It is shown that in the finite-dimensional case the extreme points of the convex set of all N-representable 2-matrices D2 are exposed points and that for the case N = 3 and N = 4 the preimage of an exposed point under the contraction map is unique. In addition, the convex structure of both D2 and its associated polar is given in the case where the 1-rank is equal to N + 2. A long list of exposed points of D2 is given which includes all previously published examples as well as some new ones. However, this list is shown not to be exhaustive.
It is shown that in the finite-dimensional case the extreme points of the convex set of all N-representable 2-matrices D2 are exposed points and that for the case N = 3 and N = 4 the preimage of an exposed point under the contraction map is unique. In addition, the convex structure of both D2 and its associated polar is given in the case where the 1-rank is equal to N + 2. A long list of exposed points of D2 is given which includes all previously published examples as well as some new ones. However, this list is shown not to be exhaustive.
In [J. Math. Phys. 13, 1608–1621 (1972)], Erdahl10.1063/1.1665885 considered the convex structure of the set of N-representable 2-body reduced density matrices in the case of fermions. Some of these results have a straightforward extension to the m-body setting and to the more general quantum marginal problem. We describe these extensions, but cannot resolve a problem in the proof of Erdahl's claim that every extreme point is exposed in finite dimensions. Nevertheless, we can show that when 2m ⩾ N every extreme point of the set of N-representable m-body reduced density matrices has a unique pre-image in both the symmetric and anti-symmetric setting. Moreover, this extends to the quantum marginal setting for a pair of complementary m-body and (N − m)-body reduced density matrices.
A new method of analysing positive bistochastic maps on the algebra of complex matrices [Formula: see text] has been proposed. By identifying the set of such maps with a convex set of linear operators on [Formula: see text], one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been shown that all positive bistochastic maps, extremal in the set of all positive maps of [Formula: see text], that are not Jordan isomorphisms of [Formula: see text] are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one-dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.
A natural and intrinsic characterization of the structure of the set C of positive unital maps is given, i.e. it is shown that C is isometrically isomorphic to the subset $\gD$ of bp-positive density matrices endowed with the geometry given by the norm α dual to the Grothendieck projective norm π, the structure of $\gD$ is determined by the set of its exposed points, and finally a characterization of exposed points of $\gD$ in terms of convex analysis is presented. This seems to be an answer to an old open problem, characterization of the structure of the set of positive maps, studied both in Quantum Information and Operator Algebras. Our arguments are based on the concept of exposed points and links between tensor products and mapping spaces. It should be emphasized that the concept of exposed point depends strongly on the geometry, hence the role of the norm α is crucial.
We construct a large class of indecomposable positive linear maps from the matrix algebra into the matrix algebra, which generate exposed extreme rays of the convex cone of all positive maps. We show that extreme points of the dual faces for separable states arising from these maps are parametrized by the Riemann sphere, and the convex hulls of the extreme points arising from a circle parallel to the equator have exactly the same properties with the convex hull of the trigonometric moment curve studied from combinatorial topology. Any interior points of the dual faces are boundary separable states with full ranks. We exhibit concrete examples of such states.
Extremal Marginal States of Maximal Rank in (d,d+m)
We study the extreme points of the convex set C(ρ1,ρ2) of bipartite quantum states with fixed marginals ρ1 and ρ2. We construct extreme points in (d,d+m) dimension, of rank d+m, matching the highest possible value, for all d≥3, m>2d2−2d−2 (when d=2, m≥1). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of C(ρ1,ρ2), it is sufficient to study the special case C(D1,D2), where the marginals are diagonal. Additionally, we observe that it is sufficient to consider d1≤d2. Thus, our results show that apart from possibly a few finite cases, for each d1, the maximal rank is achieved almost all times.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators subject to affine constraints. These examples include sets of quantum states with given expectation values of some observables of interest, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels, and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterisation of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and are used to retrieve classic results such as Choi's characterisation of the extreme quantum channels, Parhasarathy's characterisation of the extreme quantum states with given marginals, and the quantum version of Birkhoff's theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalisation, and we characterise the extremal POVMs.