New lower bounds for CDS and $f$-routing
New lower bounds tie robust CDS randomness to communication complexity and inner-product routing entanglement to sign rank.
The paper studies entanglement and randomness costs related to non-local quantum computation, focusing on f-routing and robust conditional disclosure of secrets. It proves that robust CDS shared-randomness cost is at least the logarithm of deterministic simultaneous-message-passing communication complexity, even with unlimited communication and private randomness, and that the bound is tight for equality. For one-sided-perfect f-routing, a sign-rank argument gives a linear entanglement lower bound for the inner-product function in both one-sided settings, matching the known upper bound.
- Robust CDS shared-randomness cost is at least log of deterministic SMP complexity.
- The CDS lower bound is tight for the equality function.
- One-sided-perfect inner-product routing needs linear entanglement, matching the upper bound.
Full article229 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.24291