We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely related to the decay of out-of-time-order correlators. We back up our results with numerics in two non-integrable models and analytic results in a perfect tensor network model of chaotic time evolution. These results show that the butterfly effect in quantum systems implies the information-theoretic definition of scrambling.
A. Kitaev, A simple model of quantum holography, talks given at The Kavli Institute for Theoretical Physics (KITP), University of California, Santa Barbara, U.S.A., 7 April 2015 and 27 May 2015.
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000).
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [INSPIRE].
A. Kitaev, Hidden correlations in the hawking radiation and thermal noise, talk given at The Fundamental Physics Prize Symposium, Stanford University, Standford, U.S.A., 10 November 2014.
D.A. Roberts and B. Swingle, to appear.
E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].
M.B. Hastings, Locality in quantum systems, arXiv:1008.5137.
D. Stanford, Scrambling and the entanglement wedge, unpublished.
M.C. Bañuls, J.I. Cirac and M.B. Hastings, Strong and weak thermalization of infinite nonintegrable quantum systems, Phys. Rev. Lett. 106 (2011) 050405 [arXiv:1007.3957].
F.G.S.L. Brandao, A.W. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
H. Kim and D.A. Huse, Ballistic spreading of entanglement in a diffusive nonintegrable system, Phys. Rev. Lett. 111 (2013) 127205 [arXiv:1306.4306].
S.G. Nezami, P. Hayden, X.L. Qi, N. Thomas, M. Walters and Z. Yang, Random tensor networks as models of holography, to appear.
M. Hastings, Random mera states and the tightness of the brandao-horodecki entropy bound, arXiv:1505.06468.
K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Almqvist & Wiksell (1930).
M. Fannes, A continuity property of the entropy density for spin lattice systems, Commun. Math. Phys. 31 (1973) 291.
K.M.R. Audenaert, A sharp fannes-type inequality for the von neumann entropy, J. Phys. A 40 (2007) 8127 [quant-ph/0610146].
V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps and F. Galli, Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories, Phys. Rev. D 84 (2011) 105017 [arXiv:1110.0488] [INSPIRE].
D. Fattal, T.S. Cubitt, Y. Yamamoto, S. Bravyi and I.L. Chuang, Entanglement in the stabilizer formalism, quant-ph/0406168.
B. Yoshida and I.L. Chuang, Framework for classifying logical operators in stabilizer codes, Phys. Rev. A 81 (2010) 052302 [arXiv:1002.0085].
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ArXiv ePrint: 1511.04021
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Hosur, P., Qi, XL., Roberts, D.A. et al. Chaos in quantum channels. J. High Energ. Phys. 2016, 4 (2016). https://doi.org/10.1007/JHEP02(2016)004
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