A property of quantum relative entropy with an application to privacy in quantum communication

Abstract

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 ϵ > 0, there is a state ρ′ such that the trace distance ‖ρ′ - ρ‖ tr : Tr √(ρ′ - ρ) 2 ≤ ϵ, and ρ′/2 O ( c /ϵ 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 two-party 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 O( 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.