{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T02:16:47Z","timestamp":1768443407888,"version":"3.49.0"},"reference-count":113,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,11,7]],"date-time":"2019-11-07T00:00:00Z","timestamp":1573084800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Thermodynamics is a theory of principles that permits a basic description of the macroscopic properties of a rich variety of complex systems from traditional ones, such as crystalline solids, gases, liquids, and thermal machines, to more intricate systems such as living organisms and black holes to name a few. Physical quantities of interest, or equilibrium state variables, are linked together in equations of state to give information on the studied system, including phase transitions, as energy in the forms of work and heat, and\/or matter are exchanged with its environment, thus generating entropy. A more accurate description requires different frameworks, namely, statistical mechanics and quantum physics to explore in depth the microscopic properties of physical systems and relate them to their macroscopic properties. These frameworks also allow to go beyond equilibrium situations. Given the notably increasing complexity of mathematical models to study realistic systems, and their coupling to their environment that constrains their dynamics, both analytical approaches and numerical methods that build on these models show limitations in scope or applicability. On the other hand, machine learning, i.e., data-driven, methods prove to be increasingly efficient for the study of complex quantum systems. Deep neural networks, in particular, have been successfully applied to many-body quantum dynamics simulations and to quantum matter phase characterization. In the present work, we show how to use a variational autoencoder (VAE)\u2014a state-of-the-art tool in the field of deep learning for the simulation of probability distributions of complex systems. More precisely, we transform a quantum mechanical problem of many-body state reconstruction into a statistical problem, suitable for VAE, by using informationally complete positive operator-valued measure. We show, with the paradigmatic quantum Ising model in a transverse magnetic field, that the ground-state physics, such as, e.g., magnetization and other mean values of observables, of a whole class of quantum many-body systems can be reconstructed by using VAE learning of tomographic data for different parameters of the Hamiltonian, and even if the system undergoes a quantum phase transition. We also discuss challenges related to our approach as entropy calculations pose particular difficulties.<\/jats:p>","DOI":"10.3390\/e21111091","type":"journal-article","created":{"date-parts":[[2019,11,7]],"date-time":"2019-11-07T11:17:25Z","timestamp":1573125445000},"page":"1091","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Variational Autoencoder Reconstruction of Complex Many-Body Physics"],"prefix":"10.3390","volume":"21","author":[{"given":"Ilia A.","family":"Luchnikov","sequence":"first","affiliation":[{"name":"Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, 121205 Moscow Region, Russia"},{"name":"Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, 141700 Moscow Region, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8012-406X","authenticated-orcid":false,"given":"Alexander","family":"Ryzhov","sequence":"additional","affiliation":[{"name":"Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, 121205 Moscow Region, Russia"}]},{"given":"Pieter-Jan","family":"Stas","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Stanford University 348 Via Pueblo Mall, Stanford, CA 94305, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6414-2137","authenticated-orcid":false,"given":"Sergey N.","family":"Filippov","sequence":"additional","affiliation":[{"name":"Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, 141700 Moscow Region, Russia"},{"name":"Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Pr. 34, 117218 Moscow, Russia"},{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina St. 8, 119991 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1914-0244","authenticated-orcid":false,"given":"Henni","family":"Ouerdane","sequence":"additional","affiliation":[{"name":"Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, 121205 Moscow Region, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,7]]},"reference":[{"key":"ref_1","unstructured":"Muller, I. (2007). A History of Thermodynamics, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRev.37.405","article-title":"Reciprocal Relations in Irreversible Processes. I","volume":"37","author":"Onsager","year":"1931","journal-title":"Phys. Rev."},{"key":"ref_3","unstructured":"De Groot, S.R. (1958). Thermodynamics of Irreversible Processes, Interscience."},{"key":"ref_4","unstructured":"Le Bellac, M., Mortessagne, F., and Batrouni, G.G. (2006). Equilibrium and Non-Equilibrium Statistical Thermodynamics, Cambridge University Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"012113","DOI":"10.1103\/PhysRevE.90.012113","article-title":"Revisiting Feynman\u2019s ratchet with thermoelectric transport theory","volume":"90","author":"Apertet","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"032136","DOI":"10.1103\/PhysRevE.94.032136","article-title":"Closed-loop approach to thermodynamics","volume":"94","author":"Goupil","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1002\/anie.201001411","article-title":"Current trends in finite-time thermodynamics","volume":"50","author":"Andresen","year":"2011","journal-title":"Angew. Chem.-Int. Edit."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1140\/epjst\/e2015-02431-x","article-title":"Continuity and boundary conditions in thermodynamics: From Carnot\u2019s efficiency to efficiencies at maximum power","volume":"224","author":"Ouerdane","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"022119","DOI":"10.1103\/PhysRevE.96.022119","article-title":"True nature of the Curzon-Ahlborn efficiency","volume":"96","author":"Apertet","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_10","first-page":"373","article-title":"Uber die beziehung dem zweiten Haubtsatze der mechanischen Warmetheorie und der Wahrscheinlichkeitsrechnung respektive den Satzen uber das Warmegleichgewicht","volume":"76","author":"Boltzmann","year":"1877","journal-title":"Wiener Berichte"},{"key":"ref_11","unstructured":"Gibbs, J.W. (1902). Elementary Principles in Statistical Mechanics, Charles Scribner\u2019s Sons."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1937","DOI":"10.1088\/0034-4885\/42\/12\/002","article-title":"Foundations of statistical mechanics","volume":"42","author":"Penrose","year":"1979","journal-title":"Rep. Prog. Phys."},{"key":"ref_13","unstructured":"Goldstein, S., Lebowitz, J.L., and Zangh\u00ec, N. (2019). Gibbs and Boltzmann entropy in classical and quantum mechanics. arXiv, Available online: https:\/\/arxiv.org\/abs\/1903.11870."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Labs Tech. J."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"von Neumann, J. (2018). Mathematical Foundations of Quantum Mechanics. New Edition, Princeton University Press.","DOI":"10.23943\/princeton\/9780691178561.001.0001"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Datta, S. (1995). Electronic Transport in Mesoscopic Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511805776"},{"key":"ref_17","unstructured":"Heikill\u00e4, T.T. (2013). The Physics of Nanoelectronics, Oxford University Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.physrep.2003.09.006","article-title":"Nuclear spinodal fragmentation","volume":"389","author":"Chomaz","year":"2004","journal-title":"Phys. Rep."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5345","DOI":"10.1063\/1.1455618","article-title":"Robust wave function optimization procedures in quantum Monte Carlo methods","volume":"116","author":"Bressanini","year":"2002","journal-title":"J. Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"220401","DOI":"10.1103\/PhysRevB.72.220401","article-title":"Finite-temperature density matrix renormalization using an enlarged Hilbert space","volume":"72","author":"Feiguin","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2046","DOI":"10.1103\/PhysRevA.43.2046","article-title":"Quantum statistical mechanics in a closed system","volume":"43","author":"Deutsch","year":"1991","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1103\/PhysRevE.50.888","article-title":"Chaos and quantum thermalization","volume":"50","author":"Srednicki","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1038\/nature06838","article-title":"Thermalization and its mechanism for generic isolated quantum systems","volume":"452","author":"Rigol","year":"2008","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"012140","DOI":"10.1103\/PhysRevE.97.012140","article-title":"Subsystem eigenstate thermalization hypothesis","volume":"97","author":"Dymarsky","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"224302","DOI":"10.1103\/PhysRevB.99.224302","article-title":"Mechanism of macroscopic equilibration of isolated quantum systems","volume":"99","author":"Dymarsky","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1038\/srep00243","article-title":"Localization and glassy dynamics of many-body quantum systems","volume":"2","author":"Carleo","year":"2012","journal-title":"Sci. Rep."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.chemphys.2018.08.041","article-title":"Dynamics of the spin-boson model: A comparison of the multiple Davydov D1, D1.5, D2 Ans\u00e4tze","volume":"515","author":"Chen","year":"2018","journal-title":"Chem. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1158","DOI":"10.1038\/nphys4244","article-title":"Efficient tomography of a quantum many-body system","volume":"13","author":"Lanyon","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_29","first-page":"031041","article-title":"Differentiable programming tensor networks","volume":"9","author":"Liao","year":"2019","journal-title":"Phys. Rev. X"},{"key":"ref_30","unstructured":"Fetter, A.L., and Walecka, J.D. (2003). Quantum Theory of Many-Particle Systems, Dover."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Avella, A., and Mancini, F. (2011). The pseudoparticle approach to strongly correlated electron systems. Strongly Correlated Systems, Springer.","DOI":"10.1007\/978-3-642-21831-6"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1103\/RevModPhys.68.13","article-title":"Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions","volume":"68","author":"Georges","year":"1996","journal-title":"Rev. Mod. Phys."},{"key":"ref_33","unstructured":"Negele, J.W., and Orland, H. (1998). Quantum Many-Particle Systems, Perseus Books."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1103\/RevModPhys.73.33","article-title":"Quantum Monte Carlo simulations of solids","volume":"73","author":"Foulkes","year":"2001","journal-title":"Rev. Mod. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.aop.2014.06.013","article-title":"A practical introduction to tensor networks: Matrix product states and projected entangled pair states","volume":"349","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1038\/s42254-019-0086-7","article-title":"Tensor networks for complex quantum systems","volume":"1","year":"2019","journal-title":"Nat. Rev. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.aop.2010.09.012","article-title":"The density-matrix renormalization group in the age of matrix product states","volume":"326","year":"2011","journal-title":"Ann. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"147902","DOI":"10.1103\/PhysRevLett.91.147902","article-title":"Efficient classical simulation of slightly entangled quantum computations","volume":"91","author":"Vidal","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Evenbly, G., and Vidal, G. (2013). Quantum criticality with the multiscale entanglement renormalization ansatz. Strongly Correlated Systems, Springer.","DOI":"10.1007\/978-3-642-35106-8_4"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"012127","DOI":"10.1103\/PhysRevA.97.012127","article-title":"Non-Markovian quantum processes: Complete framework and efficient characterization","volume":"97","author":"Pollock","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"160401","DOI":"10.1103\/PhysRevLett.122.160401","article-title":"Simulation complexity of open quantum dynamics: Connection with tensor networks","volume":"122","author":"Luchnikov","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","unstructured":"Taranto, P., Pollock, F.A., and Modi, K. (2019). Memory strength and recoverability of non-Markovian quantum stochastic processes. arXiv, Available online: https:\/\/arxiv.org\/abs\/1907.12583."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"012108","DOI":"10.1103\/PhysRevA.98.012108","article-title":"Reconstructing non-Markovian quantum dynamics with limited control","volume":"98","author":"Milz","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Luchnikov, I.A., Vintskevich, S.V., Grigoriev, D.A., and Filippov, S.N. (2019). Machine learning of Markovian embedding for non-Markovian quantum dynamics. arXiv, Available online: https:\/\/arxiv.org\/abs\/1902.07019.","DOI":"10.1103\/PhysRevLett.124.140502"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1080\/14789940801912366","article-title":"Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems","volume":"57","author":"Verstraete","year":"2008","journal-title":"Adv. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"120601","DOI":"10.1103\/PhysRevLett.99.120601","article-title":"Tensor renormalization group approach to two-dimensional classical lattice models","volume":"99","author":"Levin","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"180405","DOI":"10.1103\/PhysRevLett.115.180405","article-title":"Tensor network renormalization","volume":"115","author":"Evenbly","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Gemmer, J., and Michel, M. (2009). Quantum Thermodynamics, Springer.","DOI":"10.1007\/978-3-540-70510-9"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"204105","DOI":"10.1063\/1.5096173","article-title":"Quantum thermodynamics and open-systems modeling","volume":"150","author":"Kosloff","year":"2019","journal-title":"J. Phys. Chem."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"041118","DOI":"10.1103\/PhysRevE.77.041118","article-title":"Work extremum principle: Structure and function of quantum heat engines","volume":"77","author":"Allahverdyan","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"031135","DOI":"10.1103\/PhysRevE.83.031135","article-title":"Coupled quantum Otto cycle","volume":"83","author":"Thomas","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1103\/RevModPhys.73.357","article-title":"Quantum-state engineering with Josephson-junction devices","volume":"73","author":"Makhlin","year":"2001","journal-title":"Rev. Mod. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"144506","DOI":"10.1103\/PhysRevB.100.144506","article-title":"Entangling continuous variables with a qubit array","volume":"100","author":"Navez","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_54","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1093\/mind\/LIX.236.433","article-title":"Computing machinery and intelligence","volume":"59","author":"Turing","year":"1950","journal-title":"Mind"},{"key":"ref_56","unstructured":"Crevier, D. (1993). AI: The Tumultuous Search for Artificial Intelligence, BasicBooks."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1038\/nature23474","article-title":"Quantum machine learning","volume":"549","author":"Biamonte","year":"2017","journal-title":"Nature"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1126\/science.aag2302","article-title":"Solving the quantum many-body problem with artificial neural networks","volume":"355","author":"Carleo","year":"2017","journal-title":"Science"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1038\/s41567-018-0048-5","article-title":"Neural-network quantum state tomography","volume":"14","author":"Torlai","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_60","unstructured":"Tiunov, E.S., Tiunova, V.V., Ulanov, A.E., Lvovsky, A.I., and Fedorov, A.K. (2019). Experimental quantum homodyne tomography via machine learning. arXiv, Available online: https:\/\/arxiv.org\/abs\/1907.06589."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"124125","DOI":"10.1103\/PhysRevB.100.125124","article-title":"Study of the two-dimensional frustrated J1-J2 model with neural network quantum states","volume":"100","author":"Choo","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Sharir, O., Levine, Y., Wies, N., Carleo, G., and Shashua, A. (2019). Deep autoregressive models for the efficient variational simulation of many-body quantum systems. arXiv, Available online: https:\/\/arxiv.org\/abs\/1902.04057.","DOI":"10.1103\/PhysRevLett.124.020503"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"080602","DOI":"10.1103\/PhysRevLett.122.080602","article-title":"Solving statistical mechanics using variational autoregressive networks","volume":"122","author":"Wu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Kharkov, Y.A., Sotskov, V.E., Karazeev, A.A., Kiktenko, E.O., and Fedorov, A.K. (2019). Revealing quantum chaos with machine learning. arXiv, Available online: https:\/\/arxiv.org\/abs\/1902.09216.","DOI":"10.1103\/PhysRevB.101.064406"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1038\/s41534-018-0077-z","article-title":"Learning hard quantum distributions with variational autoencoders","volume":"4","author":"Rocchetto","year":"2018","journal-title":"npj Quantum Inf."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1038\/s42256-019-0028-1","article-title":"Reconstructing quantum states with generative models","volume":"1","author":"Carrasquilla","year":"2019","journal-title":"Nat. Mach. Intell."},{"key":"ref_67","unstructured":"Generative Models for Physicists (2019, November 07). Lecture note. Available online: http:\/\/wangleiphy.github.io\/lectures\/PILtutorial.pdf."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Hewson, A.C. (1993). The Kondo Problem to Heavy Fermions, Cambridge University Press.","DOI":"10.1017\/CBO9780511470752"},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Kronm\u00faller, H., and Parkin, S. (2007). Heavy fermions: Electrons at the edge of magnetism. Handbook of Magnetism and Advanced Magnetic Materials, John Wiley & Sons.","DOI":"10.1002\/9780470022184"},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Sachdev, S. (2000). Quantum Phase Transitions, Cambridge University Press.","DOI":"10.1017\/CBO9780511622540"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1038\/nature03279","article-title":"Quantum criticality","volume":"433","author":"Coleman","year":"2000","journal-title":"Nature"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1103\/PhysRev.124.41","article-title":"Localized Magnetic States in Metals","volume":"124","author":"Anderson","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.nuclphysb.2007.05.025","article-title":"Slave bosons in radial gauge: a bridge between path integral and Hamiltonian language","volume":"785","author":"Ouerdane","year":"2007","journal-title":"Nucl. Phys. B"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"31001","DOI":"10.1209\/0295-5075\/82\/31001","article-title":"Barnes slave-boson approach to the two-site single-impurity Anderson model with non-local interaction","volume":"82","author":"Ouerdane","year":"2008","journal-title":"EPL"},{"key":"ref_75","unstructured":"Diu, B., Guthmann, C., Lederer, D., and Roulet, B. (1996). Physique Statistique, \u00c9ditions Hermann."},{"key":"ref_76","unstructured":"Pavarini, E., Koch, E., and Coleman, P. (2015). Frustrated spin systems. Many-Body Physics: From Kondo to Hubbard, Verlag des Forschungszentrum J\u00fclich."},{"key":"ref_77","doi-asserted-by":"crossref","unstructured":"Refael, G., and Moore, J.E. (2004). Entanglement Entropy of Random Quantum Critical Points in One Dimension. Phys. Rev. Lett., 93.","DOI":"10.1103\/PhysRevLett.93.260602"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1103\/RevModPhys.77.259","article-title":"The density-matrix renormalization group","volume":"77","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/BF02980577","article-title":"Beitrag zur Theorie des Ferromagnetismus","volume":"31","author":"Ising","year":"1925","journal-title":"Z. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1103\/PhysRev.60.252","article-title":"Statistics of the two-dimensional ferromagnet. Part I","volume":"60","author":"Kramers","year":"1941","journal-title":"Phys. Rev."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"214406","DOI":"10.1103\/PhysRevB.68.214406","article-title":"Antiferromagnetic Ising chain in a mixed transverse and longitudinal magnetic field","volume":"68","author":"Ovchinnikov","year":"2003","journal-title":"Phys. Rev. B"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1126\/science.1180085","article-title":"Quantum criticality in an Ising chain: Experimental evidence for emergent E8 symmetry","volume":"327","author":"Coldea","year":"2010","journal-title":"Science"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1063\/1.3554314","article-title":"Quantum criticality","volume":"64","author":"Sachdev","year":"2011","journal-title":"Phys. Today"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1143\/PTP.14.351","article-title":"A new approach to quantum statistical mechanics","volume":"14","author":"Matsubara","year":"1955","journal-title":"Prog. Theor. Exp."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1103\/RevModPhys.51.659","article-title":"An introduction to lattice gauge theory and spin systems","volume":"51","author":"Kogut","year":"1979","journal-title":"Rev. Mod. Phys."},{"key":"ref_86","unstructured":"Krizhevsky, A., Sutskever, I., and Hinton, G.E. (2012, January 3\u20138). Imagenet classification with deep convolutional neural networks. Proceedings of the NIPS: Advances in Neural Information Processing Systems 25, Stateline, NV, USA."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Holevo, A.S. (2011). Probabilistic and Statistical Aspects of Quantum Theory, Springer.","DOI":"10.1007\/978-88-7642-378-9"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s10946-010-9122-x","article-title":"Inverse spin-s portrait and representation of qudit states by single probability vectors","volume":"31","author":"Filippov","year":"2010","journal-title":"J. Russ. Laser Res."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1140\/epjd\/e2017-80024-y","article-title":"Introducing the Qplex: A novel arena for quantum theory","volume":"71","author":"Appleby","year":"2017","journal-title":"Eur. Phys. J. D"},{"key":"ref_90","unstructured":"Caves, C.M. (2019, November 07). Symmetric informationally complete POVMs - UNM Information Physics Group internal report (1999). Available online: http:\/\/info.phys.unm.edu\/~caves\/reports\/infopovm.pdf."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/S0022-2496(02)00028-7","article-title":"Tutorial on maximum likelihood estimation","volume":"47","author":"Myung","year":"2003","journal-title":"J. Math. Psychol."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s10946-010-9139-1","article-title":"Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics","volume":"31","author":"Filippov","year":"2010","journal-title":"J. Russ. Laser Res."},{"key":"ref_93","unstructured":"(2019, November 07). mpnum: A Matrix Product Representation Library for Python. Available online: https:\/\/mpnum.readthedocs.io\/en\/latest\/."},{"key":"ref_94","unstructured":"Sohn, K., Lee, H., and Yan, X. (2015, January 7\u201312). Learning structured output representation using deep conditional generative models. Proceedings of the NIPS: Advances in Neural Information Processing Systems 28, Montreal, QC, Canada."},{"key":"ref_95","unstructured":"Kingma, D.P., and Welling, M. (2013). Auto-encoding variational Bayes. arXiv, Available online: https:\/\/arxiv.org\/abs\/1312.6114."},{"key":"ref_96","unstructured":"Rezende, D.J., Mohamed, S., and Wierstra, D. (2014, January 21\u201326). Stochastic backpropagation and approximate inference in deep generative models. Proceedings of the 31st International Conference on Machine Learning (ICML), Beijing, China."},{"key":"ref_97","unstructured":"Jang, E., Gu, S., and Poole, B. (2016). Categorical reparameterization with Gumbel-softmax. arXiv, Available online: https:\/\/arxiv.org\/abs\/1611.01144."},{"key":"ref_98","unstructured":"Kusner, M.J., and Hern\u00e1ndez-Lobato, J.M. (2016). Gans for sequences of discrete elements with the Gumbel-softmax distribution. arXiv, Available online: https:\/\/arxiv.org\/abs\/1611.04051."},{"key":"ref_99","unstructured":"Maddison, C.J., Mnih, A., and Teh, Y.W. (2016). The concrete distribution: A continuous relaxation of discrete random variables. arXiv, Available online: https:\/\/arxiv.org\/abs\/1611.00712."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equation of State Calculations by Fast Computing Machines","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1103\/RevModPhys.80.1061","article-title":"Colloquium: Quantum annealing and analog quantum computation","volume":"80","author":"Das","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_102","unstructured":"Yavorsky, A., Markovich, L.A., Polyakov, E.A., and Rubtsov, A.N. (2019). Highly parallel algorithm for the Ising ground state searching problem. arXiv, Available online: https:\/\/arxiv.org\/abs\/1907.05124."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"094423","DOI":"10.1103\/PhysRevB.73.094423","article-title":"Matrix product states represent ground states faithfully","volume":"73","author":"Verstraete","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1103\/RevModPhys.82.277","article-title":"Colloquium: Area laws for the entanglement entropy","volume":"82","author":"Eisert","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_105","first-page":"021021","article-title":"Quantum entanglement in neural network states","volume":"7","author":"Deng","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_106","first-page":"99","article-title":"On a measure of divergence between two statistical populations defined by their probability distributions","volume":"35","author":"Bhattacharyya","year":"1943","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1038\/nphys2900","article-title":"Evidence for quantum annealing with more than one hundred qubits","volume":"10","author":"Boixo","year":"2014","journal-title":"Nat. Phys."},{"key":"ref_108","first-page":"031015","article-title":"What is the computational value of finite-range tunneling?","volume":"6","author":"Denchev","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"064304","DOI":"10.1103\/PhysRevB.95.064304","article-title":"Propagation of fluctuations in the quantum Ising model","volume":"95","author":"Navez","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"46004","DOI":"10.1209\/0295-5075\/106\/46004","article-title":"A radically new suggestion about the electrodynamics of water: Can the pH index and the Debye relaxation be of a common origin?","volume":"106","author":"Volkov","year":"2014","journal-title":"EPL"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"8067","DOI":"10.1039\/C9CP00257J","article-title":"A unified mechanism for ice and water electrical conductivity from direct current to terahertz","volume":"21","author":"Artemov","year":"2019","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_112","unstructured":"(2019, November 07). Github Repository with Code. Available online: https:\/\/github.com\/LuchnikovI\/Representation-of-quantum-many-body-states-via-VAE."},{"key":"ref_113","unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv, Available online: https:\/\/arxiv.org\/abs\/1412.6980."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/11\/1091\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:32:40Z","timestamp":1760189560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/11\/1091"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,7]]},"references-count":113,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["e21111091"],"URL":"https:\/\/doi.org\/10.3390\/e21111091","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,7]]}}}