Compatibility of subsystem states

Abstract

We examine the possible states of subsystems of a system of bits or qubits. In the classical case (bits), this means the possible marginal distributions of a probability distribution on a finite number of binary variables; we give necessary and sufficient conditions for a set of probability distributions on all proper subsets of the variables to be the marginals of a single distribution on the full set. In the quantum case (qubits), we consider mixed states of subsets of a set of qubits; in the case of three qubits, we find quantum Bell inequalities -- necessary conditions for a set of two-qubit states to be the reduced states of a single mixed state of three qubits. We conjecture that these conditions are also sufficient.

19 pages, LaTeX. In memoriam Asher Peres. Substantial revision: one theorem removed, one author added

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Cited by in corpus (17)

• A hole-ographic spacetime
• Compatibility of subsystem states and convex geometry
• Compatible quantum correlations: on extension problems for Werner and isotropic states
• Mutually Unbiased Bases and Semi-definite Programming
• Spectrum conditions for symmetric extendible states
• Exponentially many entanglement and correlation constraints for multipartite quantum states
• On an Extension Problem for Density Matrices
• Positive maps and trace polynomials from the symmetric group
• Compatibility conditions from multipartite entanglement measures
• Symmetric extension of bipartite quantum states and its use in quantum key distribution with two-way postprocessing
• Some Ulam's reconstruction problems for quantum states
• Constraints on correlations in multiqubit systems
• A complete picture of the four-party linear inequalities in terms of the 0-entropy
• Refuting spectral compatibility of quantum marginals
• New Partial Trace Inequalities and Distillability of Werner States
• On random classical marginal problems with applications to quantum information theory
• Fully quantum inflation: quantum marginal problem constraints in the service of causal inference