Coherence and imaginarity of quantum states

TL;DR

This paper explores how quantum coherence measures relate to the imaginary parts of quantum states, proposing new ways to quantify coherence and extending results to bosonic Gaussian states.

Contribution

It demonstrates that coherence measures invariant under complex conjugation satisfy specific inequalities and introduces a symmetrized measure for states lacking this invariance.

Findings

Coherence measures satisfy C(ρ)-C(Reρ) ≥ 0 if invariant under conjugation.

A new symmetric coherence measure C' is defined for non-invariant cases.

Results extended to bosonic Gaussian states.

Abstract

Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying the coherence of quantum states [\href{http://dx.doi.org/10.1103/PhysRevLett.113.140401}{Phys. Rev. Lett. 113, 140401 (2014)}]. In BCP framework, a quantum state is called incoherent if it is diagonal in the fixed orthonormal basis, and a coherence measure should satisfy some conditions. For a fixed orthonormal basis, if a quantum state $\rho $ has nonzero imaginary part, then $\rho $ must be coherent. How to quantitatively characterize this fact? In this work, we show that any coherence measure $C$ in BCP framework has the property $C(\rho )-C($Re$\rho )\geq 0$ if $C$ is invariant under state complex conjugation, i.e., $C(\rho )=C(\rho ^{\ast })$, here $\rho ^{\ast }$ is the conjugate of $\rho ,$ Re$\rho $ is the real part of $\rho .$ If $C$ does not satisfy $C(\rho )=C(\rho ^{\ast }),$ we can define a new coherence measure $C^{\prime }(\rho )=\frac{1}{2}[C(\rho )+C(\rho ^{\ast })]$ such that $C^{\prime }(\rho )=C^{\prime }(\rho ^{\ast }).$ We also establish some similar results for bosonic Gaussian states.