Wehrl Entropy: Quantum Phase-Space Measure

• Wehrl entropy is a classical-like entropy measure defined via the Husimi Q-function, capturing phase-space localization.
• It quantifies both quantum purity and classical-like spreading, offering insights into squeezed, Gaussian, and multimode states.
• Its analytic formulation, linked to covariance matrices, facilitates practical applications in quantum optics and communication.

Wehrl entropy is a classical-like entropy measure for quantum states, formulated in phase space via the Husimi Q-function. Unlike the von Neumann entropy, which captures quantum uncertainty in the density operator, Wehrl entropy is a functional of a smooth, positive-definite phase-space distribution. Wehrl entropy plays a critical role in quantum optics, quantum information, and semiclassical analyses, providing a measure of phase-space localization that is sensitive to both quantum purity and classical-like spreading.

1. Definition and Mathematical Formulation

For a density operator $\hat\rho$ of a quantum system (typically a single or multimode bosonic field), the Husimi Q-function is defined by

$$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$

where $|\alpha\rangle$ is a multimode coherent state and $N$ is the number of modes. The Wehrl entropy is given by:

$$S_W[\hat\rho] = -\int Q_{\rho}(\alpha) \ln Q_{\rho}(\alpha)\; d^{2N}\alpha.$$

This quantity is always non-negative, and is strictly larger than the minimal entropy possible in quantum mechanics (the von Neumann entropy vanishes for a pure state, but Wehrl entropy does not).

For Gaussian states, in particular those arising from Bogoliubov transformations (i.e., squeezed or displaced vacua), $Q_\rho(\alpha)$ is a multimode Gaussian, and the Wehrl entropy computation reduces to an explicit function of the Gaussian covariance matrix. This relation is central in studies of quantum optics and many-body physics, especially when handling Gaussian channels or the output of nonlinear processes such as spontaneous parametric down-conversion (Huh, 2020, Hellebek et al., 2024).

2. Properties and Comparison with von Neumann Entropy

Wehrl entropy is strictly greater than or equal to 1 for a single-mode quantum state (known as Wehrl’s original conjecture, proved by Lieb), with equality only for a coherent state (i.e., minimum-uncertainty, classical-like states). The key properties include:

• Positivity and strict lower bound: $S_W[\hat\rho]\geq 1$ for any normalized density operator in a single mode, with equality if and only if $\hat\rho$ is a coherent state.
• Purity sensitivity: Wehrl entropy strictly increases with state mixedness but, unlike the von Neumann entropy, is nonzero even for pure states except the vacuum (coherent state). For squeezed states, $S_W$ is always strictly above 1, capturing the additional phase-space spreading due to squeezing (Huh, 2020).
• Classical-like character: Since $Q_\rho(\alpha)$ is regular (never negative), Wehrl entropy can be seen as a "classical" entropy for quantum states, circumventing the issues with negative regions in the Wigner function and the subtleties of quantum-mechanical entropy definition.

3. Multimode Bosonic Systems and Gaussian States

In multimode scenarios, Wehrl entropy plays a significant role in characterizing the phase-space delocalization induced by multimode squeezing, displacements, or Bogoliubov transformations. For a multimode Gaussian state defined by a covariance matrix $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$0, the Q-function is given by a complex multivariate Gaussian:

$$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$1

where $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$2 is related to the state's covariance. The Wehrl entropy is then expressible as:

$$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$3

up to an additive normalization, directly linking $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$4 to the degree of squeezing and correlations present in the state (Huh, 2020).

This analytic tractability extends Wehrl entropy’s utility across quantum optics, continuous-variable quantum information, and studies of photon statistics and entanglement in nonclassical light (Nam et al., 2015, Hellebek et al., 2024).

4. Wehrl Entropy in Quantum Information and Quantum Optics

Wehrl entropy provides a measure of localization and effective phase-space volume, thus characterizing quantum noise, delocalization, and the degree of nonclassicality. For single-photon sources (e.g., heralded sources based on SPDC), Wehrl entropy quantifies modal impurity and effective number of excited modes, complementing Schmidt number and purity-based metrics (Hellebek et al., 2024). In quantum communication, $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$5 serves as a measure of decoherence and "classicalization" of quantum states due to environmental coupling.

Additionally, Wehrl entropy is operationally accessible in experiments relying on state tomography via coherent-state measurements or Q-function reconstruction (Huh, 2020).

5. Computational Methods and Relation to Multivariate Hermite Polynomials

For non-Gaussian states as well as for arbitrary multimode states, Wehrl entropy evaluation typically involves the expansion of the Q-function in terms of Fock states, with matrix elements generated by multivariate Hermite polynomials (MHPs). The Husimi Q-function acts as a generating function for the Fock-basis elements:

$$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$6

The MHPs encode the complete information required to reconstruct matrix elements, and their derivatives or moments yield explicit Q-function expansions (Huh, 2020).

This structure allows highly efficient numerical evaluation of $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$7 for states resulting from arbitrary Gaussian operations (squeezing, displacement, multimode mixing) as well as for composite systems with non-Gaussian elements inserted in the mode structure.

6. Applications and Extensions

Wehrl entropy has found application across quantum statistical mechanics, phase-space information theory, and quantum control. Notably:

• Squeezed state analysis: The dependence of $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$8 on squeezing parameters provides a tool to quantify phase-space delocalization and the degree of quantum correlations injected by Bogoliubov transformations (Huh, 2020).
• Quantum source characterization: In quantum optics, Wehrl entropy acts as an indicator of the effective number of modes and purity in heralded single-photon sources (Hellebek et al., 2024).
• Quantum-to-classical transition: Wehrl entropy is frequently used to analyze the interface between quantum and classical dynamical regimes, especially for systems where decoherence maps pure states to mixtures, raising $$Q_{\rho}(\alpha) = \pi^{-N} \langle \alpha | \hat\rho | \alpha \rangle,$$9 towards the classical regime.

Extensions of the Wehrl entropy notion, such as those incorporating generalized phase-space distributions or different resolution fiducial vectors (e.g., spin coherent states for spin systems), further broaden its applicability.

7. Connections to the Broader Entropic Landscape

Wehrl entropy must be viewed as part of an ensemble of quantum and semiclassical entropy measures, each with distinct operational and conceptual interpretations. Its distinguishing features—positivity, phase-space localization, and operational measurability—complement the operator-based von Neumann entropy and Wigner-function-based Rényi entropies, providing a bridge between quantum statistical mechanics and classical information theory.

In summary, Wehrl entropy is central to the analysis of phase-space structures in quantum states, especially Gaussian states and those accessible via multimode Bogoliubov transformations. Its wide applicability and analytic tractability ensure ongoing relevance across quantum optics, continuous-variable quantum information, and quantum statistical mechanics (Huh, 2020, Hellebek et al., 2024).