  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.
