# Maximal violation of steering inequalities and the matrix cube

1QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark
2Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France

### Abstract

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.

The phenomenon of quantum steering was already discovered in the early days of quantum mechanics. Quantum states exhibiting quantum steering have correlations which are stronger than mere entanglement, but not necessarily as strong as the correlations needed to violate a Bell inequality. In a setting with two parties, Alice and Bob, Alice can use a steerable state to influence Bob's system by creating collections of states, so-called assemblages, on his side in such a manner that he has to conclude that Alice can influence his system even without trusting her. Experimentally, steering can be certified by the violation of a steering inequality.

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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