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

We prove the following theorem about relative entropy of quantum states. Substate theorem: Let ρ and σ be quantum states in the same Hilbert space with relative entropy S(ρ‖σ) := Tr ρ(log ρ− log σ) = c. Then for all > 0, there is a state ρ′ such that the trace distance ‖ρ′ − ρ‖tr := Tr p (ρ′ − ρ)2 ≤ , and ρ′/2O(c/ 2) ≤ σ. It states that if the relative entropy of ρ and σ is small, then there is a state ρ′ close to ρ, i.e. with small trace distance ‖ρ′ − ρ‖tr, that when scaled down by a factor 2 O(c) ‘sits inside’, or becomes a ‘substate’ of, σ. 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 twoparty quantum communication model. Here Alice is given a subset A ⊆ [n], Bob an input i ∈ [n], and they need to determine if i ∈ 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/2 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. Most of the results presented here have previously appeared in [JRS02].