Wigner Shannon 熵猜想的物理反例
Wigner entropy below vacuum: physical counterexamples and stability limits
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中文总结 AI 辅助
本文构造了 Wigner 函数处处为正但 Shannon 熵低于真空值的物理量子态，证明 Wigner Shannon 熵猜想存在反例，并给出最大熵亏缺的锐利尺度。
中文摘要 AI 辅助
我们构造了物理量子态，其 Wigner 函数处处为正，但 Shannon 熵低于真空值 $1+\ln\pi$。除了一个显式的有限能量反例外，我们还获得了具有解析正性和熵界的秩二有限 Fock 支撑族。对于固定的 Fock 能级 $n$ 和相干分数 $0\le\lambda<1$，熵差满足 $h(W)-(1+\ln\pi)=(2n-2^n\lambda^2)t^2+O_{n,\lambda}(t^4)$，从而在显式相干阈值之上，对每个 $n\ge3$ 都给出有限支撑反例。我们还表明，负二次项的缺失并不排除熵下降：一个完全相干的真空-单光子核心，加上消失的正热修复，给出 $h(W)-(1+\ln\pi)=-4t^6/3+o(t^6)$。对于真空-三光子构造，我们确定了 Wigner 非负性所需的最小固定热混合权重的对数渐近。最后，在平均光子数至多 $E$ 的所有单模 Wigner 非负态上优化，我们证明了最大熵亏缺具有锐利尺度 $E^\gamma/[\ln(1/E)]^\beta$，其中 $\gamma\simeq0.7412033679$ 和 $\beta\simeq0.5861054961$。因此，当 $E\to0$ 时恢复真空熵，但最优亏缺的衰减远慢于平均能量中任何普适线性修正。
英文摘要
Positive Wigner functions can have less Shannon entropy than the vacuum, even arbitrarily close to the vacuum state. We construct physical counterexamples and identify the competition that controls their entropy: the relative entropy to the vacuum phase-space density can exceed twice the mean photon number. A two-level family exhibits a finite window in which coherence lowers entropy while preserving global Wigner positivity. A complementary construction repairs a remote negative tail with an exponentially small thermal admixture; an explicit mixing weight of $2\times10^{-21}$ preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale $E^γ/[\ln(1/E)]^β$, with $γ\simeq0.7412$ and $β\simeq0.5861$. Because $γ<1$, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound.
发表机构
- University of Glasgow（格拉斯哥大学）
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