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Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture

Qipeng Qian, Christos Gagatsos

TL;DR

This work addresses the Wigner entropy conjecture for Wigner-nonnegative states by introducing a minimal, physically motivated constraint set and deriving a computable hierarchy of purity-based lower bounds on the Wigner entropy. Using a truncated-series expansion of $-\ln x$ and the moment structure of the Wigner function, it constructs bounds $B_n$ and translates them into explicit purity thresholds $\mu_n$, with $\mu_2=4-2\sqrt{3}$ and a large-$n$ limit of $\mu=2/e$. It also proves the optimality of the purity-only bound under the given constraints via a flat-top extremizer and strengthens the bound by incorporating a variance term $\mathrm{Var}(X)$, yielding improved thresholds when higher moments are constrained. The results clarify why additional physicality constraints are necessary to move toward the extremal regime $\mu\le 1$, and they outline a pathway to refine the bounds via higher moments or structural quantum constraints, potentially extending to multimode settings.

Abstract

We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value $1+\lnπ$.Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound $πW\le 1$, we construct an explicit hierarchy of lower bounds $B_n$ on $S[W]$ by combining a truncated series lower bound for $-\ln x$ with moment identities of the Wigner function.This yields closed-form purity-based sufficient conditions ensuring $S[W]\ge 1+\lnπ$.In particular, we first prove that all Wigner non-negative states with $μ\le 4-2\sqrt3$ satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, yielding the simple sufficient condition $μ\le 2/e$. On top of aforesaid results, our analysis clarifies why additional physicality constraints are necessary for purity-based approaches that aim to approach the extremal case $μ\leq1$.

Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture

TL;DR

This work addresses the Wigner entropy conjecture for Wigner-nonnegative states by introducing a minimal, physically motivated constraint set and deriving a computable hierarchy of purity-based lower bounds on the Wigner entropy. Using a truncated-series expansion of and the moment structure of the Wigner function, it constructs bounds and translates them into explicit purity thresholds , with and a large- limit of . It also proves the optimality of the purity-only bound under the given constraints via a flat-top extremizer and strengthens the bound by incorporating a variance term , yielding improved thresholds when higher moments are constrained. The results clarify why additional physicality constraints are necessary to move toward the extremal regime , and they outline a pathway to refine the bounds via higher moments or structural quantum constraints, potentially extending to multimode settings.

Abstract

We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value .Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound , we construct an explicit hierarchy of lower bounds on by combining a truncated series lower bound for with moment identities of the Wigner function.This yields closed-form purity-based sufficient conditions ensuring .In particular, we first prove that all Wigner non-negative states with satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, yielding the simple sufficient condition . On top of aforesaid results, our analysis clarifies why additional physicality constraints are necessary for purity-based approaches that aim to approach the extremal case .
Paper Structure (10 sections, 2 theorems, 34 equations)

This paper contains 10 sections, 2 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Definitions and conditions
  3. Hierarchy of lower bounds and purity thresholds
  4. Truncated-series bound and the hierarchy $B_n$
  5. Low-order examples: $n=1$ and $n=2$
  6. A systematic purity-only relaxation for all $n\ge 2$
  7. Explicit sufficient thresholds and the large-$n$ limit
  8. Optimality of the purity-only threshold under constraints \ref{['eq: Cond-nonneg']}--\ref{['eq: Cond-pointwise']}
  9. A variance-refined purity criterion at order $n=2$
  10. Conclusion

Key Result

Proposition 1

Fix any $\mu\in(0,2]$. There exists a nonnegative, normalized function $W$ obeying eq: Cond-nonneg--eq: Cond-pointwise with purity $\mu=2\pi\int dq\,dp\,W(q,p)^2$ and Wigner entropy Consequently, if $\mu>2/e$, then there exists such a $W$ for which $S[W]<1+\ln\pi$. In particular, no sufficient condition of the form "$\mu\le \mu_\star \Rightarrow S[W]\ge 1+\ln\pi$" can hold with $\mu_\star>2/e$ un

Theorems & Definitions (6)

  • Conjecture 1
  • Proposition 1: Sharpness of the purity-only bound
  • proof
  • Proposition 2: Variance-refined $n=2$ bound
  • proof
  • Remark 1