Perturbation Theory for Quantum Information

TL;DR

This paper introduces a perturbation theory for quantum states that simplifies the calculation of matrix functions like entropy and fidelity, using only eigenspectrum and matrix elements, without full state analysis.

Contribution

It develops a novel perturbation framework for quantum states that efficiently computes key information measures, considering support-preserving and extending perturbations.

Findings

Efficient formulas for Von Neumann entropy, quantum relative entropy, Chernoff bound, and fidelity.

Perturbation theory relies only on eigenspectrum and matrix elements, not full states.

Supports two classes of perturbations with distinct computational features.

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 Fr\'echet 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.