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Author Jain, Rahul ♦ Radhakrishnan, Jaikumar ♦ Sen, Pranab
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2009
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Privacy ♦ Quantum communication complexity ♦ Quantum information theory
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 $\textit{S}(ρ$ ∥ σ) ≔ Tr ρ (log ρ™ log σ) = $\textit{c}.$ 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 $ρ′/2^{O(c/ε^{2})}$ ≤ 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 $\textit{A}$ &subse; $[\textit{n}],$ Bob an input $\textit{i}$ ∈ $[\textit{n}],$ and they need to determine if $\textit{i}$ ∈ $\textit{A}.$ Privacy trade-off for set membership: In any two-party quantum communication protocol for the set membership problem, if Bob reveals only $\textit{k}$ 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.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2009-09-08
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 56
Issue Number 6
Page Count 32
Starting Page 1
Ending Page 32

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Source: ACM Digital Library