CDS 与 $f$-路由的新下界
New lower bounds for CDS and $f$-routing
- Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对非局域量子计算中的$f$-路由问题，本文建立了鲁棒CDS共享随机性代价的下界（基于确定性SMP通信复杂度对数）及单侧完美$f$-路由纠缠代价的符号秩下界，并给出内积函数的线性下界。
AI中文摘要：
理解非局域量子计算（NLQC）的纠缠代价与复杂性理论、密码学、量子引力及相关领域相关。一个核心特例是 $f$-路由，部分受量子位置验证的启发。在完全鲁棒设置下证明其纠缠代价的下界一直是 NLQC 中的一个重大开放问题。受此问题启发，我们建立了两个相关的下界。首先，我们研究了鲁棒条件秘密泄露（CDS）的共享随机性代价。Allerstorfer 等人（Quantum 2024）建立的 CDS 与 $f$-路由之间的联系，使得理解鲁棒 CDS 的随机性复杂性成为迈向完全鲁棒路由问题下界的自然一步。我们证明，即使通信和私有随机性不受限制，鲁棒 CDS 的共享随机性代价也由确定性 SMP 通信复杂度的对数下界限定。我们的下界对于相等函数是紧的。其次，我们考虑单侧完美 $f$-路由，其中协议在一类输入上是精确的，在另一类输入上具有恒定误差。通过利用 Asadi、Culf 和 May（ITCS 2025）方法中出现的低秩矩阵的正性，我们推导出基于符号秩的纠缠代价的一般下界。特别地，这为内积函数在两种单侧完美设置下的路由纠缠代价提供了线性下界，与已知上界匹配。
英文摘要：
Understanding the entanglement cost of non-local quantum computation (NLQC) is relevant to complexity theory, cryptography, quantum gravity, and related areas. A central special case is $f$-routing, motivated in part by quantum position verification. Proving lower bounds on its entanglement cost in the fully robust setting has been a major open problem in NLQC. Motivated by this problem, we establish two related lower bounds. First, we study the shared-randomness cost of robust conditional disclosure of secrets (CDS). The connection between CDS and $f$-routing established by Allerstorfer et al. (Quantum 2024) makes understanding the randomness complexity of robust CDS a natural step toward lower bounds for the fully robust routing problem. We show that the shared-randomness cost of robust CDS is lower bounded by the logarithm of deterministic SMP communication complexity, even when communication and private randomness are unrestricted. Our lower bound is tight for the equality function. Second, we consider one-sided-perfect $f$-routing, in which the protocol is exact on one input class and has constant error on the other. By exploiting the positivity of the low-rank matrix arising in the method of Asadi, Culf, and May (ITCS 2025), we derive a general lower bound on the entanglement cost in terms of sign rank. In particular, this yields a linear lower bound on the entanglement cost of routing for the inner-product function in both one-sided-perfect settings, matching the known upper bound.
