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

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.