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An easily computable measure of Gaussian quantum imaginarity

Ting Zhang, Jinchuan Hou, Xiaofei Qi

TL;DR

This work introduces $\mathcal{I}^{G_n}$, a computable measure of Gaussian imaginarity for $n$-mode continuous-variable systems that depends only on the displacement vector and covariance matrix. It proves faithfulness and monotonicity under real Gaussian channels, and demonstrates substantial computational advantages over prior measures $M_F$ and $M_{T,\mu}$, including simple single-mode reductions and scalable multi-mode expressions. The authors further show that $\mathcal{I}^{G_n}$ induces a multipartite Gaussian correlation measure $\mathcal{I}_m^{G_n}$, satisfying symmetry, unification, and hierarchy conditions, thereby treating Gaussian imaginarity as a genuine multipartite Gaussian resource. Applications to Markovian Gaussian environments illustrate the dynamic behavior of imaginarity, highlighting its potential for efficient analysis in CV quantum information tasks.

Abstract

The resource-theoretic frameworks for quantum imaginarity have been developed in recent years. Within these frameworks, many imaginarity measures for finite-dimensional systems have been proposed. However, for imaginarity of Gaussian states in continuous-variable (CV) systems, there are only two known Gaussian imaginarity measures, which exhibit prohibitive computational complexity when applied to multi-mode Gaussian states. In this paper, we propose a computable Gaussian imaginarity measure $\mathcal I^{G_n}$ for $n$-mode Gaussian systems. The value of $\mathcal I^{G_n}$ is simply formulated by the displacement vectors and covariance matrices of Gaussian states. A comparative analysis of $\mathcal{I}^{G_n}$ with existing two Gaussian imaginarity measures indicates that $\mathcal{I}^{G_n}$ can be used to detect imaginarity in any $n$-mode Gaussian states more efficiently. As an application, we study the dynamics behaviour of $(1+1)$-mode Gaussian states in Gaussian Markovian noise environments for two-mode CV system by utilizing ${\mathcal I}^{G_2}$. Moreover, we prove that, ${\mathcal I}^{G_n}$ can induce a quantification of any $m$-multipartite multi-mode CV systems which satisfies all requirements for measures of multipartite multi-mode Gaussian correlations, which unveils that, $n$-mode Gaussian imaginarity can also be regarded as a kind of multipatite multi-mode Gaussian correlation and is a multipartite Gaussian quantum resource.

An easily computable measure of Gaussian quantum imaginarity

TL;DR

This work introduces , a computable measure of Gaussian imaginarity for -mode continuous-variable systems that depends only on the displacement vector and covariance matrix. It proves faithfulness and monotonicity under real Gaussian channels, and demonstrates substantial computational advantages over prior measures and , including simple single-mode reductions and scalable multi-mode expressions. The authors further show that induces a multipartite Gaussian correlation measure , satisfying symmetry, unification, and hierarchy conditions, thereby treating Gaussian imaginarity as a genuine multipartite Gaussian resource. Applications to Markovian Gaussian environments illustrate the dynamic behavior of imaginarity, highlighting its potential for efficient analysis in CV quantum information tasks.

Abstract

The resource-theoretic frameworks for quantum imaginarity have been developed in recent years. Within these frameworks, many imaginarity measures for finite-dimensional systems have been proposed. However, for imaginarity of Gaussian states in continuous-variable (CV) systems, there are only two known Gaussian imaginarity measures, which exhibit prohibitive computational complexity when applied to multi-mode Gaussian states. In this paper, we propose a computable Gaussian imaginarity measure for -mode Gaussian systems. The value of is simply formulated by the displacement vectors and covariance matrices of Gaussian states. A comparative analysis of with existing two Gaussian imaginarity measures indicates that can be used to detect imaginarity in any -mode Gaussian states more efficiently. As an application, we study the dynamics behaviour of -mode Gaussian states in Gaussian Markovian noise environments for two-mode CV system by utilizing . Moreover, we prove that, can induce a quantification of any -multipartite multi-mode CV systems which satisfies all requirements for measures of multipartite multi-mode Gaussian correlations, which unveils that, -mode Gaussian imaginarity can also be regarded as a kind of multipatite multi-mode Gaussian correlation and is a multipartite Gaussian quantum resource.

Paper Structure

This paper contains 12 sections, 8 theorems, 107 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real Gaussian states
  4. Real Gaussian quantum channels
  5. Gaussian imaginarity measures
  6. A Gaussian imaginarity measure based on displacement vector and CM
  7. The single-mode systems
  8. The multi-mode systems
  9. Comparing $\mathcal{I}^{G_n}$ with $M_F$ and $M_{T,\mu}$
  10. Behavior of Gaussian imaginarity in Markovian environments
  11. Multi-mode Gaussian imaginarity as a multipartite multi-mode Gaussian correlation
  12. Conclusions

Key Result

Theorem 2

For any single-mode Gaussian state $\rho\in{\mathcal{S}}(H)$, $\mathcal{I}^{G}(\rho)\geq 0$ and $\mathcal{I}^{G}(\rho)=0$ if and only if $\rho$ is real.

Figures (8)

  • Figure 1: $\mathcal{I}^{G},M_F,M_{T,\mu}$ in Eqs.\ref{['eq16']}-\ref{['eq15']} as the functions of ${\rm Im}\alpha$.
  • Figure 2: Graphs of $\mathcal{I}^{G}, M_F, M_{T,\mu}$ in Eqs.\ref{['eq18']}-\ref{['eq200']}. (a) Images of $\mathcal{I}^{G}, M_F, M_{T,\mu}$ as functions of ${\rm Re}(\zeta)$ and ${\rm Im}(\zeta)$. (b) Images of $\mathcal{I}^{G},M_F,M_{T,\mu}$ as functions of $\sin^2\theta\sinh^2(2|\zeta|)$.
  • Figure 3: Behavior of $\mathcal{I}^{G_2}(\rho(t))$, with $\rho(0)$ the squeezed vacuum state of $r=1$, as a function of the parameters $t$ and $\phi$ for fixed $n_\mathrm{th}=1.5, R = 1, \lambda=0.1$. (a) Plots of $\mathcal{I}^{G_2}$ as a function of the parameter $t$ for fixed $\phi= 10, 15,20$, respectively. (b) Plots of $\mathcal{I}^{G_2}$ as a function of the parameter $\phi$ for fixed $t=1,2,3$, respectively.
  • Figure 4: Behavior of $\mathcal{I}^{G_2}(\rho(t))$, with $\rho(0)$ the squeezed vacuum state of $r=1$, as a function of the parameters $t$ and $R$ for fixed $n_\mathrm{th}=1.5, \phi=\frac{\pi}{2}, \lambda=0.1$. (a) Plots of $\mathcal{I}^{G_2}$ as a function of the parameter $t$ for fixed $R= 2, 3,4$, respectively. (b) Plots of $\mathcal{I}^{G_2}$ as a function of the parameter $R$ for fixed $t=1,2,3$, respectively.
  • Figure 5: Behavior of $\mathcal{I}^{G_2}(\rho(t))$, with $\rho(0)$ the squeezed vacuum state of $r=1$, as a function of the parameters $t$ and $n_\mathrm{th}$ for fixed $R=1, \phi=\frac{\pi}{2}, \lambda=0.1$. (a) plot of $\mathcal{I}^{G_2}$ as a function of the parameter $t$ for fixed $n_\mathrm{th}= 10, 15, \ {\rm and} \ 20$ and (b) plot of $\mathcal{I}^{G_2}$ as a function of the parameter $n_\mathrm{th}$ for fixed $t=1,2, \ {\rm and} \ 3$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Example 7
  • Example 8
  • Example 9
  • Theorem 10
  • ...and 3 more