Gaussian quantum systems and Kahler geometrical structure
Abstract
In this article, we study the phase-space distribution of the quantum state as a framework to describe the different properties of quantum systems in continuous-variable systems. The natural approach to quantum systems is given the Gaussian Wigner representation, to unify the description of bosonic and fermionic quantum states, we study the structure of the Kahler space geometry as the geometry generated by three forms under the agreement conditions depended on the nature of the state bosonic or fermionic. Multi-mode light is studied, and we established that the Fock space vacuum corresponds to a certain homogeneous Gaussian state.References
G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Systems & Information Dynamics 21 (2014) 1440001.
W. Arendt, H. Vogt, and J. Voigt, Form Methods for Evolution Equations. Lecture Notes of the 18th International Internet seminar, version: 6 March (2019).
A. Ashtekar and P. Singh, Loop Quantum Cosmology: A Status Report, Class. Quant. Grav. 28 (2011) 213001.
Budde C. and Landsman K. A bounded transform approach to self-adjoint operators: functional calculus and affiliated von Neumann algebras. Ann. Funct. Anal. 7, 3 (2016), 411--420.
C. Batty, A. Gomilko, and Y. Tomilov, Product formulas in functional calculi for sectorial operators. Math. Z. 279, 1-2 (2015), 479--507.
S. Clark, Sums of operator logarithms. Q. J. Math. 60, 4 (2009), 413--427.
F. Colombo, G. Gentili, I. Sabadini, D.C.~Struppa, Noncommutative functional calculus: unbounded operators, preprint, (2007).
R. DeLaubenfels, Automatic extensions of functional calculi. Studia Math. 114, 3 (1995), 237--259.
L. D'Alessio, Y. A. Kafri, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics 65 (2016) 239.
N. Dungey, Asymptotic type for sectorial operators and an integral of fractional powers. J. Funct. Anal. 256, 5 (2009), 1387--1407.
N. Dupuis, L. Canet, A. Eichhorn, and others, The nonperturbative functional renormalization group and its applications, Physics Reports 910, 1--114 (2021).
P. Chalupa, T. Sch¨afer, M. Reitner, and others, Fingerprints of the Local Moment Formation and its Kondo Screening in the Generalized Susceptibilities of Many-Electron Problems, Phys. Rev. Lett. 126, 056403 (2021).
F. Krien, A.I. Lichtenstein, and G. Rohringer, Fluctuation diagnostic of the nodal/antinodal dichotomy in the Hubbard model at weak coupling: A parquet dual fermion approach, Phys. Rev. B 102, 235133 (2020).
T. Schafer and A. Toschi, How to read between the lines of electronic spectra: the diagnostics of fluctuations in strongly correlated electron systems, Journal of Physics: Condensed Matter (2021).
L. D. Re and G. Rohringer, Fluctuations diagnostic of the spin susceptibility: Neel ordering revisited in dmft (2021), arXiv:2104.11737.
G. Rohringer, A. Valli, and A. Toschi, Local electronic correlation at the two-particle level, Phys. Rev. B 86, 125114 (2012).
G. Rohringer, H. Hafermann, A. Toschi, and others, Diagrammatic routes to nonlocal correlations beyond dynamical mean-field theory, Rev. Mod. Phys. 90, 025003 (2018).
N. Wentzell, G. Li, A. Tagliavini, C. Taranto, G. Rohringer, K. Held, A. Toschi, and S. Andergassen, High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation, Phys. Rev. B 102, 085106 (2020).
T. Eisner, B. Farkas, M. Haase, and R. Nagel, Operator Theoretic Aspects of Ergodic Theory. Vol. 272 of Graduate Texts in Mathematics. Springer, Cham, (2015).
Eugenio Bianchi, Lucas Hackl, Nelson Yokomizo ``Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate,'' Journal for High Energy Physics (2018) 2018: 25.
M. Haase, The Functional Calculus for Sectorial Operators. Vol. 169 of Operator Theory: Advances and Applications. Birkh¨auser Verlag, Basel, (2006).
M. Haase, Functional analysis. An Elementary Introduction. Vol. 156 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, (2014).
Lucas Hackl, Robert C. Myers ``Circuit complexity of free fermions,'' Journal for High Energy Physics (2018) 2018: 139.
Krien F., Valli A., and Capone M. Single-boson exchange decomposition of the vertex function, Phys. Rev. B 100, 155149 (2019).
J. Nokkala, R. Mart'{i}nez-Pe~{n}a, G. L. Giorgi, V. Parigi, M. C. Soriano, and R. Zambrini, Gaussian states of continuous-variable quantum systems provide universal and versatile reservoir computing, Commun. Physics 4, 53 (2021).
E. Martin-Martnez, D. Aasen and A. Kempf, Processing quantum information with the relativistic motion of atoms, Phys. Rev. Lett. 110 (2013) 160501 [1209.4948].
M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, (1980).
D. R. M. Arvidsson-Shukur, Yunger N. Halpern, H. V. Lepage, A. A. Lasek, C. H. W. Barnes, and S. Lloyd, Quantum advantage in postselected metrology, Nat. Commun. 11, 3775 (2020).
K. Schmudgen, Unbounded Self-adjoint Operators on Hilbert Space. Vol. 265 of Graduate Texts in Mathematics. Springer, Dordrecht, (2012).
T. Shi, E. Demler, and J. I. Cirac, Variational study of fermionic and bosonic systems with non-gaussian states: Theory and applications, Annals of Physics (2017).
P. Woit, Quantum theory, groups, and representations: An introduction. Springer, 2017.
D. Vilardi, P. M. Bonetti, and W. Metzner, Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model, Phys. Rev. B 102, 245128 (2020).
P. M. Bonetti, Accessing the ordered phase of correlated Fermi systems: Vertex bosonization and mean-field theory within the functional renormalization group, Phys. Rev. B 102, 235160 (2020).
G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dynamics 21, 1440001 (2014).
M. Qin, T. Sch¨afer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective (2021).
M. Walschaers, N. Treps, B. Sundar, L. D. Carr, and V. Parigi, Emergent complex quantum networks in continuous-variables non-gaussian states, arXiv:2012. 15608 [quant-ph] (2021).
M.I. Yaremenko, Calderon-Zygmund Operators and Singular Integrals,~Applied Mathematics & Information Sciences: Vol. 15: Iss. 1, Article 13, (2021).
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
