德·菲内蒂层次结构中的不动点
Fixed points in de Finetti hierarchies
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中文总结 AI 辅助
研究德·菲内蒂层次结构中可行态为量子信道不动点的情况，结合平均遍历定理等证明相关界和法则，推导出多个德·菲内蒂定理，利用对偶性等表明舍入方案可在多项式时间实现，应用于双线性优化和量子纠错。
中文摘要 AI 辅助
德·菲内蒂定理将置换对称性转化为乘积态的近似混合，从而在经典和量子统计中证明了广泛的简化。本文研究德·菲内蒂层次结构，其中可行态还被约束为量子信道的不动点，该条件包含任意紧致对称群下的不变性。结合平均遍历定理和条件期望的结构理论，我们证明了条件期望对偶的纠缠辅助经典容量的紧界、适用于不动点代数的信息完备测量的逐块失真界以及置换不变态链式法则的精确基于类型的细化。从这些工具中我们推导出几个德·菲内蒂定理：一个具有\(O(\sqrt{\log n}/n)\)收敛的双边扩展定理、一个维度依赖仅由不动点代数的块结构决定且恢复最大环面的维度无关经典行为的插值定理以及一个玻色对称变体。利用舒尔 - 外尔对偶性和盖尔范德 - 采特林基，我们进一步表明，对于固定的局部维度，为（受约束的）可分性问题生成可认证良好可分内近似的舍入方案可以在\(1/\epsilon\)的多项式时间内实现，补充了已知的高效外层次结构。还讨论了在对称下的双线性优化和近似量子纠错中的应用。
英文摘要
De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with $O\left(\sqrt{\log n}/n\right)$ convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in $1/ε$ for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.
