Perturbation Theory for Quantum Information

Abstract

We report lowest-order series expansions for primary matrix functions of quantum states based on a perturbation theory for functions of linear operators. Our theory enables efficient computation of functions of perturbed quantum states that assume only knowledge of the eigenspectrum of the zeroth order state and the density matrix elements of a zero-trace, Hermitian perturbation operator, not requiring analysis of the full state or the perturbation term. We develop theories for two classes of quantum state perturbations, perturbations that preserve the vector support of the original state and perturbations that extend the support beyond the support of the original state. We highlight relevant features of the two situations, in particular the fact that functions and measures of perturbed quantum states with preserved support can be elegantly and efficiently represented using Fréchet derivatives. We apply our perturbation theories to find simple expressions for four of the most important quantities in quantum information theory that are commonly computed from density matrices: the Von Neumann entropy, the quantum relative entropy, the quantum Chernoff bound, and the quantum fidelity.

21 pages, updated to clarify that the perturbation theory results are only proven for states on finite-dimensional Hilbert spaces, or state spaces that can be truncated to finite dimensions; it is likely possible to prove that the same results hold for states spanning an infinite-dimensional Hilbert space

References in corpus (5)

• The Quantum Chernoff Bound
• Computable bounds for the discrimination of Gaussian states
• Classical capacity of bosonic broadcast communication and a new minimum output entropy conjecture
• Perturbation Theory for Quantum Information
• An exact correspondence between the quantum Fisher information and the Bures metric

Cited by in corpus (5)

• In-in formalism for the entropy of quantum fields in curved spacetimes
• Perturbation Theory for Quantum Information
• Demonstration of quantum-limited discrimination of multi-copy pure versus mixed states
• Reservoir-Free Decoherence in Flying Qubits
• Quantum Illumination Advantage for Classification Among an Arbitrary Library of Targets