从微分几何出发的量子态扩散时间反演
Reverse quantum state diffusion from differential geometry
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中文总结 AI 辅助
从微分几何角度推导量子态扩散的时间反演，得到适用于任意维数和Lindblad算符的后向随机微分方程，并用于生成新系综，在退极化信道下给出误差界。
中文摘要 AI 辅助
量子态扩散将Lindblad主方程分解为纯态的随机轨迹。轨迹系综由纯态流形上的概率密度描述，该密度包含比密度矩阵（其第一矩）丰富得多的信息。我们推导了该扩散的时间反演。将流形上一般扩散的反演写成Stratonovich形式，并经由纯态流形，我们获得了适用于任意希尔伯特空间维数和任意Lindblad算符的状态向量后向随机微分方程。该后向随机方程提供了逆Lindblad算符的一种物理的、非线性的解纠缠，该解纠缠仅对构造它的系综成立。遵循基于分数的生成模型，我们随后利用后向方程生成一个接近原始系综的新系综。对于退极化信道，解纠缠是复射影空间上的布朗运动，因此在任意维数下纯态的均匀分布是其平稳分布。从该先验出发启动后向方程，我们在不同设置下（精确与学习到的分数、连续与离散时间）界定了所得分布与原始分布之间的误差。
英文摘要
Quantum state diffusion unravels a Lindblad master equation into stochastic trajectories of pure states. The ensemble of trajectories is described by a probability density on the manifold of pure states, which contains much more information than the density matrix, its first moment. We derive the time reversal of this diffusion. Writing the reversal of a general diffusion on a manifold in Stratonovich form, and passing through the manifold of pure states, we obtain a backward stochastic differential equation for the state vector, valid for any Hilbert space dimension and any Lindbladian. The backward stochastic equation provides a physical, nonlinear unravelling of the inverse Lindbladian that holds only for the ensemble from which it was constructed. Following score-based generative models, we then use the backward equation to generate a new ensemble close to the original. For the depolarizing channel the unravelling is a Brownian motion on complex projective space, so the uniform distribution of pure states is its stationary law in any dimension. Starting the backward equation from that prior, we bound the error of the resulting distribution compared to the original one in different settings: exact and learned scores, continuous and discrete time.
发表机构
- National University of Singapore（新加坡国立大学）
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