Minimal and maximal matrix convex sets

Abstract

Abstract. To every convex body K ⊆ R d , one may associate a minimal matrix convex set W min (K), and a maximal matrix convex set W max (K), which have K as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies This constant is sharp, and it is new for all p = 2. Moreover, for some sets K we find a minimal set L for whichIn particular, we obtain that a convex body K satisfies W max (K) = W min (K) if and only if K is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every d-tuple of self-adjoint operators of norm less than or equal to 1, can be dilated to a commuting family of self-adjoints, each of norm at most √ d. We also introduce new explicit constructions of these (and other) dilations.

Compatibility of Quantum Measurements and Inclusion Constants for the Matrix Jewel

“…Contrary to the matrix diamond appearing in the study of binary measurements [BN18], the matrix jewel has not been studied in the literature on free spectrahedra. In algebraic convexity, the matrix convex sets having received the most attention are the matrix cube [BTN02,HKMS19], the different matricial notions of sphere [HKMS19,DDOSS17], and the maximal spectrahedra built upon p spaces [PSS18]. These examples have symmetries that the matrix jewel lacks, rendering its structure more involved.…”

“…The right hand side was studied in [BN18]. From [BN18, Theorem VIII.8], which uses results from [PSS18], we obtain ∆(g, d, k) ⊆ QC g ∀d ≥ 2 g−1 2…”

“…As B was arbitrary, this proves the first assertion. The second follows from [BN18, Theorem VII.7], which adapts results from [PSS18].…”

Joint measurability of quantum effects and the matrix diamond

“…We present in this section two upper bounds (i.e. containing sets) for the Γ and Γ 0 sets, one coming from quantum information theory [Zhu15] and another one coming from matrix convex set theory [PSS18]. These two upper bounds are interesting in two different regimes: the first one applies when the number of POVMs is larger than the dimension of the quantum system, while the second one applies in the complementary regime, where the dimension is large with respect to the number of POVMs.…”

“…which is the conclusion we aimed for. In the equation above, we have used the following fact (see [PSS18,equation (5.4)] for the corresponding statement): for non-negative scalars a 1 , . .…”