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In this work, we establish the connection between the study of freespectrahedra and the compatibility of quantum measurements with an arbitrarynumber of outcomes. This generalizes previous results by the authors formeasurements with two outcomes. Free spectrahedra arise from matricialrelaxations of linear matrix inequalities. A particular free spectrahedronwhich we define in this work is the matrix jewel. We find that thecompatibility of arbitrary measurements corresponds to the inclusion of thematrix jewel into a free spectrahedron defined by the effect operators of themeasurements under study. We subsequently use this connection to bound the setof (asymmetric) inclusion constants for the matrix jewel using results fromquantum information theory and symmetrization. The latter translate to newlower bounds on the compatibility of quantum measurements. Among the techniqueswe employ are approximate quantum cloning and mutually unbiased bases.

Compatibility of quantum measurements and inclusion constants for the matrix jewel