{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T01:21:38Z","timestamp":1772155298165,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,31]],"date-time":"2021-10-31T00:00:00Z","timestamp":1635638400000},"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>The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.<\/jats:p>","DOI":"10.3390\/e23111445","type":"journal-article","created":{"date-parts":[[2021,11,1]],"date-time":"2021-11-01T22:21:08Z","timestamp":1635805268000},"page":"1445","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5489-698X","authenticated-orcid":false,"given":"Julio A.","family":"L\u00f3pez-Sald\u00edvar","sequence":"first","affiliation":[{"name":"Instituto de Ciencias Nucleares, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Apdo. Postal 70-543, Ciudad de M\u00e9xico 04510, Mexico"},{"name":"Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow 141700, Russia"},{"name":"Russian Quantum Center, Skolkovo, Moscow 143025, Russia"}]},{"given":"Margarita A.","family":"Man\u2019ko","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"}]},{"given":"Vladimir I.","family":"Man\u2019ko","sequence":"additional","affiliation":[{"name":"Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow 141700, Russia"},{"name":"Russian Quantum Center, Skolkovo, Moscow 143025, Russia"},{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,31]]},"reference":[{"key":"ref_1","unstructured":"Landau, L.D., and Lifshitz, E.M. (1965). Quantum Mechanics, Pergamon Press. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Schr\u00f6dinger, E. (1926). Quantisierung als Eigenwertproblem (Zweite Mitteilung). Ann. Phys., 384.","DOI":"10.1002\/andp.19263840602"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1007\/BF01343064","article-title":"Das D\u00e4mpfungsproblem in der Wellenmechanik","volume":"45","author":"Landau","year":"1927","journal-title":"Z. Phys."},{"key":"ref_4","first-page":"245","article-title":"Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik","volume":"1927","year":"1927","journal-title":"Mathematisch-Physikalische Klasse"},{"key":"ref_5","unstructured":"Von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/0034-4877(72)90010-9","article-title":"On quantum statistical mechanics of non-Hamiltonian systems","volume":"3","author":"Kossakowski","year":"1972","journal-title":"Rep. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01608499","article-title":"On the generators of quantum dynamical semigroups","volume":"48","author":"Lindblad","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1063\/1.522979","article-title":"Completely positive semigroups of N-level systems","volume":"17","author":"Gorini","year":"1976","journal-title":"J. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the quantum correction for thermodynamic equilibrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_10","first-page":"264","article-title":"Some Formal Properties of the Density Matrix","volume":"22","author":"Husimi","year":"1940","journal-title":"Proc. Phys. Math. Soc. Jpn."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2766","DOI":"10.1103\/PhysRev.131.2766","article-title":"Coherent and incoherent states of the radiation field","volume":"131","author":"Glauber","year":"1963","journal-title":"Phys. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1103\/PhysRevLett.10.277","article-title":"Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams","volume":"10","author":"Sudarshan","year":"1963","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1007\/BF01397280","article-title":"\u00dcber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik","volume":"43","author":"Heisenberg","year":"1927","journal-title":"Z. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0375-9601(96)00107-7","article-title":"Symplectic Tomography as Classical Approach to Quantum Systems","volume":"213","author":"Mancini","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1109\/TMI.1986.4307775","article-title":"On the determination of functions from their integral values along certain manifolds","volume":"5","author":"Radon","year":"1986","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_16","unstructured":"Dodonov, V.V., and Man\u2019ko, V.I. (1989). Invariants and the evolution of nonstationary quantum systems. Proceedings of the Lebedev Physical Institute, Nova Science Publishers."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Pitaevskii, L., and Stringari, S. (2016). Bose\u2013Einstein Condensation and Superfluidity, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198758884.001.0001"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"041104","DOI":"10.1063\/1.5046663","article-title":"Recent advances in Wigner function approaches","volume":"5","author":"Weinbub","year":"2018","journal-title":"Appl. Phys. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Man\u2019ko, V.I., Marmo, G., Porzio, A., Solimeno, S., and Ventriglia, F. (2010). Homodyne estimation of quantum states purity by exploiting covariant uncertainty relation. arXiv.","DOI":"10.1088\/0031-8949\/83\/04\/045001"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1166\/asl.2009.1060","article-title":"A possible experimental check of the uncertainty relations by means of homodyne measuring field quadrature","volume":"2","author":"Marmo","year":"2009","journal-title":"Adv. Sci. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Schleich, W.P. (2001). Quantum Optics in Phase Space, Wiley-VCH Verlag.","DOI":"10.1002\/3527602976"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"L19","DOI":"10.1088\/0305-4470\/8\/2\/001","article-title":"The Green function and thermodynamical properties of quadratic systems","volume":"8","author":"Dodonov","year":"1975","journal-title":"J. Phys. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0378-4371(82)90137-6","article-title":"Wigner functions of quadratic systems","volume":"115","author":"Akhundova","year":"1982","journal-title":"Physica A"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Dodonov, V.V. (2021). Invariant Quantum States of Quadratic Hamiltonians. Entropy, 23.","DOI":"10.3390\/e23050634"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-Sald\u00edvar, J.A., Man\u2019ko, M.A., and Man\u2019ko, V.I. (2020). Differential parametric formalism for the evolution of Gaussian states: Nonunitary evolution and invariant states. Entropy, 22.","DOI":"10.3390\/e22050586"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"040007","DOI":"10.1063\/5.0054953","article-title":"Nonlinear differential equations of Gaussian states","volume":"2362","year":"2021","journal-title":"AIP Conf. Proc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2722","DOI":"10.1103\/PhysRevLett.84.2722","article-title":"Inseparability Criterion for Continuous Variable Systems","volume":"84","author":"Duan","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2726","DOI":"10.1103\/PhysRevLett.84.2726","article-title":"Peres\u2013Horodecki Separability Criterion for Continuous Variable Systems","volume":"84","author":"Simon","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"82","DOI":"10.3390\/quantum1010009","article-title":"Solution to the time-dependent coupled harmonic oscillators Hamiltonian with arbitrary interactions","volume":"1","year":"2019","journal-title":"Quantum Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2075001","DOI":"10.1142\/S0218301320750014","article-title":"Time-dependent coupled harmonic oscillators: Classical and quantum solutions","volume":"29","year":"2020","journal-title":"Int. J. Mod. Phys. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"042203","DOI":"10.1103\/PhysRevE.97.042203","article-title":"Coupled harmonic oscillators and their quantum entanglement","volume":"97","author":"Makarov","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"052213","DOI":"10.1103\/PhysRevE.102.052213","article-title":"Quantum entanglement and reflection coefficient for coupled harmonic oscillators","volume":"102","author":"Makarov","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"12557","DOI":"10.1038\/s41598-020-68309-3","article-title":"Quantum entanglement maintained by virtual excitations in an ultrastrongly coupled oscillator system","volume":"10","author":"Zhou","year":"2020","journal-title":"Sci. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1445\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:23:48Z","timestamp":1760167428000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1445"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,31]]},"references-count":33,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["e23111445"],"URL":"https:\/\/doi.org\/10.3390\/e23111445","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,31]]}}}