Compatibility of quantum measurements and inclusion constants for the matrix jewel

Abstract

In this work, we establish the connection between the study of free spectrahedra and the compatibility of quantum measurements with an arbitrary number of outcomes. This generalizes previous results by the authors for measurements with two outcomes. Free spectrahedra arise from matricial relaxations of linear matrix inequalities. A particular free spectrahedron which we define in this work is the matrix jewel. We find that the compatibility of arbitrary measurements corresponds to the inclusion of the matrix jewel into a free spectrahedron defined by the effect operators of the measurements under study. We subsequently use this connection to bound the set of (asymmetric) inclusion constants for the matrix jewel using results from quantum information theory and symmetrization. The latter translate to new lower bounds on the compatibility of quantum measurements. Among the techniques we employ are approximate quantum cloning and mutually unbiased bases.

arXiv commentv5: section 3.3 has been expanded significantly to incorporate the generalization of the Cartesian product and the direct sum to matrix convex sets. Many other minor modifications. Closed to the published version

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Cited by in corpus (12)

• Quantum random access codes and incompatibility of measurements
• Witnessing incompatibility of quantum channels
• Diagonal unitary and orthogonal symmetries in quantum theory
• Incompatibility in general probabilistic theories, generalized spectrahedra, and tensor norms
• The compatibility dimension of quantum measurements
• Maximal violation of steering inequalities and the matrix cube
• A tensor norm approach to quantum compatibility
• Dilations of unitary tuples
• A Fisher information-based incompatibility criterion for quantum channels
• On compatibility of binary qubit measurements
• Beyond Operator Systems
• Polytope compatibility -- from quantum measurements to magic squares