Perturbation Theory for Quantum Information

We report a lowest-order Taylor-like series expansion that enables efficient analytical computation of primary matrix functions of perturbed quantum states whose perturbation preserves the vector support of the original state. We apply our theory to find simple expressions for four important quantities in quantum information theory: the von Neumann entropy, the quantum relative entropy, the quantum Chernoff bound, and the quantum fidelity. Our results, which we elegantly represent using Fréchet derivatives, require only knowledge of the eigenspectrum of the unperturbed state and the density matrix elements of the perturbation, bypassing eigenanalysis of the full perturbed state. These results were recently used to derive the fundamental quantum limits of identifying diffraction-limited objects in passive incoherent imaging [1] and in an approach to quantify the covert communications capacity of a bosonic channel [2]. We discuss other avenues where our results could be applied.