A theorem about relative entropy of quantum states with an application to privacy in quantum communication
Abstract
We prove the following theorem about relative entropy of quantum states. "Substate theorem: Let rho and sigma be quantum states in the same Hilbert space with relative entropy S(rho|sigma) = Tr rho (log rho - log sigma) = c. Then for all epsilon > 0, there is a state rho' such that the trace distance ||rho' - rho||t = Tr sqrt(rho' - rho)2 <= epsilon, and rho'/2O(c/epsilon2) <= sigma." It states that if the relative entropy of rho and sigma is small, then there is a state rho' close to rho, i.e. with small trace distance ||rho' - rho||t, that when scaled down by a factor 2O(c) `sits inside', or becomes a `substate' of, sigma. This result has several applications in quantum communication complexity and cryptography. Using the substate theorem, we derive a privacy trade-off for the set membership problem in the two-party quantum communication model. Here Alice is given a subset A of [n], Bob an input i in [n], and they need to determine if i in A. "Privacy trade-off for set membership: In any two-party quantum communication protocol for the set membership problem, if Bob reveals only k bits of information about his input, then Alice must reveal at least n/2O(k) bits of information about her input." We also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem.