Wigner entropy conjecture and the interference formula in quantum phase space

Abstract

Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states -- called Wigner entropy for brevity -- emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the $p$-norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter $α\geq 1/2$.

Figures (6)

• Figure 1: Illustrative Venn diagram of the overlap between the set of phase-space probability distributions $W(\alpha)$ and the set of quantum states $\hat{\rho}$. The ambient space is the set of Hermitian trace-one operators (not necessarily density operators), corresponding to the set of real normalized functions in phase space (not necessarily probability distributions). The set of quantum states $\mathcal{Q}$ contains all positive semi-definite operators $\hat{\rho}$, which are associated with either a Wigner-positive probability distribution or a Wigner-negative quasi-probability distribution. The set of phase-space probability distributions $W(\alpha)$ is associated to the set of operators $\mathcal{P}$ and contains all Wigner-positive operators, which can be either states or quasi-states. The intersection of both sets gives the set of Wigner-positive states $\mathcal{W}_+$. The hatched blue region denotes positive Wigner functions that tend to saturate the positivity condition in phase space (hence becoming quasi-probability distributions beyond the border). The solid blue region denotes the set of Wigner-positive states that tend to saturate the positive semi-definiteness operator condition (hence becoming quasi-states beyond the border).
• Figure 2: Schematic of a beam splitter with transmittance $\eta$ acting on a product input state $\hat{\rho}_1\otimes\hat{\rho}_2$. The reduced state of output 1 is $\hat{\sigma}(\hat{\rho}_1,\hat{\rho}_2)=\mathrm{Tr}_2\left[\hat{U}_{\eta}\left(\hat{\rho}_1\otimes\hat{\rho}_2\right)\hat{U}^\dagger_\eta\right]$ and is described by the Wigner function $W_{\hat{\sigma}}=\mathcal{L}_{\sqrt{\eta}}[W_1]\ast\mathcal{L}_{\sqrt{1-\eta}}[W_2]$.
• Figure 3: Elementary beam-splitter state $\hat{\sigma}_\ast(\psi,\varphi)$ produced from the pure input states $\ket{\psi}$ and $\ket{\varphi}$. The output Wigner function is computed as $W_{\hat{\sigma}}=\mathcal{L}_{1/\sqrt{2}}[W_\psi\ast W_\varphi]$, where $W_\psi$ and $W_\varphi$ are the input Wigner functions. As a consequence of the outer interference formula (see Sec. \ref{['sec:interference_formula']}), the output Wigner function is also equal to $W_{\hat{\sigma}}=2\pi \, \mathcal{L}_{\sqrt{2}}[\abs{W_{\psi\varphi}}^2]$, where $W_{\psi\varphi}$ is the cross Wigner function.
• Figure 4: Illustration of the inner interference formula \ref{['eq:inner_interference_formula']} with the pure state $\ket{\psi}=(\ket{0}-\ket{1}+i\ket{2})/\sqrt{3}$. From its Wigner function $W_\psi$, we compute $W_\psi^2$ and $W_\psi\ast W_\psi$, which are both non-negative. We observe that, up to a rescaling and a multiplicative constant, the two distributions are equal, confirming that $W_\psi\ast W_\psi=2\pi\, \mathcal{L}_{2}[W^2_\psi]$.
• Figure 5: Schematic representation of the proven validity regions of the Wigner-Rényi entropy conjecture as a function of the parameter $\alpha$ for different classes of states. (1) Husimi BS states $\hat{\sigma}_\mathrm{H} \vcentcolon= \hat{\sigma}(\hat{\rho},0)$, $\forall \hat{\rho}$, see Ref. Lieb2014-qx; (2) BS states $\hat{\sigma}\in\mathcal{B}$, see Theorem \ref{['theorem-Wigner-Renyi']}; (3) Wigner-positive state $\hat{\rho}_+\in\mathcal{W}_+$, see Ref. Dias2023-qr.

Theorems & Definitions (10)

• Definition 1: Wigner entropy Van_Herstraeten2021-nj
• Conjecture 1: Wigner entropy conjecture Van_Herstraeten2021-nj
• Definition 2: Wigner-Rényi entropy Van_Herstraeten2021-nj
• Conjecture 2: Wigner-Rényi entropy conjecture Van_Herstraeten2021-nj
• Definition 3: Beam-splitter state
• Definition 4: Matched Gaussian pair
• Lemma 1: Restatement of Lieb1990-ev
• Lemma 2
• Theorem 1: Wigner-Rényi entropy of BS states
• Theorem 2: Wigner entropy of BS states