Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics, the canonical model of continuous-variable decoherence, we prove that the unrestricted instantaneous reverse optimum exhibits a noiseless-to-noisy transition: below a critical squeezing-to-thermal ratio, reversal can be noiseless; above it, complete positivity forces irreducible reverse noise whose minimum cost we determine in closed form. The optimal reverse diffusion is uniquely covariance-aligned and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact Gaussian reversal of a pure quantum state is dynamically unattainable. These results establish the exact Gaussian benchmark against which any broader theory of quantum reverse diffusion must be measured.

翻译：经典反向扩散是通过在固定噪声下改变漂移而产生的。我们证明这一原理的量子版本遵循一个具有尖锐相边界的精确定律。对于高斯纯损耗动力学（连续变量退相干的规范模型），我们证明无限制的瞬时反向最优解呈现无噪声到有噪声的相变：当压缩与热比率低于临界值时，反演可以是无噪声的；高于该临界值时，完全正定性迫使产生不可逆的反向噪声，我们以闭式形式确定了其最小代价。最优反向扩散具有唯一的协方差对齐特性，并同时最小化了反演的几何、计量和热力学代价。对于多模轨迹，精确代价在模式分辨数据的规范集合中具有可加性，且一个全局连续协议在每个混合态区间上达到该最优解。若包含纯非经典终点，则在每个 $t>0$ 时刻相同点态定律成立，但最优解以 $2/t$ 的形式发散：纯量子态的精确高斯反演在动力学上不可实现。这些结果建立了一个精确的高斯基准，任何更广泛的量子反向扩散理论都必须以此进行衡量。