Abstract
We probe the multipartite entanglement structure of the vacuum state of a CFT in 1+1 dimensions, using recovery operations that attempt to reconstruct the density matrix in some region from its reduced density matrices on smaller subregions. We use an explicit recovery channel known as the twirled Petz map, and study distance measures such as the fidelity, relative entropy, and trace distance between the original state and the recovered state. One setup we study in detail involves three contiguous intervals A, B and C on a spatial slice, where we can view these quantities as measuring correlations between A and C that are not mediated by the region B that lies between them. We show that each of the distance measures is both UV finite and independent of the operator content of the CFT, and hence depends only on the central charge and the cross-ratio of the intervals. We evaluate these universal quantities numerically using lattice simulations in critical spin chain models, and derive their analytic forms in the limit where A and C are close using the OPE expansion. In the case where A and C are far apart, we find a surprising non-commutativity of the replica trick with the OPE limit. For all values of the cross-ratio, the fidelity is strictly better than a general information-theoretic lower bound in terms of the conditional mutual information. We also compare the mutual information between various subsystems in the original and recovered states, which leads to a more qualitative understanding of the differences between them. Further, we introduce generalizations of the recovery operation to more than three adjacent intervals, for which the fidelity is again universal with respect to the operator content.
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C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
J.I. Latorre, E. Rico and G. Vidal, Ground state entanglement in quantum spin chains, Quant. Inf. Comput. 4 (2004) 48 [quant-ph/0304098] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech. 1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
H. Casini, E. Teste and G. Torroba, Modular Hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov Property of the Conformal Field Theory Vacuum and the a Theorem, Phys. Rev. Lett. 118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
H. Casini and M. Huerta, Lectures on entanglement in quantum field theory, PoS TASI2021 (2023) 002 [arXiv:2201.13310] [INSPIRE].
P. Hayden, R. Jozsa, D. Petz and A. Winter, Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality, Commun. Math. Phys. 246 (2004) 359 [quant-ph/0304007] [INSPIRE].
D. Petz, Sufficiency of channels over Von Neumann algebras, Quart. J. Math. Oxford Ser. 39 (1988) 97 [INSPIRE].
D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Commun. Math. Phys. 105 (1986) 123 [INSPIRE].
H. Reeh and S. Schlieder, Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern, Nuovo Cim. 22 (1961) 1051 [INSPIRE].
E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
R. Verch and R.F. Werner, Distillability and positivity of partial transposes in general quantum field systems, Rev. Math. Phys. 17 (2005) 545 [quant-ph/0403089] [INSPIRE].
D. Sutter, Approximate quantum Markov chains, arXiv:1802.05477 [https://doi.org/10.48550/ARXIV.1802.05477].
M. Junge et al., Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy, Annales Henri Poincare 19 (2018) 2955 [arXiv:1509.07127] [INSPIRE].
M.M. Wilde, Recoverability in quantum information theory, Proc. Roy. Soc. Lond. A 471 (2015) 20150338 [arXiv:1505.04661] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for orbifolds of the type M(N)/S(N), Nucl. Phys. B Proc. Suppl. 101 (2001) 296 [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge University Press (2011).
T. Faulkner, S. Hollands, B. Swingle and Y. Wang, Approximate Recovery and Relative Entropy I: General von Neumann Subalgebras, Commun. Math. Phys. 389 (2022) 349 [arXiv:2006.08002] [INSPIRE].
E. O’Brien and P. Fendley, Lattice supersymmetry and order-disorder coexistence in the tricritical Ising model, Phys. Rev. Lett. 120 (2018) 206403 [arXiv:1712.06662] [INSPIRE].
F.C. Alcaraz, M. Baake, U. Grimm and V. Rittenberg, Operator Content of the Xxz Chain, J. Phys. A 21 (1988) L117 [INSPIRE].
Y. Zou, A. Milsted and G. Vidal, Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states, Phys. Rev. Lett. 121 (2018) 230402 [arXiv:1710.05397] [INSPIRE].
M. Hauru and G. Vidal, Uhlmann fidelities from tensor networks, Phys. Rev. A 98 (2018) 042316.
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
A. Dei and L. Eberhardt, Correlators of the symmetric product orbifold, JHEP 01 (2020) 108 [arXiv:1911.08485] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
S.G. Avery, Using the D1D5 CFT to understand black holes, Ph.D. thesis, The Ohio State University, U.S.A. (2010).
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, Combinatorial Expansions of Conformal Blocks, Theor. Math. Phys. 164 (2010) 831 [arXiv:0907.3946] [INSPIRE].
K. Roumpedakis, Comments on the SN orbifold CFT in the large N-limit, JHEP 07 (2018) 038 [arXiv:1804.03207] [INSPIRE].
B.A. Burrington, I.T. Jardine and A.W. Peet, The OPE of bare twist operators in bosonic SN orbifold CFTs at large N, JHEP 08 (2018) 202 [arXiv:1804.01562] [INSPIRE].
E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742] [INSPIRE].
A. Uhlmann, Relative Entropy and the Wigner-Yanase-Dyson-Lieb Concavity in an Interpolation Theory, Commun. Math. Phys. 54 (1977) 21 [INSPIRE].
D. Petz, Quasi-entropies for finite quantum systems, Rept. Math. Phys. 23 (1986) 57 [INSPIRE].
F. Hiai and D. Petz, The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability, in H. Masahito ed., Asymptotic Theory of Quantum Statistical Inference: Selected Papers, World Scientific Publishing Co. Pte. Ltd (2005) pp. 43–63, https://doi.org/10.1142/9789812563071_0004.
T. Ogawa and M. Hayashi, A new Proof of the Direct Part of Stein’s Lemma in Quantum Hypothesis Testing, quant-ph/0110125 [https://doi.org/10.48550/ARXIV.QUANT-PH/0110125].
M. Mosonyi and T. Ogawa, Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies, Commun. Math. Phys. 334 (2015) 1617 [arXiv:1309.3228].
J. Kudler-Flam, Rényi Mutual Information in Quantum Field Theory, Phys. Rev. Lett. 130 (2023) 021603 [arXiv:2211.01392] [INSPIRE].
J. Zhang, P. Ruggiero and P. Calabrese, Subsystem Trace Distance in Quantum Field Theory, Phys. Rev. Lett. 122 (2019) 141602 [arXiv:1901.10993] [INSPIRE].
A. Winter, Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints, Commun. Math. Phys. 347 (2016) 291 [arXiv:1507.07775] [INSPIRE].
B. Swingle and J. McGreevy, Mixed s-sourcery: Building many-body states using bubbles of Nothing, Phys. Rev. B 94 (2016) 155125 [arXiv:1607.05753] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [INSPIRE].
B.G. Swingle and Y. Wang, Recovery map for fermionic Gaussian channels, J. Math. Phys. 60 (2019) 072202 [arXiv:1811.04956] [INSPIRE].
R. Longo and F. Xu, Relative Entropy in CFT, Adv. Math. 337 (2018) 139 [arXiv:1712.07283] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
S.R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69 (1992) 2863 [INSPIRE].
F. Verstraete and J.I. Cirac, Matrix product states represent ground states faithfully, Phys. Rev. B 73 (2006) 094423 [INSPIRE].
F. Verstraete et al., Renormalization group transformations on quantum states, Phys. Rev. Lett. 94 (2005) 140601 [quant-ph/0410227] [INSPIRE].
Y. Zou et al., Universal tripartite entanglement in one-dimensional many-body systems, Phys. Rev. Lett. 126 (2021) 120501 [arXiv:2011.11864] [INSPIRE].
V. Eisler and I. Peschel, Analytical results for the entanglement Hamiltonian of a free-fermion chain, J. Phys. A 50 (2017) 284003 [arXiv:1703.08126] [INSPIRE].
K.M.R. Audenaert and N. Datta, α-z-Rényi relative entropies, J. Math. Phys. 56 (2015) 022202 [arXiv:1310.7178].
Acknowledgments
We would like to thank Gauri Batra, Lorenz Eberhardt, Thomas Faulkner, Abhijit Gadde, Jeongwan Haah, Patrick Hayden, Jonah Kudler-Flam, Raghu Mahajan, Alex May, Mukund Rangamani, Stephen Shenker, Douglas Stanford, and Jinzhao Wang for helpful discussions. SV is supported by Google and the Stanford Institute for Theoretical Physics. YZ is supported by the Stanford Q-FARM Postdoctoral Fellowship.
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Vardhan, S., Wei, A.Y. & Zou, Y. Petz recovery from subsystems in conformal field theory. J. High Energ. Phys. 2024, 16 (2024). https://doi.org/10.1007/JHEP03(2024)016
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DOI: https://doi.org/10.1007/JHEP03(2024)016