A Property of Quantum Relative Entropy with an Application to Privacy in Quantum Communication
We prove the following information-theoretic property about quantum states. Substate theorem: Let rho and sigma be quantum states in the same Hilbert space with relative entropy S(rho parallel to sigma) := Tr rho(log rho - log sigma) = c. Then for all epsilon > 0, there is a state rho' such that the trace distance parallel to rho' - rho parallel to(tr) := Tr root(rho' - rho)(2) <= epsilon, and rho'/2(O(c/epsilon 2)) <= 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 parallel to rho' - rho parallel to(tr), that when scaled down by a factor 2(O(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 subset of [n], Bob an input i epsilon [n], and they need to determine if i epsilon 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.
privacy quantum communication complexity theory quantum information theory