Rank lower bounds on non-local quantum computation

A non-local quantum computation (NLQC) replaces an interaction between two quantum systems with a single simultaneous round of communication and shared entanglement. We study two classes of NLQC, f-routing and f-BB84, which are of relevance to classical information theoretic cryptography and quantum position-verification. We give the first non-trivial lower bounds on entanglement in both settings, but are restricted to lower bounding protocols with perfect correctness. Our technique is based on the rank of a function g that is zero if and only if the function f which defines the given non-local quantum computation is zero. For the equality, non-equality, and greater-than functions we obtain explicit linear lower bounds on entanglement for f-routing and f-BB84 in the perfect setting. Because of a relationship between f-routing and the conditional disclosure of secrets (CDS) primitive studied in information theoretic cryptography, we also obtain a new technique for lower bounding the randomness complexity of CDS.