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Electronic Journal of Linear Algebra

Totally Positive Density Matrices and Linear Preservers

David W. Kribs, Jeremy Levick, Rajesh Pereira

Published Feb 5, 2016• 1 citation
Publisher

Abstract

The intersection between the set of totally nonnegative matrices, which are of interest in many areas of matrix theory and its applications, and the set of density matrices, which provide the mathematical description of quantum states, are investigated. The single qubit case is characterized, and several equivalent conditions for a quantum channel to preserve the set in that case are given. Higher dimensional cases are also discussed.

DOI
10.13001/1081-3810.3129

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The intersection between the set of totally nonnegative matrices, which are of interest in many areas of matrix theory and its applications, and the set of density matrices, which provide the mathematical description of quantum states, are investigated. The single qubit case is characterized, and several equivalent conditions for a quantum channel to preserve the set in that case are given. Higher dimensional cases are also discussed.
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The most general evolution of the density matrix of a quantum system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of Hermiticity of density matrices renders an associated supermatrix Hermitian and hence diagonalizable. The positivity of the density matrix does not make the associated supermatrix positive though. If the map itself is positive, it is called completely positive and it has a simple parametrization. This is extended to all positive (not completely positive) maps. A general dynamical map that does not preserve the norm of the density matrices it acts on can be thought of as the contraction of a norm-preserving map of an extended system. The reconstruction of such extended dynamics is also given.
It is a hard and important problem to find the criterion of a set of positive-definite matrices which can be written as reduced density operators of a multipartite global quantum state. This problem is closely related to the study of many-body quantum entanglement, which is one of the focuses of current quantum-information theory. We give several results on the necessary compatibility relations between a set of reduced and global density matrices, including (i) compatibility conditions for the one-party reduced density matrices of any $({N}_{A}\ifmmode\times\else\texttimes\fi{}{N}_{B})$-dimensional bipartite mixed quantum state, (ii) compatibility conditions for the one-party reduced matrices of any MMM-partite quantum states with the dimension ${N}^{\ensuremath{\bigotimes}M}$, and (iii) compatibility conditions for the two-party reduced density matrixes of $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2)$-dimensional tripartite mixed quantum state.
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This preprint studies quantum Markov dynamics and quantum channels that preserve the Lee–Yang property. For systems of two or more qubits, preservation for all Lee–Yang density matrices forces tensor-product dynamics, with local generators characterized by explicit semidefinite conditions. This ordinary preservation property is equivalent to preservation in the presence of arbitrary inert reference systems. For systems containing higher spins, preservation of all pure Lee–Yang inputs excludes intersite interactions and forces isotropic diffusion with axial drift at higher-spin sites. The paper also develops classifications of static maps with stable full Choi tensors, including singular one-qubit maps and independently covariant maps between finite spins, and establishes quantitative rigidity results for diagonal Hamiltonians. The proofs combine positivity of quantum maps, polynomial root-collision arguments, and coherent tensor contraction. The manuscript includes a disclosure of the AI tools used in its development.