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 authors would like to thank Franck Barthe for providing us with the proof of Lemma 6.2 in the even case, as well as MathOverflow users “Fedor Petrov” and “Steve” for providing simpler, more conceptual proofs for parts of Lemma 6.2. A.B. acknowledges financial support from the VILLUM FONDEN via the QMATH Centre of Excellence (Grant No.10059) and the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) via the Innovation Fund Denmark. I.N. was supported by the ANR project “ESQuisses” (grant number ANR-20-CE47-0014-01).