Abstract
We introduce a framework to study the emergence of time and causal structure in quantum many-body systems. In doing so, we consider quantum states which encode spacetime dynamics, and develop information theoretic tools to extract the causal relationships between putative spacetime subsystems. Our analysis reveals a quantum generalization of the thermodynamic arrow of time and begins to explore the roles of entanglement, scrambling and quantum error correction in the emergence of spacetime. For instance, exotic causal relationships can arise due to dynamically induced quantum error correction in spacetime: there can exist a spatial region in the past which does not causally influence any small spatial regions in the future, but yet it causally influences the union of several small spatial regions in the future. We provide examples of quantum causal influence in Hamiltonian evolution, quantum error correction codes, quantum teleportation, holographic tensor networks, the final state projection model of black holes, and many other systems. We find that the quantum causal influence provides a unifying perspective on spacetime correlations in these seemingly distinct settings. In addition, we prove a variety of general structural results and discuss the relation of quantum causal influence to spacetime quantum entropies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, in Statistical Mechanics, pp. 425-431, Springer (1972).
O. Oreshkov, F. Costa and Č. Brukner, Quantum correlations with no causal order, Nature Commun.3 (2012) .
Č. Brukner, Quantum causality, Nature Phys.10 (2014) 259.
Y. Aharonov, S. Popescu and J. Tollaksen, Each instant of time a new universe, in Quantum theory: a two-time success story, pp. 21-36, Springer (2014).
J. F. Fitzsimons, J. A. Jones and V. Vedral, Quantum correlations which imply causation, Sci. Rept.5 (2015) 18281.
K. Ried, M. Agnew, L. Vermeyden, D. Janzing, R. W. Spekkens and K. J. Resch, A quantum advantage for inferring causal structure, Nature Phys.11 (2015) 414.
J. Pienaar and Č. Brukner, A graph-separation theorem for quantum causal models, New J. Phys.17 (2015) 073020.
Č. Brukner, Bounding quantum correlations with indefinite causal order, New J. Phys.17 (2015) 083034 [arXiv:1404.0721] [INSPIRE].
F. Costa and S. Shrapnel, Quantum causal modelling, New J. Phys.18 (2016) 063032.
O. Oreshkov and C. Giarmatzi, Causal and causally separable processes, New J. Phys.18 (2016) 093020.
M. Ringbauer, C. Giarmatzi, R. Chaves, F. Costa, A.G. White and A. Fedrizzi, Experimental test of nonlocal causality, Sci. Adv.2 (2016) e1600162 [arXiv:1602.02767].
J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee and R. W. Spekkens, Quantum common causes and quantum causal models, Phys. Rev.X 7 (2017) 031021.
J.-P.W. MacLean, K. Ried, R.W. Spekkens and K.J. Resch, Quantum-coherent mixtures of causal relations, Nature Commun.8 (2017) 15149 [arXiv:1606.04523].
E. Castro-Ruiz, F. Giacomini and Č. Brukner, Dynamics of quantum causal structures, Phys. Rev.X 8 (2018) 011047 [arXiv:1710.03139] [INSPIRE].
O. Oreshkov and N.J. Cerf, Operational formulation of time reversal in quantum theory, Nature Phys.11 (2015) 853 [arXiv:1507.07745] [INSPIRE].
O. Oreshkov and N.J. Cerf, Operational quantum theory without predefined time, New J. Phys.18 (2016) 073037 [arXiv:1406.3829] [INSPIRE].
L. Hardy, The operator tensor formulation of quantum theory, Phil. Trans. Roy. Soc. Lond.A 370 (2012) 3385 [arXiv:1201.4390].
L. Hardy, Operational General Relativity: Possibilistic, Probabilistic and Quantum, arXiv:1608.06940 [INSPIRE].
D. Jia, Generalizing Entanglement, Phys. Rev.A 96 (2017) 062132 [arXiv:1707.07340] [INSPIRE].
D. Jia and N. Sakharwade, Tensor products of process matrices with indefinite causal structure, Phys. Rev.A 97 (2018) 032110 [arXiv:1706.05532] [INSPIRE].
D. Jia, Quantum theories from principles without assuming a definite causal structure, Phys. Rev.A 98 (2018) 032112 [arXiv:1808.00898] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
G.T. Horowitz and J.M. Maldacena, The Black hole final state, JHEP02 (2004) 008 [hep-th/0310281] [INSPIRE].
J. Cotler, C.-M. Jian, X.-L. Qi and F. Wilczek, Superdensity Operators for Spacetime Quantum Mechanics, JHEP09 (2018) 093 [arXiv:1711.03119] [INSPIRE].
G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett.91 (2003) 147902 [quant-ph/0301063].
F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066.
M. Levin and C.P. Nave, Tensor renormalization group approach to 2D classical lattice models, Phys. Rev. Lett.99 (2007) 120601 [cond-mat/0611687] [INSPIRE].
X.-L. Qi and Z. Yang, Space-time random tensor networks and holographic duality, arXiv:1801.05289 [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett.83 (1999) 648 [quant-ph/9901025] [INSPIRE].
C. Bény, A. Kempf and D.W. Kribs, Generalization of quantum error correction via the heisenberg picture, Phys. Rev. Lett.98 (2007) 100502 [quant-ph/0608071].
C. Bény, A. Kempf and D.W. Kribs, Quantum error correction of observables, Phys. Rev.A 76 (2007) 042303 [arXiv:0705.1574].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys.291 (2009) 257 [arXiv:0802.1919].
W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].
J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, Complexity and Random Matrices, JHEP11 (2017) 048 [arXiv:1706.05400] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP05 (2017) 065 [arXiv:1608.05101] [INSPIRE].
C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett.70 (1993) 1895 [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 25 [hep-th/9802150] [INSPIRE].
G. Vidal, Class of Quantum Many-Body States That Can Be Efficiently Simulated, Phys. Rev. Lett.101 (2008) 110501 [quant-ph/0610099] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
X.-L. Qi and Z. Yang, Butterfly velocity and bulk causal structure, arXiv:1705.01728 [INSPIRE].
D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, Rev. Mod. Phys.88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].
S. Lloyd, Almost certain escape from black holes, Phys. Rev. Lett.96 (2006) 061302 [quant-ph/0406205] [INSPIRE].
E. Verlinde and H. Verlinde, Black Hole Entanglement and Quantum Error Correction, JHEP10 (2013) 107 [arXiv:1211.6913] [INSPIRE].
D. Gottesman and J. Preskill, Comment on ‘The Black hole final state’, JHEP03 (2004) 026 [hep-th/0311269] [INSPIRE].
R. Bousso and D. Stanford, Measurements without Probabilities in the Final State Proposal, Phys. Rev.D 89 (2014) 044038 [arXiv:1310.7457] [INSPIRE].
D. Gottesman, Stabilizer codes and quantum error correction, quant-ph/9705052.
D. Fattal, T.S. Cubitt, Y. Yamamoto, S. Bravyi and I.L. Chuang, Entanglement in the stabilizer formalism, quant-ph/0406168.
M.M. Wolf, F. Verstraete, M.B. Hastings and J.I. Cirac, Area Laws in Quantum Systems: Mutual Information and Correlations, Phys. Rev. Lett.100 (2008) 070502 [arXiv:0704.3906] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav.42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 [INSPIRE].
X.-L. Qi, Z. Yang and Y.-Z. You, Holographic coherent states from random tensor networks, JHEP08 (2017) 060 [arXiv:1703.06533] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1811.05485
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cotler, J., Han, X., Qi, XL. et al. Quantum causal influence. J. High Energ. Phys. 2019, 42 (2019). https://doi.org/10.1007/JHEP07(2019)042
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2019)042