Incompatibility in general probabilistic theories, generalized spectrahedra, and tensor norms

Abstract

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements.

60 pages, 4 figures. Changed the notation of the norm characterizing compatibility from rho-norm to c-norm

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Cited by in corpus (13)

• General probabilistic theories: An introduction
• Assemblages and steering in general probabilistic theories
• Measurement incompatibility vs. Bell non-locality: an approach via tensor norms
• Maximal violation of steering inequalities and the matrix cube
• A tensor norm approach to quantum compatibility
• A simple formulation of no-cloning and no-hiding that admits efficient and robust verification
• Symmetries and Wigner representations of operational theories
• Extreme points of matrix convex sets and their spanning properties
• Polytope compatibility -- from quantum measurements to magic squares
• On compatibility of binary qubit measurements
• Encoding and decoding of information in general probabilistic theories
• Beyond Operator Systems
• Matrix Extreme Points and Free extreme points of Free spectrahedra