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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.
Highlights
Quantum steering is a phenomenon which was already discovered in the early days of quantum mechanics
Already in [32, 54], it has been shown that the correlations needed for quantum states to exhibit steering are stronger than mere entanglement, but weaker than the correlations required to violate a Bell inequality
Theorem 3.1 implies in particular that we can determine whether VQ(F) ≤ 1 by means of a semidefinite program, since the inclusion of free spectrahedra can be checked in this way [19, 25, 26]
Summary
Quantum steering is a phenomenon which was already discovered in the early days of quantum mechanics. In the same spirit as for Bell inequalities, we can consider steering inequalities [17] and compare their classical value (more precisely the value obtained using local hidden states (LHS) models) to their quantum value These are the objects of study in [37, 55]. Similar connections have been established previously by the authors of this work in the context of the compatibility of quantum measurements [8–10] This connection allows us to find tight upper bounds on the violation of steering inequalities for ensembles of fixed dimension and generated by a fixed number of dichotomic measurements on Alice’s side. The value above is achieved by a sequence of steering inequalities obtained by discretizing the Haar measure on the unitary group U(d) This type of result is reminiscent of [20], where the certification of genuine high-dimensional steering was considered, building on earlier work on dimension witnesses [13, 18] in the framework of Bell inequalities.
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