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.Comment: v5: 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 versio
