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Quantifying complexity of continuous-variable quantum states via Wehrl entropy and Fisher information

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

Abstract

The notion of complexity of quantum states is quite different from uncertainty or information contents, and involves the tradeoff between its classical and quantum features. In this work, we we introduce a quantifier of complexity of continuous-variable states, e.g. quantum optical states, based on the Husimi quasiprobability distribution. This quantity is built upon two functions of the state: the Wehrl entropy, capturing the spread of the distribution, and the Fisher information with respect to location parameters, which captures the opposite behaviour, i.e. localization in phase space. We analyze the basic properties of the quantifier and illustrate its features by evaluating complexity of Gaussian states and some relevant non-Gaussian states. We further generalize the quantifier in terms of s-ordered phase-space distributions and illustrate its implications.

Quantifying complexity of continuous-variable quantum states via Wehrl entropy and Fisher information

Abstract

The notion of complexity of quantum states is quite different from uncertainty or information contents, and involves the tradeoff between its classical and quantum features. In this work, we we introduce a quantifier of complexity of continuous-variable states, e.g. quantum optical states, based on the Husimi quasiprobability distribution. This quantity is built upon two functions of the state: the Wehrl entropy, capturing the spread of the distribution, and the Fisher information with respect to location parameters, which captures the opposite behaviour, i.e. localization in phase space. We analyze the basic properties of the quantifier and illustrate its features by evaluating complexity of Gaussian states and some relevant non-Gaussian states. We further generalize the quantifier in terms of s-ordered phase-space distributions and illustrate its implications.
Paper Structure (10 sections, 69 equations, 5 figures, 2 tables)

This paper contains 10 sections, 69 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Complexity of CV states via Wehrl entropy and Fisher information
  3. Examples
  4. Gaussian states
  5. Photon-added thermal states
  6. Photon-added coherent states
  7. Cat states
  8. Comparison with other quantifiers
  9. Generalization to the $s$-ordered complexity
  10. Summary and discussion

Figures (5)

  • Figure 1: (a) The complexity ${\cal C}( \rho_{\rm g} )$ of the Gaussian state $\rho_{\rm g}$ defined by Eq. (\ref{['eq:gaussian_complexity_power']}) as a function of $\bar{n}$ with $r=0.5,1,1.5,2$ (from bottom to top). (b) Semilog plot of complexity ${\cal C}( \rho_{\rm g} )$ as a function of $r$ with $\bar{n}=0.1,1,10$ (from bottom to top).
  • Figure 2: The complexity ${\cal C}(\psi _{\rm pac} )$ of the photon-added coherent state $|\psi _{\rm pac}\rangle$ as a function of the amplitude $\beta$. The complexity does not depend on the phase of $\beta$.
  • Figure 3: The complexity ${\cal C}(m_\beta)$ of a mixture of two coherent states (black line) versus the complexity ${\cal C}(\psi_{\rm cat})$ of cat states (colored lines) as functions of the coherent state amplitude $\beta$. We show results for different values of the relative phase $\phi$ defined in Eq. (\ref{['CAT']}): $\phi=0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4},\frac{9\pi}{10},\pi$ (from bottom to top). The odd-cat state is not well defined as $\beta \to 0.$
  • Figure 4: The $s$-ordered complexity ${\cal C}_s(\rho_{\beta})$ of the phase-averaged coherent state $\rho_{\beta}$ as a function of the ordering $s$, with $|\beta|=0.5,1,1.5$ (from bottom to top).
  • Figure 5: The $s$-ordered complexity ${\cal C}_s(k)$ of the Fock state $|k\rangle$ as a function of the ordering $s$, with $k=1,2,3,4$ (from bottom to top).