Matthew Gray

We give a meta-complexity characterization of EFI pairs, which are considered the “minimal” primitive in quantum cryptography (due to their equivalence to quantum commitments and for being implied by almost all other known quantum cryptographic primitives). More precisely, we show that the existence of EFI pairs is equivalent to the following: there exists a non-uniformly samplable distribution over pure states such that the problem of estimating a certain Kolmogorov-like complexity measure is hard given a single copy. The complexity measure that we consider is a smoothed version of the algorithmic entropy notion introduced by Gács [Gác01]. A key technical step in our proof, which may be of independent interest, is to show that the existence of EFI pairs is equivalent to the existence of non-uniform single-copy secure pseudorandom state generators (nu 1-PRS). As a corollary, we get an alternative, arguably simpler, construction of a universal EFI pair.