In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension.
In this work, we look at how robust quantum steering is to noise. We have found that noise robustness in this context can be studied using free spectrahedra. These are objects which arise in optimization theory as relaxations of linear matrix inequalities. Such relaxations are used to find tractable approximations to intractable problems phrased as inclusion problems of spectrahedra. Inclusion constants for these objects quantify how well the relaxation captures the original problem. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, a free spectrahedron which is well-studied in optimization. This allows us to find new upper bounds on the maximal violation of steering inequalities and hence to quantify the amount of steerability available for fixed physical parameters such as the dimension of Bob's system. Finally, we show that previously obtained violations are optimal.
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