AI 中文总结
该研究通过将de Finetti近似转化为对称扩展半定规划的整度间隙并应用argmax取整方法，证明了最优有限量子de Finetti上界，解决了遗留的维度依赖性问题，还反驳了解纠缠器猜想并获得了相关算法。
AI 中文摘要
我们证明了最优有限量子de Finetti上界。给定玻色态ρ_N∈D(Sym^N(ℂ^d))，存在单位球上的概率测度ν，使得‖ρ_N^(2)−∫|u⟩⟨u|^(⊗2)dν(u)‖₁≤√(d−1)/(N−1)。通过纯化，该玻色定理也为任意可交换态给出了最优O(d/N)上界。这些结果解决了Christandl、König、Mitchison和Renner（CMP 2007）留下的维度依赖性问题。该证明将de Finetti近似转化为平方和（sum-of-squares）取整，并应用了Jeronimo、Wu和Xu（2026年手稿）的argmax方法。更一般地，t位点边际满足O(t√d/N)的玻色界和O(td/N)的置换不变界。我们的证明将de Finetti近似表述为对称扩展半定规划的整度间隙，并通过argmax原理对最优解取整。这些尖锐的界有若干推论：对于每个固定的ε∈(0,1)，我们构造了一个输入维度D=exp(O_ε(√d log d))=exp(o(d))的信道，其输出与局部维度为d的可分态ε-接近，且其像包含所有此类可分态，从而反驳了Watrous的解纠缠器猜想。我们还获得了确定性的exp(Õ(√d/ε))时间算法，用于处理无完美完备性的显式最佳可分态问题以及迹距离可分性测试。最后，谱截断给出了希尔伯特-施密特距离下首个无维度依赖的玻色de Finetti定理，当维度可增长时，其最优率为Θ(N^(-1/2))。
英文摘要
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state $ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))$, there is a probability measure $ν$ on the unit sphere such that \[ \left\| ρ_N^{(2)}-\int |u\rangle\langle u|^{\otimes 2}\,dν(u) \right\|_1 \le \frac{\sqrt{d-1}}{N-1}. \] By purification, the bosonic theorem also gives the optimal $O(d/N)$ upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026). More generally, $t$-site marginals satisfy $O(t\sqrt d/N)$ bosonic and $O(td/N)$ permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed $\varepsilon\in(0,1)$, we construct a channel with input dimension $D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))$ whose outputs are $\varepsilon$-close to separable states of local dimension $d$ and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic $\exp(\widetilde O(\sqrt d/\varepsilon))$-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing. Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate $Θ(N^{-1/2})$ when the dimension may grow.
