Abstract In this note we study completely positive, trace-preserving (CPT) maps M 4 ( C ) ↦ M 4 ( C ) , in particular the class of examples given by quantum Markov chains (QMCs) describing the evolution of one qubit on a 2-vertex graph. For this class of channels, and regarding its convex structure, we examine the problem of finding the extreme points both with respect to CPT maps and with respect to the set of QMCs. This is done by considering the class of maps whose image is contained in the set of block-diagonal matrices. Motivated by previous results in the literature, a practical criterion for extremality is obtained via the Choi–Jamiołkowski isomorphism and appropriate direct-sum subspaces. The study of extreme maps is natural from a structural point of view, as these play the role of indecomposable operations and provide fundamental information about the geometry of the set of quantum channels.
Abstract In this note we study completely positive, trace-preserving (CPT) maps M 4 ( C ) ↦ M 4 ( C ) , in particular the class of examples given by quantum Markov chains (QMCs) describing the evolution of one qubit on a 2-vertex graph. For this class of channels, and regarding its convex structure, we examine the problem of finding the extreme points both with respect to CPT maps and with respect to the set of QMCs. This is done by considering the class of maps whose image is contained in the set of block-diagonal matrices. Motivated by previous results in the literature, a practical criterion for extremality is obtained via the Choi–Jamiołkowski isomorphism and appropriate direct-sum subspaces. The study of extreme maps is natural from a structural point of view, as these play the role of indecomposable operations and provide fundamental information about the geometry of the set of quantum channels.
Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space X to another one Y is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.
Abstract Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly.
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.
A linear map L from Cn×n into Cn×n is called a quantum channel if it is completely positive and trace preserving. The set Ln of all such quantum channels is known to be a compact convex set. While the extreme points of Ln can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix Z(L) associated with the quantum channel L, we compute the maximum dimension of a proper face of Ln, and in addition the possible dimensions of faces generated by L when $rank \ Z(L)=2 $.
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.
In this thesis, quantum channels are studied from the point of view of Mathematics. We studied the CPTP (completely positive trace preserving) and UCPTP channels, the latter being the unital channels. In both cases, the sets are compact convex, so they are the closure of their extreme points. We constructed extreme CPTP maps for every possible rank. We also constructed extreme UCPTP maps with rank 2 for each possible dimension. For ranks of at least 3, we constructed extreme UCPTP maps for many dimensions and ranks. We also proved that the tensor product of extreme CPTP maps is an extreme CPTP map. For UCPTP maps, we proved that the same is not always true, with many counterexamples. It is known that the category of CPTP maps is semicartesian. We add to this a classification of binary products. We proved that products only exist in trivial cases. For binary coproducts, we obtained the dimensions they should have, if they exist. In both cases, topological techniques were used. For this reason, we also investigated the topology of the closure of the set of extreme points of the CPTP maps. Topological invariants can be computed from decompositions of a space such as CW structures. These are a type of decomposition in which the space is expressed as a gluing of balls. In the search for a CW structure for the closure of the set of extreme CPTP maps, we obtained a partial decomposition that resembles a CW structure of the complex projective space.