Department of Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
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Abstract
A physical system is said to satisfy a thermal area law if the mutual information between two adjacent regions in the Gibbs state is controlled by the area of their boundary. Lattice bosons have recently gained significant interest because they can be precisely tuned in experiments and bosonic codes can be employed in quantum error correction to circumvent classical no-go theorems. However, the proofs of many basic information-theoretic inequalities such as the thermal area law break down for bosons because their interactions are unbounded. Here, we rigorously derive a thermal area law for a class of bosonic Hamiltonians in any dimension which includes the paradigmatic Bose-Hubbard model. The main idea to go beyond bounded interactions is to introduce a quasi-free reference state with artificially decreased chemical potential by means of a double Peierls-Bogoliubov estimate.
Featured image: Decomposition of a system into two regions A and B. An area law is satisfied if the correlations or entanglement between A and B grow at most like the size of the green boundary region.
Popular summary
Thermal area laws control the amount of quantum entanglement and correlation. Consequently, they are expected to be potentially very useful in practice for the efficient approximation of Gibbs states by tensor network states (matrix product operators) with fixed bond dimension. The complexity of such states grows only polynomially and not exponentially in the system size, which is crucial for any numerical algorithm.
While earlier works established thermal area laws for quantum spin systems, our paper presents the first thermal area law for bosonic lattice systems. Our proof is technically more involved than in the spin case due to unbounded interactions and infinite-dimensional Hilbert spaces, which require special trace inequalities. Notably, our findings encompass the Bose-Hubbard model, a versatile description of optical lattices hosting cold neutral atoms trapped in laser-induced interference patterns. This model’s fine tunability holds promising applications in quantum simulation and computation.
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[1] Steven R White “Density-matrix algorithms for quantum renormalization groups” Phys. Rev. B 48, 10345 (1993).
https://doi.org/10.1103/PhysRevB.48.10345
[2] Frank Verstraeteand J Ignacio Cirac “Renormalization algorithms for quantum-many body systems in two and higher dimensions” arXiv:cond-mat/0407066 (2004).
https://doi.org/10.48550/arXiv.cond-mat/0407066
[3] Ulrich Schollwöck “The density-matrix renormalization group: a short introduction” Philos. Trans. Royal Soc. A 369, 2643–2661 (2011).
https://doi.org/10.1098/rsta.2010.0382
[4] Edwin M Stoudenmireand Steven R White “Studying two-dimensional systems with the density matrix renormalization group” Annu. Rev. Condens. Matter Phys. 3, 111–128 (2012).
https://doi.org/10.1146/annurev-conmatphys-020911-125018
[5] Alexander M Dalzelland Fernando GSL Brandão “Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians” Quantum 3, 187 (2019).
https://doi.org/10.22331/q-2019-09-23-187
[6] Frank Verstraeteand J Ignacio Cirac “Matrix product states represent ground states faithfully” Phys. Rev. B 73, 094423 (2006).
https://doi.org/10.1103/PhysRevB.73.094423
[7] Jens Eisert, Marcus Cramer, and Martin B Plenio, “Colloquium: Area laws for the entanglement entropy” Rev. Mod. Phys. 82, 277 (2010).
https://doi.org/10.1103/RevModPhys.82.277
[8] Yimin Geand Jens Eisert “Area laws and efficient descriptions of quantum many-body states” New J. Phys. 18, 083026 (2016).
https://doi.org/10.1088/1367-2630/18/8/083026
[9] Raphael Bousso “The holographic principle” Rev. Mod. Phys. 74, 825 (2002).
https://doi.org/10.1007/978-94-010-0211-0_3
[10] Matthew B Hastings “An area law for one-dimensional quantum systems” J. Stat. Mech: Theory Exp. 2007, P08024 (2007).
https://doi.org/10.1088/1742-5468/2007/08/P08024
[11] Itai Arad, Zeph Landau, and Umesh Vazirani, “Improved one-dimensional area law for frustration-free systems” Phys. Rev. B 85, 195145 (2012).
https://doi.org/10.1103/PhysRevB.85.195145
[12] Itai Arad, Alexei Kitaev, Zeph Landau, and Umesh Vazirani, “An area law and sub-exponential algorithm for 1D systems” (2013).
https://doi.org/10.48550/arXiv.1301.1162
[13] Itai Arad, Zeph Landau, Umesh Vazirani, and Thomas Vidick, “Rigorous RG algorithms and area laws for low energy eigenstates in 1D” Commun. Math. Phys. 356, 65–105 (2017).
https://doi.org/10.1007/s00220-017-2973-z
[14] Yichen Huang “Area law in one dimension: Degenerate ground states and Renyi entanglement entropy” arXiv:1403.0327 (2014).
https://doi.org/10.48550/arXiv.1403.0327
[15] Fernando GSL Brandãoand Michal Horodecki “An area law for entanglement from exponential decay of correlations” Nat. Phys. 9, 721–726 (2013).
https://doi.org/10.1038/nphys2747
[16] Jaeyoon Cho “Realistic area-law bound on entanglement from exponentially decaying correlations” Phys. Rev. X 8, 031009 (2018).
https://doi.org/10.1103/PhysRevX.8.031009
[17] K Audenaert, J Eisert, MB Plenio, and RF Werner, “Entanglement properties of the harmonic chain” Phys. Rev. A 66, 042327 (2002).
https://doi.org/10.1103/PhysRevA.66.042327
[18] Martin B Plenio, Jens Eisert, J Dreissig, and Marcus Cramer, “Entropy, entanglement, and area: analytical results for harmonic lattice systems” Phys. Rev. Lett. 94, 060503 (2005).
https://doi.org/10.1103/PhysRevLett.94.060503
[19] Marcus Cramerand Jens Eisert “Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices” New J. Phys. 8, 71 (2006).
https://doi.org/10.1088/1367-2630/8/5/071
[20] Marcus Cramer, Jens Eisert, Martin B Plenio, and J Dreissig, “Entanglement-area law for general bosonic harmonic lattice systems” Phys. Rev. A 73, 012309 (2006).
https://doi.org/10.1103/PhysRevA.73.012309
[21] Karel Van Acoleyen, Michaël Mariën, and Frank Verstraete, “Entanglement rates and area laws” Phys. Rev. Lett. 111, 170501 (2013).
https://doi.org/10.1103/PhysRevLett.111.170501
[22] Michaël Mariën, Koenraad MR Audenaert, Karel Van Acoleyen, and Frank Verstraete, “Entanglement rates and the stability of the area law for the entanglement entropy” Commun. Math. Phys. 346, 35–73 (2016).
https://doi.org/10.1007/s00220-016-2709-5
[23] Matthew B Hastings “Entropy and entanglement in quantum ground states” Phys. Rev. B 76, 035114 (2007).
https://doi.org/10.1103/PhysRevB.76.035114
[24] Lluís Masanes “Area law for the entropy of low-energy states” Phys. Rev. A 80, 052104 (2009).
https://doi.org/10.1103/PhysRevA.80.052104
[25] N de Beaudrap, M Ohliger, TJ Osborne, and J Eisert, “Solving frustration-free spin systems” Phys. Rev. Lett. 105, 060504 (2010).
https://doi.org/10.1103/PhysRevLett.105.060504
[26] Nilin Abrahamsen “A polynomial-time algorithm for ground states of spin trees” arXiv:1907.04862 (2019).
https://doi.org/10.48550/arXiv.1907.04862
[27] Spyridon Michalakis “Stability of the area law for the entropy of entanglement” arXiv:1206.6900 (2012).
https://doi.org/10.48550/arXiv.1206.6900
[28] Fernando GSL Brandaoand Marcus Cramer “Entanglement area law from specific heat capacity” Phys. Rev. B 92, 115134 (2015).
https://doi.org/10.1103/PhysRevB.92.115134
[29] Jaeyoon Cho “Sufficient condition for entanglement area laws in thermodynamically gapped spin systems” Phys. Rev. Lett. 113, 197204 (2014).
https://doi.org/10.1103/PhysRevLett.113.197204
[30] Anurag Anshu, Itai Arad, and David Gosset, “Entanglement subvolume law for 2D frustration-free spin systems” Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing 868–874 (2020).
https://doi.org/10.1145/3357713.3384292
[31] Anurag Anshu, Itai Arad, and David Gosset, “An area law for 2d frustration-free spin systems” Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing 12–18 (2022).
https://doi.org/10.1145/3519935.3519962
[32] Zhe-Xuan Gong, Michael Foss-Feig, Fernando G. S. L. Brandão, and Alexey V. Gorshkov, “Entanglement Area Laws for Long-Range Interacting Systems” Phys. Rev. Lett. 119, 050501 (2017).
https://doi.org/10.1103/PhysRevLett.119.050501
[33] Tomotaka Kuwaharaand Keiji Saito “Area law of noncritical ground states in 1D long-range interacting systems” Nat. Commun. 11, 1–7 (2020).
https://doi.org/10.1038/s41467-020-18055-x
[34] Tomotaka Kuwahara, Álvaro M Alhambra, and Anurag Anshu, “Improved thermal area law and quasilinear time algorithm for quantum Gibbs states” Phys. Rev. X 11, 011047 (2021).
https://doi.org/10.1103/PhysRevX.11.011047
[35] Samuel O. Scalet, Álvaro M. Alhambra, Georgios Styliaris, and J. Ignacio Cirac, “Computable Rényi mutual information: Area laws and correlations” Quantum 5, 541 (2021).
https://doi.org/10.22331/q-2021-09-14-541
[36] Álvaro M. Alhambraand J. Ignacio Cirac “Locally Accurate Tensor Networks for Thermal States and Time Evolution” PRX Quantum 2, 040331 (2021).
https://doi.org/10.1103/PRXQuantum.2.040331
[37] Michael A Nielsenand Isaac Chuang “Quantum computation and quantum information” (2002).
https://doi.org/10.1119/1.1463744
[38] Álvaro M Alhambra “Quantum many-body systems in thermal equilibrium” arXiv:2204.08349 (2022).
https://doi.org/10.48550/arXiv.2204.08349
[39] Michael M Wolf, Frank Verstraete, Matthew B Hastings, and J Ignacio Cirac, “Area laws in quantum systems: mutual information and correlations” Phys. Rev. Lett. 100, 070502 (2008).
https://doi.org/10.1103/PhysRevLett.100.070502
[40] Daniel Gottesmanand Matthew B Hastings “Entanglement versus gap for one-dimensional spin systems” New J. Phys. 12, 025002 (2010).
https://doi.org/10.1088/1367-2630/12/2/025002
[41] H Bernigau, M J Kastoryano, and J Eisert, “Mutual information area laws for thermal free fermions” J. Stat. Mech: Theory Exp. 2015, P02008 (2015).
https://doi.org/10.1088/1742-5468/2015/02/p02008
[42] Nicholas E Sherman, Trithep Devakul, Matthew B Hastings, and Rajiv RP Singh, “Nonzero-temperature entanglement negativity of quantum spin models: Area law, linked cluster expansions, and sudden death” Phys. Rev. E. 93, 022128 (2016).
https://doi.org/10.1103/PhysRevE.93.022128
[43] Michael J Kastoryanoand Jens Eisert “Rapid mixing implies exponential decay of correlations” J. Math. Phys. 54, 102201 (2013).
https://doi.org/10.1063/1.4822481
[44] Fernando GSL Brandao, Toby S Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia, “Area law for fixed points of rapidly mixing dissipative quantum systems” J. Math. Phys. 56, 102202 (2015).
https://doi.org/10.1063/1.4932612
[45] Marko Žnidarič, Tomaž Prosen, and Iztok Pižorn, “Complexity of thermal states in quantum spin chains” Phys. Rev. A 78, 022103 (2008).
https://doi.org/10.1103/PhysRevA.78.022103
[46] Mohammadamin Tajik, Ivan Kukuljan, Spyros Sotiriadis, Bernhard Rauer, Thomas Schweigler, Federica Cataldini, João Sabino, Frederik Møller, Philipp Schüttelkopf, and Si-Cong Ji, “Verification of the area law of mutual information in a quantum field simulator” Nat. Phys. 1–5 (2023).
https://doi.org/10.1038/s41567-023-02027-1
[47] Huzihiro Araki “Gibbs states of a one dimensional quantum lattice” Commun. Math. Phys. 14, 120–157 (1969).
https://doi.org/10.1007/BF01645134
[48] Leonard Gross “Decay of correlations in classical lattice models at high temperature” Commun. Math. Phys. 68, 9–27 (1979).
https://doi.org/10.1007/BF01562538
[49] Yong Moon Parkand Hyun Jae Yoo “Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems” J. Stat. Phys. 80, 223–271 (1995).
https://doi.org/10.1007/BF02178359
[50] D. Ueltschi “Cluster Expansions and Correlation Functions” Moscow Math. J. 4, 511–522 (2004).
https://doi.org/10.17323/1609-4514-2004-4-2-511-522
[51] Martin Kliesch, Christian Gogolin, MJ Kastoryano, A Riera, and J Eisert, “Locality of temperature” Phys. Rev. X 4, 031019 (2014).
https://doi.org/10.1103/PhysRevX.4.031019
[52] Jürg Fröhlichand Daniel Ueltschi “Some properties of correlations of quantum lattice systems in thermal equilibrium” J. Math. Phys. 56, 053302 (2015).
https://doi.org/10.1063/1.4921305
[53] Marco Lenciand Luc Rey-Bellet “Large deviations in quantum lattice systems: one-phase region” J. Stat. Phys. 119, 715–746 (2005).
https://doi.org/10.1007/s10955-005-3015-3
[54] K Netočnỳand F Redig “Large deviations for quantum spin systems” J. Stat. Phys. 117, 521–547 (2004).
https://doi.org/10.1007/s10955-004-3452-4
[55] Tomotaka Kuwaharaand Keiji Saito “Gaussian concentration bound and ensemble equivalence in generic quantum many-body systems including long-range interactions” Ann. Phys. 421, 168278 (2020).
https://doi.org/10.1016/j.aop.2020.168278
[56] Kohtaro Katoand Fernando GSL Brandao “Quantum approximate Markov chains are thermal” Commun. Math. Phys. 370, 117–149 (2019).
https://doi.org/10.1007/s00220-019-03485-6
[57] Tomotaka Kuwahara, Kohtaro Kato, and Fernando GSL Brandao, “Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature” Phys. Rev. Lett. 124, 220601 (2020).
https://doi.org/10.1103/PhysRevLett.124.220601
[58] Francisco Barahona “On the computational complexity of Ising spin glass models” J. Phys. A: Math. Gen. 15, 3241 (1982).
https://doi.org/10.1088/0305-4470/15/10/028
[59] Leslie Ann Goldbergand Mark Jerrum “A complexity classification of spin systems with an external field” Proc. Natl. Acad. Sci. U.S.A. 112, 13161–13166 (2015).
https://doi.org/10.1073/pnas.1505664112
[60] Mohammad H Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, and Roger Melko, “Quantum Boltzmann machine” Phys. Rev. X 8, 021050 (2018).
https://doi.org/10.1103/PhysRevX.8.021050
[61] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar, “Sample-efficient learning of interacting quantum systems” Nat. Phys. 17, 931–935 (2021).
https://doi.org/10.1038/s41567-021-01232-0
[62] Fernando GSL Brandaoand Krysta M Svore “Quantum speed-ups for solving semidefinite programs” 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) 415–426 (2017).
https://doi.org/10.1109/FOCS.2017.45
[63] Joran Van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, “Quantum SDP-solvers: Better upper and lower bounds” Quantum 4, 230 (2020).
https://doi.org/10.1109/FOCS.2017.44
[64] Mario Motta, Chong Sun, Adrian TK Tan, Matthew J O’Rourke, Erika Ye, Austin J Minnich, Fernando GSL Brandão, and Garnet Kin Chan, “Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution” Nat. Phys. 16, 205–210 (2020).
https://doi.org/10.1038/s41567-020-0798-8
[65] Henry Lammand Scott Lawrence “Simulation of nonequilibrium dynamics on a quantum computer” Phys. Rev. Lett. 121, 170501 (2018).
https://doi.org/10.1103/PhysRevLett.121.170501
[66] Matthew JS Beach, Roger G Melko, Tarun Grover, and Timothy H Hsieh, “Making trotters sprint: A variational imaginary time ansatz for quantum many-body systems” Phys. Rev. B 100, 094434 (2019).
https://doi.org/10.1103/PhysRevB.100.094434
[67] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin, “Theory of variational quantum simulation” Quantum 3, 191 (2019).
https://doi.org/10.22331/q-2019-10-07-191
[68] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan, “Variational ansatz-based quantum simulation of imaginary time evolution” Npj Quantum Inf. 5, 1–6 (2019).
https://doi.org/10.1038/s41534-019-0187-2
[69] Kübra Yeter-Aydeniz, Raphael C Pooser, and George Siopsis, “Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and Lanczos algorithms” Npj Quantum Inf. 6, 1–8 (2020).
https://doi.org/10.1038/s41534-020-00290-1
[70] Peter J Love “Cooling with imaginary time” Nat. Phys. 16, 130–131 (2020).
https://doi.org/10.1038/s41567-019-0709-z
[71] Jiří Guth Jarkovský, András Molnár, Norbert Schuch, and J. Ignacio Cirac, “Efficient Description of Many-Body Systems with Matrix Product Density Operators” PRX Quantum 1, 010304 (2020).
https://doi.org/10.1103/PRXQuantum.1.010304
[72] Mario Berta, Fernando G. S. L. Brandão, Jutho Haegeman, Volkher B. Scholz, and Frank Verstraete, “Thermal states as convex combinations of matrix product states” Phys. Rev. B 98, 235154 (2018).
https://doi.org/10.1103/PhysRevB.98.235154
[73] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger, “Many-body physics with ultracold gases” Rev. Mod. Phys. 80, 885 (2008).
https://doi.org/10.1103/RevModPhys.80.885
[74] Andrew M Childs, David Gosset, and Zak Webb, “The Bose-Hubbard model is QMA-complete” International Colloquium on Automata, Languages, and Programming 308–319 (2014).
https://doi.org/10.1007/978-3-662-43948-7_26
[75] Victor V Albert “Bosonic coding: introduction and use cases” arXiv:2211.05714 (2022).
https://doi.org/10.48550/arXiv.2211.05714
[76] Steven M. Girvin “Introduction to quantum error correction and fault tolerance” SciPost Phys. Lect. Notes 70 (2023).
https://doi.org/10.21468/SciPostPhysLectNotes.70
[77] Marcus Cramer, Jens Eisert, and MB Plenio, “Statistics dependence of the entanglement entropy” Phys. Rev. Lett. 98, 220603 (2007).
https://doi.org/10.1103/PhysRevLett.98.220603
[78] Vincenzo Alba, Masudul Haque, and Andreas M Läuchli, “Entanglement spectrum of the two-dimensional Bose-Hubbard model” Phys. Rev. Lett. 110, 260403 (2013).
https://doi.org/10.1103/PhysRevLett.110.260403
[79] Max A Metlitskiand Tarun Grover “Entanglement entropy of systems with spontaneously broken continuous symmetry” arXiv:1112.5166 (2011).
https://doi.org/10.48550/arXiv.1112.5166
[80] Ann B Kallin, Matthew B Hastings, Roger G Melko, and Rajiv RP Singh, “Anomalies in the entanglement properties of the square-lattice Heisenberg model” Phys. Rev. B 84, 165134 (2011).
https://doi.org/10.1103/PhysRevB.84.165134
[81] H Francis Song, Nicolas Laflorencie, Stephan Rachel, and Karyn Le Hur, “Entanglement entropy of the two-dimensional Heisenberg antiferromagnet” Phys. Rev. B 83, 224410 (2011).
https://doi.org/10.1103/PhysRevB.83.224410
[82] Nilin Abrahamsen, Yuan Su, Yu Tong, and Nathan Wiebe, “Entanglement area law for 1D gauge theories and bosonic systems” arXiv:2203.16012 (2022).
https://doi.org/10.48550/arXiv.2203.16012
[83] Yu Tong, Victor V Albert, Jarrod R McClean, John Preskill, and Yuan Su, “Provably accurate simulation of gauge theories and bosonic systems” Quantum 6, 816 (2022).
https://doi.org/10.22331/q-2022-09-22-816
[84] Jérémy Faupin, Marius Lemm, and Israel Michael Sigal, “Maximal speed for macroscopic particle transport in the Bose-Hubbard model” Phys. Rev. Lett. 128, 150602 (2022).
https://doi.org/10.1103/PhysRevLett.128.150602
[85] Eric Carlen “Trace inequalities and quantum entropy: an introductory course” Amer. Math. Soc., Providence, RI (2010).
https://doi.org/10.1090/conm/529/10428
[86] Norbert Schuch, Sarah K Harrison, Tobias J Osborne, and Jens Eisert, “Information propagation for interacting-particle systems” Phys. Rev. A 84, 032309 (2011).
https://doi.org/10.1103/PhysRevA.84.032309
[87] Zhiyuan Wangand Kaden RA Hazzard “Tightening the Lieb-Robinson bound in locally interacting systems” PRX Quantum 1, 010303 (2020).
https://doi.org/10.1103/PRXQuantum.1.010303
[88] Tomotaka Kuwaharaand Keiji Saito “Lieb-Robinson bound and almost-linear light cone in interacting boson systems” Phys. Rev. Lett. 127, 070403 (2021).
https://doi.org/10.1103/PhysRevLett.127.070403
[89] Jérémy Faupin, Marius Lemm, and Israel Michael Sigal, “On Lieb-Robinson Bounds for the Bose-Hubbard Model” Commun. Math. Phys. 394, 1011–1037 (2022).
https://doi.org/10.1007/s00220-022-04416-8
[90] Chao Yinand Andrew Lucas “Finite speed of quantum information in models of interacting bosons at finite density” Phys. Rev. X 12, 021039 (2022).
https://doi.org/10.1103/PhysRevX.12.021039
[91] Tomotaka Kuwahara, Tan Van Vu, and Keiji Saito, “Optimal light cone and digital quantum simulation of interacting bosons” arXiv:2206.14736 (2022).
https://doi.org/10.48550/arXiv.2206.14736
[92] Mary Beth Ruskai “Inequalities for traces on von Neumann algebras” Commun. Math. Phys. 26, 280–289 (1972).
https://doi.org/10.1007/BF01645523
[93] David Ruelle “Statistical Mechanics: Rigorous Results” New York: W.A. Benjamin (1969).
https://doi.org/10.1142/4090
[94] O. Bratteliand D.W. Robinson “Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in Quantum Statistical Mechanics” Springer Berlin Heidelberg (2003).
https://doi.org/10.1007/978-3-662-09089-3
[95] Masanori Ohyaand Dénes Petz “Quantum entropy and its use” Springer Science & Business Media (2004).
https://doi.org/10.1007/978-3-642-57997-4
[96] Göran Lindblad “Completely positive maps and entropy inequalities” Commun. Math. Phys. 40, 147–151 (1975).
https://doi.org/10.1007/BF01609396
Cited by
[1] Vanja Marić and Maurizio Fagotti, “Universality in the tripartite information after global quenches: (generalised) quantum XY models”, Journal of High Energy Physics 2023 6, 140 (2023).
[2] Álvaro M. Alhambra, “Quantum many-body systems in thermal equilibrium”, arXiv:2204.08349, (2022).
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