Preprint typeset in JHEP style - HYPER VERSION BRX-TH-638 Holographic mutual information is monogamous Patrick Hayden School of Computer Science & Department of Physics, McGill University, Montreal, Quebec, Canada
[email protected] Matthew Headrick Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts, USA
[email protected] Alexander Maloney Department of Physics, McGill University, Montreal, Quebec, Canada
[email protected] Abstract: We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions A; B; C obey the inequality I(A : B [ C) ≥ I(A : B) + I(A : C), provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are characteristic of measures of quantum entanglement. This suggests that correlations in holographic theories arise primarily from entanglement rather than classical correlations. We also show that the Ryu-Takayanagi formula is consistent with all known general inequalities obeyed by the entanglement entropy, including an infinite set arXiv:1107.2940v2 [hep-th] 2 Feb 2013 recently discovered by Cadney, Linden, and Winter; this constitutes strong evidence in favour of its validity. Contents 1. Introduction 1 2. Tripartite information in quantum field theory 4 3. Holographic mutual information is monogamous 7 3.1 Proof 7 3.2 Higher-curvature corrections and general large-N theories 10 4. Ryu-Takayanagi entropy as entanglement entropy 11 5. Holography and the structure of correlations 12 5.1 Absence of quantum Markov chains 13 5.2 Monogamy and quantum cryptography 13 6. Open questions 17 6.1 Additional inequalities 17 6.2 Finite-N corrections 18 1.