AI 中文总结
该研究针对由局部分量指定的指数级大乘积量子态，给出有理乘积输入下迹距离的通用常数因子确定性多项式时间近似算法，同时证明精确计算为#P难。
AI 中文摘要
我们研究当两个指数级大的量子态由其局部分量指定时的迹距离 $D_{\mathrm{tr}}(\rho,\sigma) =\frac12\\|\rho-\sigma\\|_1$，其中 $\rho=\bigotimes_{i=1}^n\rho_i$，$\sigma=\bigotimes_{i=1}^n\sigma_i$。对于有理乘积输入，我们给出了一个通用常数因子内的确定性近似算法，其运行时间在分量数量、局域维度和输入比特长度上均为多项式时间。与之相对，即使是对角量子比特态，精确计算也是#P难的，这源于乘积分布之间总变差距离的对应困难性。该证明使用局域Uhlmann最优纯化，将问题简化为估计乘积保真度缺陷和结构化一阶算子的迹范数。尽管该算子作用于指数级大的空间，我们通过一个具有多项式规模经典锥公式的局域凸替代物来近似其迹范数。平方函数估计表明该替代物是此迹范数的上界；反之，对偶性和局域退相将反向比较简化为独立中心随机变量的头尾不等式，证明该替代物至多是与维度无关的常数倍的同一范数。
英文摘要
We study the trace distance \[D_{\mathrm{tr}}(ρ,σ) =\frac12\|ρ-σ\|_1, ρ=\bigotimes_{i=1}^nρ_i,\quad σ=\bigotimes_{i=1}^nσ_i, \] when the two exponentially large states are specified by their local factors. We give a deterministic approximation within a universal constant factor for rational product inputs. Its running time is polynomial in the number of factors, the local dimension, and the input bit length. In the opposite direction, exact computation is $\#\mathsf P$-hard even for diagonal qubit states, by the corresponding hardness of total variation distance between product distributions. The proof uses local Uhlmann-optimal purifications to reduce the problem to estimating the product-fidelity defect and the trace norm of a structured first-order operator. Although this operator acts on an exponentially large space, we approximate its trace norm by a local convex surrogate that admits a polynomial-size classical conic formulation. A square-function estimate shows that the surrogate upper-bounds this trace norm. Conversely, duality and local dephasing reduce the reverse comparison to a head--tail inequality for independent centered random variables, showing that the surrogate is at most a dimension-free constant times the same norm.
Comments32 pages
