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We prove the following theorem about relative entropy of quantum states."Substate theorem: Let rho and sigma be quantum states in the same Hilbertspace 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, andrho'/2^{O(c/epsilon^2)} <= sigma."  It states that if the relative entropy of rho and sigma is small, then thereis a state rho' close to rho, i.e. with small trace distance ||rho' - rho||_t,that when scaled down by a factor 2^{O(c)} `sits inside', or becomes a`substate' of, sigma. This result has several applications in quantumcommunication complexity and cryptography. Using the substate theorem, wederive a privacy trade-off for the set membership problem in the two-partyquantum communication model. Here Alice is given a subset A of [n], Bob aninput i in [n], and they need to determine if i in A.  "Privacy trade-off for set membership: In any two-party quantum communicationprotocol for the set membership problem, if Bob reveals only k bits ofinformation about his input, then Alice must reveal at least n/2^{O(k)} bits ofinformation about her input."  We also discuss relationships between various information theoreticquantities that arise naturally in the context of the substate theorem.

A theorem about relative entropy of quantum states with an application to privacy in quantum communication