Birkhoff-James Orthogonality in the Trace Norm, with Applications to Quantum Resource Theories

TL;DR

This paper characterizes Birkhoff-James orthogonality in the trace norm for Hermitian matrices and applies these results to quantum resource theories, including coherence and entanglement measures.

Contribution

It introduces a simple criterion for Birkhoff-James orthogonality to positive semidefinite matrices and connects this to quantum resource quantification.

Findings

Characterizes when Hermitian matrices are orthogonal to positive semidefinite matrices in trace norm.

Provides a criterion for orthogonality to diagonal positive semidefinite matrices.

Links trace distance of coherence and 2-entanglement to quantum resource problems.

Abstract

We develop numerous results that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, we develop a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices. We then explore applications of our work in the theory of quantum resources. For example, we characterize exactly which quantum states have modified trace distance of coherence equal to 1 (the maximal possible value), and we establish a connection between the modified trace distance of 2-entanglement and the NPPT bound entanglement problem.