Lower bounds reveal limits for quantum secret sharing and routing
New lower bounds for CDS and $f$-routing
Cryptography and Security
Summary
Sending secret information securely and routing quantum data are challenging tasks in quantum computing. The authors found new limits on how much shared randomness and entanglement are needed to do these tasks under strict reliability requirements. They linked these limits to well-known communication complexity concepts, showing that some problems require a significant amount of resources. Specifically, they proved concrete lower bounds on the shared randomness for secret disclosure and on entanglement for quantum routing of certain functions. These results help understand the fundamental costs of quantum networks and protocols.
What this means in practice
- •For quantum network designers: Assess resource requirements and limitations when designing secure multi-party quantum communication protocols under robustness constraints.
- •For quantum cryptography engineers: Evaluate minimum entanglement and randomness costs for implementing quantum position verification and secret sharing schemes with strict error bounds.
A theory result. No direct application yet.
Authors
Atsuya Hasegawa, Ranitha Mataraarachchi
Abstract
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.