On pseudo-stochastic matrices and pseudo-positive maps

TL;DR

This paper introduces pseudo-stochastic matrices and pseudo-positive maps, extending classical and quantum transformations to include matrices with negative elements while maintaining key structural properties.

Contribution

It defines pseudo-stochastic matrices and pseudo-positive maps, exploring their algebraic structures and quantum state transformation capabilities, expanding the framework of classical and quantum state analysis.

Findings

Pseudo-stochastic matrices can have negative elements but still preserve column sums.

Pseudo-positive maps transform specific subsets of quantum states, such as those with maximal purity.

Examples include qubit dynamics related to diamond sets of matrices.

Abstract

Stochastic matrices and positive maps in matrix algebras proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with "diamond" sets of stochastic matrices and pseudo-positive maps are dealt with.