Hilbert-Schmidt quantum coherence in multi-qudit systems

TL;DR

This paper introduces a geometric approach to quantify Hilbert-Schmidt quantum coherence in multi-qudit systems using Bloch parametrization and Gell Mann matrices, analyzing coherence control and dynamics.

Contribution

It develops a novel Euclidean vector representation for Hilbert-Schmidt distance and coherence, applying it to multi-qudit states and exploring coherence control and non-monotonicity properties.

Findings

Hilbert-Schmidt quantum coherence can be expressed as Euclidean distances in observable mean value space.

Local and non-local coherences can be controlled by tuning local populations.

Non-monotonicity of Hilbert-Schmidt distance under tensor products affects coherence quantification.

Abstract

Using Bloch's parametrization for qudits ($d$-level quantum systems), we write the Hilbert-Schmidt distance (HSD) between two generic $n$-qudit states as an Euclidean distance between two vectors of observables mean values in $\mathbb{R}^{\Pi_{s=1}^{n}d_{s}^{2}-1}$, where $d_{s}$ is the dimension for qudit $s$. Then, applying the generalized Gell Mann's matrices to generate $SU(d_{s})$, we use that result to obtain the Hilbert-Schmidt quantum coherence (HSC) of $n$-qudit systems. As examples, we consider in details one-qubit, one-qutrit, two-qubit, and two copies of one-qubit states. In this last case, the possibility for controlling local and non-local coherences by tuning local populations is studied and the contrasting behaviors of HSC, $l_{1}$-norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate the decoherent dynamics of these coherence functions under the action of qutrit dephasing and dissipation channels. At last, we analyze the non-monotonicity of HSD under tensor products and report the first instance of a consequence (for coherence quantification) of this kind of property of a quantum distance measure.