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.

arXiv comment18 pages

References in corpus (19)

• Quantum entanglement
• Quantifying Coherence
• Entanglement detection
• Quantum Resource Theories
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• A Schmidt number for density matrices
• Trace-distance measure of coherence
• Evidence for Bound Entangled States with Negative Partial Transpose
• Alternative framework for quantifying coherence
• Quantifying Superposition
• Convex geometry of quantum resource quantification
• A few steps more towards NPT bound entanglement
• Many copies may be required for entanglement distillation
• Certification and Quantification of Multilevel Quantum Coherence
• A Family of Norms With Applications In Quantum Information Theory
• Orthogonality of matrices in the Ky Fan $k$-norms
• Five open problems in quantum information
• Some geometric interpretations of quantum fidelity
• A Family of Norms With Applications In Quantum Information Theory II