A new information-theoretic property about quantum states with an application to privacy in quantum communication∗

We prove the following information-theoretic property about 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 √ (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 ‘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.