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

We prove the following information-theoretic property about quantum states. <i>Substate theorem:</i> Let ρ and σ be quantum states in the same Hilbert space with relative entropy <i>S</i>(ρ ∥ σ) ≔ Tr ρ (log ρ− log σ) = <i>c</i>. Then for all ε > 0, there is a state ρ′ such that the trace distance ∥ρ′ − ρ∥_tr ≔ Tr &sqrt;(ρ′ − ρ)² ≤ ε, and ρ′/2^{<i>O</i>(<i>c</i>/ε²)} ≤ 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^{<i>O</i>(<i>c</i>)} ‘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 <i>set membership problem</i> in the two-party quantum communication model. Here Alice is given a subset <i>A</i> &subse; [<i>n</i>], Bob an input <i>i</i> ∈ [<i>n</i>], and they need to determine if <i>i</i> ∈ <i>A</i>. <i>Privacy trade-off for set membership:</i> In any two-party quantum communication protocol for the set membership problem, if Bob reveals only <i>k</i> bits of information about his input, then Alice must reveal at least <i>n</i>/2^O(<i>k</i>) 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.