We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to the set of quantum states on a separable Hilbert space. Among others, we show that every state having finite expected values of any two (not necessarily bounded) positive operators admits a decomposition into pure states with the same expected values. We discuss applications in quantum information theory.
We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to the set of quantum states on a separable Hilbert space. Among others, we show that every state having finite expected values of any two (not necessarily bounded) positive operators admits a decomposition into pure states with the same expected values. We discuss applications in quantum information theory.
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 satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, 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 characterization 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 then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization 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 normalization, and we characterize the extremal POVMs.
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.
Let K be a convex subset of the state space of a finite-dimensional C*-algebra. We study the properties of channels on K, which are defined as affine maps from K into the state space of another algebra, extending to completely positive maps on the subspace generated by K. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on K. Moreover, if K contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process positive operator valued measures (POVMs).
Let Sk be the set of separable states on B(Cm⊗Cn) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n>1, and k⩽max(m,n), we show that Sk admits a subset Vk such that Vk is dense and open in Sk, and such that each state in Vk has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains Vk. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of B(Cm⊗Cn) that preserve entanglement and separability.
For Hilbert spaces H⊆L2(R) we consider the convex sets D+(H) of Wigner-positive states (WPS), i.e.~density matrices over H with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate D+(H) through convex combinations. If H is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case H=L2(R) this is not so, due to lack of compactness of the set D+(H). Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators.
We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of k≤d2 points of the outcome space, d<∞ being the dimension of the Hilbert space. We prove that for second-countable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein–Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on k≤d2 points of the outcome space.