Abstract
Negativity is a measure of entanglement that can be used both in pure and mixed states. The negativity spectrum is the spectrum of eigenvalues of the partially transposed density matrix, and characterizes the degree and “phase” of entanglement. For pure states, it is simply determined by the entanglement spectrum. We use a diagrammatic method complemented by a modification of the Ford-Fulkerson algorithm to find the negativity spectrum in general random tensor networks with large bond dimensions. In holography, these describe the entanglement of fixed-area states. It was found that many fixed-area states have a negativity spectrum given by a semi-circle. More generally, we find new negativity spectra that appear in random tensor networks, as well as in phase transitions in holographic states, wormholes, and holographic states with bulk matter. The smallest random tensor network is the same as a micro-canonical version of Jackiw-Teitelboim (JT) gravity decorated with end-of-the-world branes. We consider the semi-classical negativity of Hawking radiation and find that contributions from islands should be included. We verify this in the JT gravity model, showing the Euclidean wormhole origin of these contributions.
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References
E.H. Fradkin, Field Theories of Condensed Matter Physics, Cambridge University Press, Cambridge U.K. (2013) [Front. Phys. 82 (2013) 1] [INSPIRE].
B. Zeng, X. Chen, D.-L. Zhou and X.-G. Wen, Quantum information meets quantum matter — from quantum entanglement to topological phase in many-body systems, arXiv:1508.02595.
P. Calabrese and J. Cardy, Quantum quenches in 1 + 1 dimensional conformal field theories, J. Stat. Mech. 1606 (2016) 064003 [arXiv:1603.02889] [INSPIRE].
R. Nandkishore and D.A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6 (2015) 15.
M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, in Lecture Notes in Physics 931, Springer (2017).
K. Życzkowski, P. Horodecki, A. Sanpera and M. Lewenstein, On the volume of the set of mixed entangled states, Phys. Rev. A 58 (1998) 883 [quant-ph/9804024] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996) 1413 [quant-ph/9604005] [INSPIRE].
J. Eisert and M.B. Plenio, A Comparison of entanglement measures, J. Mod. Opt. 46 (1999) 145 [quant-ph/9807034] [INSPIRE].
M.B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
R. Simon, Peres-Horodecki Separability Criterion for Continuous Variable Systems, Phys. Rev. Lett. 84 (2000) 2726 [quant-ph/9909044] [INSPIRE].
M. Horodecki, P. Horodecki and R. Horodecki, On the necessary and sufficient conditions for separability of mixed quantum states, Phys. Lett. A 223 (1996) 1 [quant-ph/9605038] [INSPIRE].
L. Gurvits, Classical deterministic complexity of Edmonds’ problem and quantum entanglement, in STOC ’03, proceedings of the 35th Annual ACM Symposium on Theory of Computing, San Diego, CA, U.S.A., 9–11 June 2003, quant-ph/0303055.
S. Gharibian, Strong NP-Hardness of the Quantum Separability Problem, Quantum Info. Comput. 10 (2010) 343 [arXiv:0810.4507].
S. Ryu and Y. Hatsugai, Entanglement entropy and the berry phase in the solid state, Phys. Rev. B 73 (2006) 245115 [cond-mat/0601237].
H. Li and F.D.M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett. 101 (2008) 010504 [arXiv:0805.0332] [INSPIRE].
F. Pollmann, A.M. Turner, E. Berg and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81 (2010) 064439 [arXiv:0910.1811].
P. Ruggiero, V. Alba and P. Calabrese, Negativity spectrum of one-dimensional conformal field theories, Phys. Rev. B 94 (2016) 195121 [arXiv:1607.02992] [INSPIRE].
G.B. Mbeng, V. Alba and P. Calabrese, Negativity spectrum in 1D gapped phases of matter, J. Phys. A 50 (2017) 194001 [arXiv:1612.05172] [INSPIRE].
H. Shapourian, P. Ruggiero, S. Ryu and P. Calabrese, Twisted and untwisted negativity spectrum of free fermions, SciPost Phys. 7 (2019) 037 [arXiv:1906.04211] [INSPIRE].
K. Inamura, R. Kobayashi and S. Ryu, Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories, JHEP 01 (2020) 121 [arXiv:1911.00653] [INSPIRE].
H. Shapourian, S. Liu, J. Kudler-Flam and A. Vishwanath, Entanglement Negativity Spectrum of Random Mixed States: A Diagrammatic Approach, PRX Quantum 2 (2021) 030347 [arXiv:2011.01277] [INSPIRE].
B. Collins, I. Nechita and K. Życzkowski, Random graph states, maximal flow and Fuss-Catalan distributions, J. Phys. A 43 (2010) 275303 [arXiv:1003.3075] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
J. Kudler-Flam, V. Narovlansky and S. Ryu, Distinguishing Random and Black Hole Microstates, PRX Quantum 2 (2021) 040340 [arXiv:2108.00011] [INSPIRE].
G. Aubrun, Partial transposition of random states and non-centered semicircular distributions, arXiv:1011.0275.
T. Banica and I. Nechita, Asymptotic eigenvalue distributions of block-transposed Wishart matrices, arXiv:1105.2556.
M. Fukuda and P. Śniady, Partial transpose of random quantum states: Exact formulas and meanders, J. Math. Phys. 54 (2013) 042202 [arXiv:1211.1525].
X. Dong, X.-L. Qi and M. Walter, Holographic entanglement negativity and replica symmetry breaking, JHEP 06 (2021) 024 [arXiv:2101.11029] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
L.R. Ford and D.R. Fulkerson, Maximal flow through a network, Can. J. Math. 8 (1956) 399.
P. Elias, A. Feinstein and C. Shannon, A note on the maximum flow through a network, IRE Trans. Inf. Theory 2 (1956) 117.
T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Introduction to algorithms, MIT Press (2009).
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
X. Dong, S. McBride and W. Weng, private communication.
J. Kudler-Flam, H. Liu, H. Shapourian and S. Vardhan, to appear.
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [INSPIRE].
J. Kudler-Flam, Relative Entropy of Random States and Black Holes, Phys. Rev. Lett. 126 (2021) 171603 [arXiv:2102.05053] [INSPIRE].
C. Akers and P. Rath, Holographic Renyi Entropy from Quantum Error Correction, JHEP 05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, JHEP 10 (2019) 240 [arXiv:1811.05382] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972) 333.
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 105012 [arXiv:1710.09516] [INSPIRE].
S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit Threads and Holographic Monogamy, Commun. Math. Phys. 376 (2019) 609 [arXiv:1808.05234] [INSPIRE].
J.K. Basak, D. Basu, V. Malvimat, H. Parihar and G. Sengupta, Islands for entanglement negativity, SciPost Phys. 12 (2022) 003 [arXiv:2012.03983] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99 (2019) 106014 [arXiv:1808.00446] [INSPIRE].
Y. Kusuki, J. Kudler-Flam and S. Ryu, Derivation of holographic negativity in AdS3/CFT2, Phys. Rev. Lett. 123 (2019) 131603 [arXiv:1907.07824] [INSPIRE].
M. Kulaxizi, A. Parnachev and G. Policastro, Conformal Blocks and Negativity at Large Central Charge, JHEP 09 (2014) 010 [arXiv:1407.0324] [INSPIRE].
W.-z. Guo, Entanglement spectrum of geometric states, JHEP 02 (2021) 085 [arXiv:2008.12430] [INSPIRE].
W.-z. Guo, The area operator and fixed area states in conformal field theories, arXiv:2108.03346 [INSPIRE].
R. Nandkishore and D.A. Huse, Many body localization and thermalization in quantum statistical mechanics, Ann. Rev. Condens. Matter Phys. 6 (2015) 15 [arXiv:1404.0686] [INSPIRE].
D.A. Abanin, E. Altman, I. Bloch and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91 (2019) 021001 [arXiv:1804.11065].
B. Skinner, J. Ruhman and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Phys. Rev. X 9 (2019) 031009 [arXiv:1808.05953] [INSPIRE].
Y. Li, X. Chen and M.P.A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98 (2018) 205136 [arXiv:1808.06134] [INSPIRE].
A. Chan, R.M. Nandkishore, M. Pretko and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99 (2019) 224307 [arXiv:1808.05949] [INSPIRE].
R. Vasseur, A.C. Potter, Y.-Z. You and A.W.W. Ludwig, Entanglement Transitions from Holographic Random Tensor Networks, Phys. Rev. B 100 (2019) 134203 [arXiv:1807.07082] [INSPIRE].
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Kudler-Flam, J., Narovlansky, V. & Ryu, S. Negativity spectra in random tensor networks and holography. J. High Energ. Phys. 2022, 76 (2022). https://doi.org/10.1007/JHEP02(2022)076
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DOI: https://doi.org/10.1007/JHEP02(2022)076