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Statistical phase-space complexity of continuous-variable quantum channels

Siting Tang, Francesco Albarelli, Yue Zhang, Shunlong Luo, Matteo G. A. Paris

TL;DR

The paper defines a channel-level statistical complexity for continuous-variable quantum systems via the Husimi $Q$-function, with state complexity $\mathcal{C}(\rho) = e^{S_W(\rho)-1} I(\rho)$ based on Wehrl entropy $S_W$ and Fisher information $I(\rho)$. The channel complexity $\mathcal{C}(\mathcal{E}) = \sup_{\mathcal{C}(\rho_0)=1} \mathcal{C}(\mathcal{E}(\rho_0))$ measures how much complexity a channel can generate from a minimal-complex input, and is studied for single-mode Gaussian channels and selected non-Gaussian channels. A closed-form, time-dependent expression is derived for Gaussian channels, revealing that non-squeezed baths cannot generate complexity and that the maximum is bounded by $\mathcal{C}(\mathcal{E}_\infty) \le \sqrt{N+1}$, attained by pure squeezed asymptotic states. In contrast, the phase-diffusion (non-Gaussian) channel can produce unbounded complexity as the input displacement increases, with a nontrivial dependence on the diffusion strength; photon-addition and -subtraction sub-channels yield the same finite bound $e^\gamma$ for their maximal complexity. Overall, non-Gaussianity emerges as a crucial resource for high channel complexity, with potential implications for quantum metrology and communication.

Abstract

The statistical complexity of continuous-variable quantum states can be characterized with a quantifier defined in terms of information-theoretic quantities derived from the Husimi Q-function. In this work, we utilize this complexity quantifier of quantum states to study the complexity of single-mode bosonic quantum channels. We define the complexity of quantum channels as the maximal amount of complexity they can generate from an initial state with the minimal complexity. We illustrate this concept by evaluating the complexity of Gaussian channels and some examples of non-Gaussian channels.

Statistical phase-space complexity of continuous-variable quantum channels

TL;DR

The paper defines a channel-level statistical complexity for continuous-variable quantum systems via the Husimi -function, with state complexity based on Wehrl entropy and Fisher information . The channel complexity measures how much complexity a channel can generate from a minimal-complex input, and is studied for single-mode Gaussian channels and selected non-Gaussian channels. A closed-form, time-dependent expression is derived for Gaussian channels, revealing that non-squeezed baths cannot generate complexity and that the maximum is bounded by , attained by pure squeezed asymptotic states. In contrast, the phase-diffusion (non-Gaussian) channel can produce unbounded complexity as the input displacement increases, with a nontrivial dependence on the diffusion strength; photon-addition and -subtraction sub-channels yield the same finite bound for their maximal complexity. Overall, non-Gaussianity emerges as a crucial resource for high channel complexity, with potential implications for quantum metrology and communication.

Abstract

The statistical complexity of continuous-variable quantum states can be characterized with a quantifier defined in terms of information-theoretic quantities derived from the Husimi Q-function. In this work, we utilize this complexity quantifier of quantum states to study the complexity of single-mode bosonic quantum channels. We define the complexity of quantum channels as the maximal amount of complexity they can generate from an initial state with the minimal complexity. We illustrate this concept by evaluating the complexity of Gaussian channels and some examples of non-Gaussian channels.
Paper Structure (5 sections, 45 equations, 4 figures)

This paper contains 5 sections, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Gaussian channels
  3. Phase diffusion channels
  4. Photon addition and photon subtraction
  5. Conclusions

Figures (4)

  • Figure 1: (a) The complexity of the phase-diffused state $\mathcal{C} (\rho_\kappa (\xi,0))$ as a function of $\xi$ with $\kappa=0.001,3,10$ (from top to bottom). (b) Blowup of (a) for small $\xi$, and the dashed lines are the fitted functions $1+\gamma_\kappa \xi^4$.
  • Figure 2: The complexity of the phase-diffused state $\mathcal{C} (\rho_\kappa (\xi,0))$ as a function of $\ln\frac{1}{\kappa} =-\ln\kappa$ with $\xi=1,2,3$ (from bottom to top).
  • Figure 3: (a) The complexity of the photon-added displaced thermal state $\mathcal{C} (\rho_+ (\xi,\bar{n}))$ as a function of $\xi$ with $\bar{n}=0.1,1,10$ (from bottom to top). (b) The complexity of the photon-added displaced thermal state $\mathcal{C} (\rho_+ (\xi,\bar{n}))$ as a function of $\bar{n}$ with $\xi=0.1,1,3$ (from top to bottom).
  • Figure 4: (a) The complexity of the photon-subtracted displaced thermal state $\mathcal{C} (\rho_- (\xi,\bar{n}))$ as a function of $\xi$ with $\bar{n}=0.1,1,10$ (from bottom to top). (b) The complexity of the photon-subtracted displaced thermal state $\mathcal{C} (\rho_- (\xi,\bar{n}))$ as a function of $\bar{n}$ with $\xi=0.1,1,3$ (from top to bottom).