Fisher discord as a quantifier of quantum complexity

TL;DR

This paper introduces a quantum complexity quantifier defined as $C(\rho,H)=I_{\rm F}(\rho,H)-I_{\rm W}(\rho,H)$, the difference between the symmetric-logarithmic-derivative quantum Fisher information and the Wigner–Yanase skew information, relative to a driving Hamiltonian $H$. It establishes fundamental properties, including nonnegativity, unitary and shift invariances, additivity, and a vanishing condition when the state is pure or commutes with $H$. The authors provide extensive discrete- and continuous-variable illustrations showing how complexity arises from noncommuting dynamics, displacement, and squeezing, and they compare this quantifier to Manzano’s Fisher–Shannon and Tang’s Fisher–Wehrl measures to highlight distinct perspectives on quantum complexity. The framework yields analytic expressions for many prototypical states, demonstrates zero complexity for equilibrium/pure cases, and suggests further exploration of Fisher-discord-based complexity in metrology and quantum correlations. Overall, the work offers a dynamic, Hamiltonian-aware lens on quantum complexity that complements existing static or phase-space based measures.

Abstract

Two classically equivalent expressions of mutual information of probability distributions (classical bipartite states) diverge when extended to quantum systems, and this difference has been employed to define quantum discord, a quantifier of quantum correlations beyond entanglement. Similarly, equivalent expressions of classical Fisher information of parameterized probability distributions diverge when extended to quantum states, and this difference may be exploited to characterize the complex nature of quantum states. By complexity of quantum states, we mean some hybrid nature which intermingles the classical and quantum features. It is desirable to quantify complexity of quantum states from various perspectives. In this work, we pursue the idea of discord and introduce an information-theoretic quantifier of complexity for quantum states (relative to the Hamiltonian that drives the evolution of quantum systems) via the notion of Fisher discord, which is defined by the difference between two important versions of quantum Fisher information: the quantum Fisher information defined via the symmetric logarithmic derivatives and the Wigner-Yanase skew information defined via the square roots of quantum states. We reveal basic properties of the quantifier of complexity, and compare it with some other quantifiers of complexity. In particular, we show that equilibrium states (or stable states, which commute with the Hamiltonian of the quantum system) and all pure states exhibit zero complexity in this setting. As illustrations, we evaluate the complexity for various prototypical states in both discrete and continuous-variable quantum systems.

Figures (8)

• Figure 1: Complexity $C(\rho_{g},a^\dag a)$ of Gaussian state $\rho_{g}$ relative to Hamiltonian $a^\dag a$ versus the squeezing parameter $|\zeta|$ for $\lambda=0.01$ (black solid line), $\lambda_0$ (blue dashed line, defined by Eq. (\ref{['lambda0']})), $0.3$ (green dash-dotted line), and $0.8$ (red dotted line) with fixed $|z|=1$ and $2\arg z-\arg\zeta=0$.
• Figure 2: Complexity $C(\rho_{g},a^\dag a)$ of Gaussian state $\rho_{g}$ relative to Hamiltonian $a^\dag a$ versus the noise parameter $\lambda$ for $|\zeta|=0.01$ (black solid line), $0.1$ (blue dashed line), $0.3$ (green dash-dotted line), $0.5$ (red dotted line) with fixed $|z|=1$ and $2\arg z-\arg\zeta=0$.
• Figure 3: Complexity $C(\rho_{p,0,1},a^\dag a)$ of the mixture state $\rho_{p,0,1}$ relative to Hamiltonian $a^\dag a$ versus the parameter $p\in[0,1]$.
• Figure 4: Complexity $C(\rho_p,X_\theta)=g_p$ of the state $\rho_{p}$ relative to Hamiltonian $X_\theta$ versus the parameter $p\in[0,1]$.
• Figure 5: Complexity $C(\tau_\lambda,X_\theta)$ for thermal state $\tau_\lambda$ (red dotted line), $C(\tau^{\rm t}_\lambda,X_\theta)$ for truncated thermal state $\tau^{\rm t}_\lambda$ (blue dashed line) and $C(\tau^{\rm pa}_\lambda,X_\theta)$ for photon-added thermal state $\tau^{\rm pa}_\lambda$ (black solid line) versus noise parameter $\lambda\in(0,1)$.