Uncertainty Relations for Quantum Coherence

TL;DR

This paper establishes uncertainty-like trade-off relations for quantum coherence in multiple incompatible bases, highlighting the role of entanglement and providing bounds for bipartite systems.

Contribution

It introduces uncertainty relations for quantum coherence across incompatible bases and explores the influence of entanglement on these relations.

Findings

Coherence measures obey uncertainty-like relations in incompatible bases.

Entanglement can tighten the coherence trade-off relations.

Upper bounds are derived for sums and differences of coherence in bipartite systems.

Abstract

Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can~there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.