State Evolution and Trace-Preserving Completely Positive Maps

Masahito Hayashi

Mary Beth Ruskai, Stanisław J. Szarek, Elisabeth M. Werner

Zejun Huang, Chi-Kwong Li, Edward Poon +1 more

Given two sets of quantum states {A1, …, Ak} and {B1, …, Bk}, represented as sets as density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(Ai) = Bi for i = 1, …, k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.

James Miller S. T. da Silva

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.

Arindam Mitra, Jatin Ghai

In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels $Λ_1, Λ_2$ and an arbitrary unknown input state, we show that the output state of the quantum channel $Λ_2$ from the pair can be uniquely identified from the output statistics of the other channel $Λ_1$ from the pair, using some POVM, iff $Λ_2$ can be obtained from $Λ_1$ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, $Λ_2$ may not always be obtained from $Λ_1$ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing $Λ_2$ given that $Λ_1$ has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.

Erling Størmer