Maximal coherence in generic basis

TL;DR

This paper investigates the maximum quantum coherence a state can have relative to a basis, identifying the Fourier matrix as a key stationary point, but not necessarily the global maximum, using the relative entropy measure.

Contribution

It characterizes the basis that maximizes quantum coherence for a given state and highlights the role of Fourier and complex Hadamard matrices in this optimization.

Findings

Fourier basis is a stationary point for the $l_1$ coherence norm.

The basis associated with the Fourier matrix is not always globally optimal.

Complex Hadamard matrices are crucial in determining maximal coherence bases.

Abstract

Since quantum coherence is an undoubted characteristic trait of quantum physics, the quantification and application of quantum coherence has been one of the long-standing central topics in quantum information science. Within the framework of a resource theory of quantum coherence proposed recently, a \textit{fiducial basis} should be pre-selected for characterizing the quantum coherence in specific circumstances, namely, the quantum coherence is a \textit{basis-dependent} quantity. Therefore, a natural question is raised: what are the maximum and minimum coherences contained in a certain quantum state with respect to a generic basis? While the minimum case is trivial, it is not so intuitive to verify in which basis the quantum coherence is maximal. Based on the coherence measure of relative entropy, we indicate the particular basis in which the quantum coherence is maximal for a given state, where the Fourier matrix (or more generally, \textit{complex Hadamard matrices}) plays a critical role in determining the basis. Intriguingly, though we can prove that the basis associated with the Fourier matrix is a stationary point for optimizing the $l_1$ norm of coherence, numerical simulation shows that it is not a global optimal choice.