论文标题
热混合量子状态的纯度
Purity of thermal mixed quantum states
论文作者
论文摘要
我们开发了一个公式来评估一系列热平衡状态的纯度，这些状态可以在数值实验中计算出，而不知道量子状态\ textit {a先验}的确切形式。规范的典型性保证了这种状态上有许多显微镜不同的表达式，我们称之为热混合量子（TMQ）状态。假设我们通过$ n_ \ mathrm {samp} $独立纯状态的混合物构建了一个TMQ状态。每个纯状态的重量由其规范给出，分区函数由规范的平均值给出。为了确定混合物的有效性，我们引入了称为“分区函数（NFPF）归一化波动”的量子统计量。对于较小的NFPF，TMQ状态更接近纯状态的同等加权混合物，这意味着较高的效率，需要较小的$ N_ \ Mathrm {samp} $。最大的NFPF是在Gibbs状态下实现的，具有Purity-0，并且指数级的$ N_ \ Mathrm {samp} $，而最小的NFPF则用于具有Purity-1和$ n_ \ Mathrm {samp} = 1 $的热纯量子状态。纯度仅使用NFPF制定，并大致给出$ n_ \ mathrm {samp}^{ - 1} $。我们的分析结果经过数值测试和确认，并通过基于矩阵基于矩阵状态的波浪功能的两种随机抽样方法进行了确认。
We develop a formula to evaluate the purity of a series of thermal equilibrium states that can be calculated in numerical experiments without knowing the exact form of the quantum state \textit{a priori}. Canonical typicality guarantees that there are numerous microscopically different expressions of such states, which we call thermal mixed quantum (TMQ) states. Suppose that we construct a TMQ state by a mixture of $N_\mathrm{samp}$ independent pure states. The weight of each pure state is given by its norm, and the partition function is given by the average of the norms. To qualify how efficiently the mixture is done, we introduce a quantum statistical quantity called "normalized fluctuation of partition function (NFPF)". For smaller NFPF, the TMQ state is closer to the equally weighted mixture of pure states, which means higher efficiency, requiring a smaller $N_\mathrm{samp}$. The largest NFPF is realized in the Gibbs state with purity-0 and exponentially large $N_\mathrm{samp}$, while the smallest NFPF is given for thermal pure quantum state with purity-1 and $N_\mathrm{samp}=1$. The purity is formulated using solely the NFPF and roughly gives $N_\mathrm{samp}^{-1}$. Our analytical results are numerically tested and confirmed by the two random sampling methods built on matrix-product-state-based wave functions.
