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Coherence and imaginarity of quantum states

Jianwei Xu

TL;DR

The paper investigates how imaginarity (the presence of an imaginary component in a quantum state) necessarily implies coherence when a fixed basis is chosen. It proves that any coherence measure C in the Baumgratz–Cramer–Plenio framework that is invariant under complex conjugation satisfies C(ρ)−C(Re ρ)≥0; if a given C does not, one can form C′(ρ)=1/2[C(ρ)+C(ρ*)] which then obeys the property. The authors extend these ideas to bosonic Gaussian states, introducing Gaussian analogs CG1–CG6 and showing that a real Gaussian state ρ′ induced from ρ satisfies C_G(ρ)≥C_G(ρ′) under suitable conditions, and that the Gaussian relative entropy C_Gr(ρ) also complies with a similar nonnegativity relation. They provide explicit examples with Gaussian states, including coherent and squeezed states, to illustrate the nonnegativity of the imaginarity–coherence gap and discuss implications for resource theories of imaginarity and coherence. The results highlight a quantitative link between imaginarity and coherence and suggest practical methods to construct coherence measures that respect this link in both finite-dimensional and continuous-variable settings.

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 $ρ$ has nonzero imaginary part, then $ρ$ 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 $C(ρ)-C($Re$ρ)\geq 0$ if $C$ is invariant under state complex conjugation, i.e., $C(ρ)=C(ρ^{\ast })$, here $ρ^{\ast }$ is the conjugate of $ρ,$ Re$ρ$ is the real part of $ρ.$ If $C$ does not satisfy $C(ρ)=C(ρ^{\ast }),$ we can define a new coherence measure $C^{\prime }(ρ)=\frac{1}{2}[C(ρ)+C(ρ^{\ast })]$ such that $C^{\prime }(ρ)=C^{\prime }(ρ^{\ast }).$ We also establish some similar results for bosonic Gaussian states.

Coherence and imaginarity of quantum states

TL;DR

The paper investigates how imaginarity (the presence of an imaginary component in a quantum state) necessarily implies coherence when a fixed basis is chosen. It proves that any coherence measure C in the Baumgratz–Cramer–Plenio framework that is invariant under complex conjugation satisfies C(ρ)−C(Re ρ)≥0; if a given C does not, one can form C′(ρ)=1/2[C(ρ)+C(ρ*)] which then obeys the property. The authors extend these ideas to bosonic Gaussian states, introducing Gaussian analogs CG1–CG6 and showing that a real Gaussian state ρ′ induced from ρ satisfies C_G(ρ)≥C_G(ρ′) under suitable conditions, and that the Gaussian relative entropy C_Gr(ρ) also complies with a similar nonnegativity relation. They provide explicit examples with Gaussian states, including coherent and squeezed states, to illustrate the nonnegativity of the imaginarity–coherence gap and discuss implications for resource theories of imaginarity and coherence. The results highlight a quantitative link between imaginarity and coherence and suggest practical methods to construct coherence measures that respect this link in both finite-dimensional and continuous-variable settings.

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 has nonzero imaginary part, then must be coherent. How to quantitatively characterize this fact? In this work, we show that any coherence measure in BCP framework has the property Re if is invariant under state complex conjugation, i.e., , here is the conjugate of Re is the real part of If does not satisfy we can define a new coherence measure such that We also establish some similar results for bosonic Gaussian states.
Paper Structure (10 sections, 11 theorems, 64 equations, 3 figures)

This paper contains 10 sections, 11 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $C(\rho )-C(\text{Re}\rho)$ for any quantum state
  3. Invariance of coherence under state complex conjugation
  4. Coherence measures satisfying (C6)
  5. Examples
  6. Coherence and imaginarity of Gaussian states
  7. Background of (bosonic) Gaussian states
  8. Coherence and imaginarity of Gaussian states
  9. Examples
  10. Summary

Key Result

Theorem 1

Suppose the coherence measure $C$ is invariant under state complex conjugation, then $C(\rho )-C($Re$\rho )\geq 0.$

Figures (3)

  • Figure 1: $C_{l_{1}}(\rho )-C_{l_{1}}(\text{Re}\rho )$ versus $x$ and $y$ in Eq. (\ref{['eq2C-4']}).
  • Figure 2: $C_{\text{Gr}}(\rho )-C_{\text{Gr}}(\rho ^{\prime })$ versus $\text{Re}\alpha$ and $\text{Im}\alpha$ in Eq. (\ref{['eq3C-2']}).
  • Figure 3: $C_{\text{Gr}}(\rho )-C_{\text{Gr}}(\rho ^{\prime })$ versus $\text{Re}\zeta$ and $\text{Im}\zeta$ in Eq. (\ref{['eq3C-5']}).

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Theorem 3
  • Example 1
  • Example 2
  • Example 3
  • Definition 2
  • Corollary 1
  • Definition 3
  • ...and 9 more