Compatibility of subsystem states and convex geometry

Abstract

The subsystem compatibility problem, which concerns the question of whether a set of subsystem states are compatible with a state of the entire system, has received much study. Here we attack the problem from a new angle, utilising the ideas of convexity that have been successfully employed against the separability problem. Analogously to an entanglement witness, we introduce the idea of a compatibility witness, and prove a number of properties about these objects. We show that the subsystem compatibility problem can be solved numerically and efficiently using semidefinite programming, and that the numerical results from this solution can be used to extract exact analytic results, an idea which we use to disprove a conjecture about the subsystem problem made by Butterley et al. [Found. Phys. 36 83 (2006)]. Finally, we consider how the ideas can be used to tackle some important variants of the compatibility problem; in particular, the case of identical particles (known as N-representability in the case of fermions) is considered.

12 pages APS format, some minor changes made

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• Spectrum conditions for symmetric extendible states
• Pinning of Fermionic Occupation Numbers: General Concepts and One Dimension
• Quantum Convex Support
• Exponentially many entanglement and correlation constraints for multipartite quantum states
• On an Extension Problem for Density Matrices
• The Complexity of the Consistency and N-representability Problems for Quantum States
• Positive maps and trace polynomials from the symmetric group
• Symmetric extension of bipartite quantum states and its use in quantum key distribution with two-way postprocessing
• Fast quantum simulation of electronic structure by spectrum amplification
• Resource Marginal Problems
• Reducible Correlations in Dicke States
• Certifying nonlocality from separable marginals
• The Local Consistency Problem for Stoquastic and 1-D Quantum Systems
• Negativity-Mutual Information conversion and coherence in two-coupled harmonic oscillators
• Recoverability from direct quantum correlations
• Refuting spectral compatibility of quantum marginals
• Minimum Relative Entropy for Quantum Estimation: Feasibility and General Solution
• Fully quantum inflation: quantum marginal problem constraints in the service of causal inference