AI 中文总结
该论文在任意距离度量下证明了态与信道的量子 de Finetti 定理，通过算子不等式形式改进了误差界，并解决了信道表示中的开放问题。
AI 中文摘要
标准有限量子 de Finetti 定理通常以迹距离将 n 系统置换不变态的 k 系统边缘近似为独立同分布（iid）态的混合。我们证明了标准形式和 Renner 指数形式的 de Finetti 定理，其形式为更强的算子不等式，从而在每一个 Schatten 范数和每一个满足数据处理性质的量子 Rényi 散度下都给出界。在标准情形下，对于固定局部维度，我们的 k/n 误差界（以最大相对熵度量）改进了先前已知的最佳 k^2/n 标度，即使在经典情形下也是如此。算子不等式方法特别适合研究信道 de Finetti 表示，因为 Choi 态之间的算子序等价于底层信道之间的完全正（CP）序。对于置换协变信道 N^(n):A^(⊗n)→B^(⊗n)，其中 d_A=dim A，我们证明 k 系统约化信道被张量幂信道混合 CP 支配，误差为 O(k/√n) 且对局部维度具有多项式依赖，解决了 Berta 等人 [Math. Program. 194, 781-829 (2022)] 提出的问题。在无信号条件下，我们还证明了一个指数信道 de Finetti 定理，其中近似混合由 Choi-几乎-iid 信道组成，其归一化 Choi 态在 Mazzola-Sutter-Renner 意义下是几乎-iid 的。在 r 个缺陷的情形下，表示误差至多为 poly(n)(2d_A^4k^3/(nr^2))^((r+1)/2)，并且对于适当的参数 r 和 k 选择，随 n 指数衰减。
英文摘要
Standard finite quantum de Finetti theorems approximate the $k$-system marginals of permutation-invariant states of $n$-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our $k/n$ error bound in max-relative entropy improves on the previously best known $k^2/n$ scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels $N^{(n)}:A^{\otimes n}\to B^{\otimes n}$, where $d_A=\dim A$, we prove that the $k$-system reduced channel is CP-dominated by a mixture of tensor-power channels with error $O(k/\sqrt{n})$ and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of $r$ defects, the representation error is at most $\mathrm{poly}(n)\bigl(2d_A^4k^3/(nr^2)\bigr)^{(r+1)/2}$ and decays exponentially in $n$ for a suitable choice of parameters $r$ and $k$.
Comments52 pages
