Frozen condition of quantum coherence

TL;DR

This paper investigates conditions under which quantum coherence remains unchanged during evolution, providing formulas for qubits, geometric conditions for higher dimensions, and classifying coherent states within operational coherence theory.

Contribution

It offers a comprehensive analysis of coherence freezing conditions, formulas for qubits, geometric criteria for higher dimensions, and a classification of coherent states.

Findings

Derived formulas for freezing coherence in single qubits

Identified universal geometric conditions for higher-dimensional systems

Classified coherent states within operational coherence framework

Abstract

Quantum coherence as an important physical resource plays the key role in implementing various quantum tasks, whereas quantum coherence is often deteriorated due to the noise. In this paper, we analyse under which dynamical conditions the $l_1$-norm or the relative entropy of coherence can remain unchanged during the whole evolution (freezing coherence). For single qubit systems, a nice formula is given to realize freezing coherence. Conversely, for a $d\ (d>2)$ dimensional system, we identify universal geometric conditions of freezing coherence. This offers an affirmative answer to the open question: how can one determine whether a unital quantum operation can be decomposed as a convex combination of unitary operations [M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000)]. Based on this analysis, we also give a complete classification of coherent states from operational coherence theory. This builds the counterpart of entanglement classification under LOCC.