热混合量子态的纯度
Purity of thermal mixed quantum states
摘要：
我们开发了一个公式来评估一系列热平衡态的纯度，这些热平衡态可以在不知道量子态\textit{a priori}的确切形式的情况下在数值实验中计算。正则典型性保证了这些态有许多微观上不同的表达，我们称之为热混合量子（TMQ）态。假设我们用$N_\mathrm{samp}$独立纯态的混合态构造TMQ态。每个纯态的权由其范数给出，配分函数由范数的平均给出。为了证明混合的效率，我们引入了一个量子统计量，称为“配分函数的归一化波动"。对于较小的NFPF，TMQ状态更接近纯状态的等权重混合，这意味着更高的效率，需要更小的$N_\mathrm{samp}$。最大的NFPF是在纯度为0和指数大的$N_\mathrm{samp}$的Gibbs态中实现的，而最小的NFPF是在纯度为1和$N_\mathrm{samp}=1$的热纯量子态中实现的。仅使用NFPF配制纯度，大致得到$N_\mathrm{samp}^{-1}$。我们的分析结果进行了数值测试，并证实了两个随机抽样方法建立在矩阵乘积状态为基础的波函数。
英文摘要：
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.
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