theory. The relative strength of quantum and classical correlations is usually studied in the context of Bell inequalities, but this tells us little about the geometry of the quantum set of correlations. In other words, we do not have good intuition about what the quantum set actually looks like. In this paper we study the geometry of the quantum set using standard tools from convex geometry. We find explicit examples of rather counter-intuitive features in the simplest non-trivial Bell scenario (two parties, two inputs and two outputs) and illustrate them using 2-dimensional slice plots. We also show that even more complex features appear in Bell scenarios with more inputs or more parties. Finally, we discuss the limitations that the geometry of the quantum set imposes on the task of self-testing.

arXiv comment11 + 8 pages, 6 figures, v2: added an argument relating self-testing and extremality, v3: typos corrected, results unchanged, published version

References in corpus (57)

• Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km
• Bell nonlocality
• Significant-loophole-free test of Bell's theorem with entangled photons
• A strong loophole-free test of local realism
• Information Causality as a Physical Principle
• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Non-local correlations as an information theoretic resource
• All multipartite Bell correlation inequalities for two dichotomic observables per site
• Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• Robust Self Testing of the Singlet
• A glance beyond the quantum model
• Local orthogonality as a multipartite principle for quantum correlations
• Almost quantum correlations
• Symmetries of the Bell correlation inequalities
• Sum-of-squares decompositions for a family of CHSH-like inequalities and their application to self-testing
• Guess your neighbour's input: a multipartite non-local game with no quantum advantage
• Popescu-Rohrlich correlations as a unit of nonlocality
• A Combinatorial Approach to Nonlocality and Contextuality
• Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities
• Information-Theoretic Implications of Quantum Causal Structures
• Local Quantum Measurement and No-Signaling Imply Quantum Correlations
• Interconversion of Nonlocal Correlations
• Robust Self Testing of Unknown Quantum Systems into Any Entangled Two-Qubit States
• Lifting Bell inequalities
• Recovering part of the quantum boundary from information causality
• Asymptotic violation of Bell inequalities and distillability
• Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality
• Extremal correlations of the tripartite no-signaling polytope
• Self-testing of binary observables based on commutation
• All the self-testings of the singlet for two binary measurements
• Device-independent bounds for Hardy's experiment
• Revealing non-classicality of inaccessible objects
• Classifying 50 years of Bell inequalities
• Closed sets of non-local correlations
• Nonclassical correlations from randomly chosen local measurements
• Device-independent tomography of multipartite quantum states
• Detection Loophole in Bell experiments: How post-selected local correlations can look non-local
• Identifying Nonconvexity in the Sets of Limited-Dimension Quantum Correlations
• Extremal quantum correlations for N parties with two dichotomic observables per site
• On the local and non-local content of bipartite qubit and qutrit correlations
• Exploring the limits of quantum nonlocality with entangled photons
• Lower bounds on the entanglement needed to play XOR non-local games
• Self-testing properties of Gisin's elegant Bell inequality
• On the security and degradability of Gaussian channels
• Quantum Online Memory Checking
• Generalized self-testing and the security of the 6-state protocol
• How much larger quantum correlations are than classical ones
• No quantum realization of extremal no-signaling boxes
• New Quantum Bounds for Inequalities involving Marginal Expectations
• Quantum Bell Inequalities from Macroscopic Locality
• Geometric Decompositions of Bell Polytopes with Practical Applications
• Simple conditions constraining the set of quantum correlations
• A unified view on Hardy's paradox and the CHSH inequality
• Trade-offs in multi-party Bell inequality violations in qubit networks
• Non-closure of the set of quantum correlations via graphs
• Many-box locality

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