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

  • Authors:

  • Rahul Jain;Jaikumar Radhakrishnan;Pranab Sen

  • Affiliations:

  • National University of Singapore, Singapore;Tata Institute of Fundamental Research, Mumbai, India;Tata Institute of Fundamental Research, Mumbai, India

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 2009

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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 &sqrt;(ρ′ − ρ)2 ≤ ε, and ρ′/2O(c/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 2O(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 &subse; [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/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.