Abstract
Dynamical aspects of information-theoretic and entropic measures of quantum systems are studied. First, we show that for the time-dependent harmonic oscillator, as well as for the charged particle in certain time-varying electromagnetic fields, the increase of the entropy and dynamics of the Fisher information can be directly described and related. To illustrate these results, we have considered several examples for which all the relations take the elementary form. Moreover, we show that the integrals of (geodesic) motion associated with some conformal Killing vectors lead to the Ermakov–Lewis invariants for the considered electromagnetic fields. Next, we explicitly work out the dynamics of the entanglement entropy of the oscillators coupled by a continuous time-dependent parameter as well as we analyse some aspects of quantum-classical transition (in particular decoherence). Finally, we study in some detail the behaviour of quantum quenches (in the presence of the critical points) for the case of mutually non-interacting non-relativistic fermions in a harmonic trap.
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References
Shannon, C., Weaver, W.: Math. Theory Commun. The University of Illinois Press, Urbana, Illinois (1949)
Rényi, A.: “On measures of entropy and information’’. Proc. Fourth Berkeley Symp. Math. Stat. Probab. 1, 547 (1961)
Fisher, R.: “Theory of statistical estimation’’. Proc. Cambridge Philos. Soc. 22, 700 (1925)
Białynicki-Birula, I., Mycielski, J.: “Uncertainty relations for information entropy in wave mechanics’’. Commun. Math. Phys. 44, 129 (1975)
Sánchez-Moreno, P., Plastino, A., Dehesa, J.: A quantum uncertainty relation based on Fisher’s information. J. Phys. A: Math. Theor. 44, 065301 (2011)
Stam, A.: “Some inequalities satisfied by the quantities of information of Fisher and Shannon’’. Inf. Control. 2, 101 (1959)
Dembo, A., Cover, T., Thomas, J.: “Information theoretic inequalities’’. IEEE Trans. Inform. Theory 37, 1501 (1991)
Vignat, C., Bercher, J.-F.: “Analysis of signals in the Fisher-Shannon information plane’’. Phys. Lett. A 312, 27 (2003)
Angulo, J., Antolín, J., Sen, K.: “Fisher-Shannon plane and statistical complexity of atoms’’. Phys. Lett. A 372, 670 (2008)
Sobrino-Coll, N., Puertas-Centeno, D., Toranzo, I., Dehesa, J.: “Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems” J. Stat. Mech. (2017) 083102
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: “Quantum entanglement’’. Rev. Mod. Phys. 81, 865 (2009)
Lewis, H.: “Classical and quantum systems with time-dependent harmonic-oscillator-type hamiltonians’’. Phys. Rev. Lett. 18, 510 (1967)
Lewis, H., Riesenfeld, W.: Class of exact invariants for classical and quantum time-dependent harmonic oscillators. J. Math. Phys. 10, 1458 (1969)
Ermakov, V.: “Second order differential equations. Conditions of complete integrability” Univ. Izv. Kiev, Series III 9 (1880) 1 (English translation: A. Harin, under redaction by P. Leach, Appl. Anal. Discrete Math. 2 (2008) 123)
Milne, E.: “The numerical determination of characteristic numbers’’. Phys. Rev. 35, 863 (1930)
Pinney, E.: “The nonlinear differential equation \(y^{\prime \prime }+p(x)y+cy^ {-3}=0\)’’. Proc. Am. Math. Soc. 1, 681 (1950)
Choi, J., Kim, M.-S., Kim, D., Maamache, M., Menouar, S., Nahme, I.: “Information theories for time-dependent harmonic oscillator’’. Ann. Phys. 326, 1381 (2011)
Aktürk, E., Özcan, Ö., Sever, R.: “Joint entropy of the harmonic oscillator with time dependent mass and/or frequency’’. Int. J. Mod. Phys. B 23, 2449 (2009)
Aguiar, V., Guedes, I.: “Fisher information of quantum damped harmonic oscillators’’. Phys. Scr. 90, 045207 (2015)
Najafizade, S.A., Hassanabadi, H., Zarrinkamar, S.: Theoretical information measurement in nonrelativistic time-dependent approach. Indian J. Phys. 92, 183 (2018)
Aguiar, V., Guedes, I.: “Joint entropy of quantum damped harmonic oscillators” Physica A 401 (2014) 159
Fotue, A., Wirngo, A., Keumo Tsiaze, R., Hounkonnou, M.: Joint entropy and decoherence without dissipation in a driven harmonic oscillator. Eur. Phys. J. Plus 136, 131 (2016)
Aguiar, V., Guedes, I., Pedrosa, I.: “Tsallis, Rényi, and Shannon entropies for time-dependent mesoscopic RLC circuits” PTEP (2015) 113A01
Aguiara, V., Guedes, I.: Entropy and information of a spinless charged particle in time-varying magnetic fields. J. Math. Phys. 57, 092103 (2016)
Burdet, G., Duval, C., Perrin, M.: “Time-dependent quantum systems and chronoprojective geometry’’. Lett. Math. Phys. 10, 255 (1985)
Duval, C., Burdet, G., Kunzle, H., Perrin, M.: “Bargmann structures and Newton-Cartan theory’’. Phys. Rev. D 31, 1841 (1985)
Cariglia, M., Gibbons, G., van Holten, J.-W., Horvathy, P., Zhang, P.-M.: Conformal killing tensors and covariant hamiltonian dynamics. J. Math. Phys. 55, 122702 (2014)
Cariglia, M., Galajinsky, A., Gibbons, G., Horvathy, P.: “Cosmological aspects of the Eisenhart-Duval lift’’. Eur. Phys. J. C 78, 314 (2018)
Cariglia, M., Duval, C., Gibbons, G., Horvathy, P.: “Eisenhart lifts and symmetries of time-dependent systems’’. Ann. Phys. 373, 631 (2016)
Ghosh, S., Gupta, K., Srivastava, S.: Entanglement dynamics following a sudden quench: an exact solution. EPL 120, 50005 (2017)
Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I., Zeh, H.: Decoherence and the appearance of a classical world in quantum theory. Springer, Berlin (1996)
Zurek, W.: “Decoherence, einselection, and the quantum origins of the classical’’. Rev. Mod. Phys. 75, 715 (2003)
Halliwell, J.: “Decoherence in quantum cosmology’’. Phys. Rev. D 39, 2912 (1989)
Morikawa, M.: “Quantum decoherence and classical correlation in quantum mechanics’’. Phys. Rev. D 42, 2929 (1990)
Nielsen, M., Chuang, I.: Quantum computation and quantum information. CUP, Cambridge (2011)
Haroche, S.: “Entanglement, decoherence and the quantum/classical boundary’’. Phys. Today 51, 36 (1998)
Schlosshauer, M.: “Quantum decoherence’’. Phys. Rep. 831, 1 (2019)
Bokulich, A., Jaeger, G.: Philosophy of quantum information and entanglement. CUP, Cambridge (2010)
Shor, P.: “Scheme for reducing decoherence in quantum computer memory’’. Phys. Rev. A 52, R2493(R) (1995)
Bombelli, L., Koul, R., Lee, J., Sorkin, R.: “Quantum source of entropy for black holes’’. Phys. Rev. D 34, 373 (1986)
Srednicki, M.: “Entropy and Area’’. Phys. Rev. Lett. 71, 666 (1993)
Klich, I., Levitov, L.: “Quantum noise as an entanglement meter’’. Phys. Rev. Lett. 102, 100502 (2009)
Song, H., Rachel, S., Flindt, C., Klich, I., Laflorencie, N., Le Hur, K.: “Bipartite fluctuations as a probe of many-body entanglement’’. Phys. Rev. B 85, 035409 (2012)
Das, S., Hampton, S., Liu, S.: Quantum quench in non-relativistic fermionic field theory: harmonic traps and 2d string theory. JHEP 08, 176 (2019)
Ciftja, O.: A simple derivation of the exact wavefunction of a harmonic oscillator with time-dependent mass and frequency. J. Phys. A: Math. Gen. 32, 6385 (1999)
Niederer, U.: “The maximal kinematical invariance group of the harmonic oscillator’’. Helv. Phys. Acta 46, 191 (1973)
Dhasmana, S., Sen, A., Silagadze, Z.: “Equivalence of a harmonic oscillator to a free particle and Eisenhart lift’’. Ann. Phys. 434, 168623 (2021)
Kim, S., Lee, C.: “Nonequilibrium quantum dynamics of second order phase transitions’’. Phys. Rev. D 62, 125020 (2000)
Dehesa, J., Guerrero, A., Sánchez-Moreno, P.: “Information-theoretic-based spreading measures of orthogonal polynomials’’. Complex Anal. Oper. Theory 6, 585 (2012)
Maartens, R., Maharaj, S.: “Conformal symmetries of pp-waves’’. Class. Quant. Grav. 8, 503 (1991)
Keane, A., Tupper, B.: “Conformal symmetry classes for pp-wave spacetimes’’. Class. Quant. Grav. 21, 2037 (2004)
Bastidas, V., Reina, J., Emary, C., Brandes, T.: “Entanglement and parametric resonance in driven quantum systems’’. Phys. Rev. A 81, 012316 (2010)
Chen, R.-X., Shen, L.-T., Yang, Z.-B., Wu, H.-Z.: “Transition of entanglement dynamics in an oscillator system with weak time-dependent coupling’’. Phys. Rev. A 91, 012312 (2015)
Park, D.: “Dynamics of entanglement and uncertainty relation in coupled harmonic oscillator system: exact results’’. Quantum Inf. Process. 17, 147 (2018)
Ghosh, S., Gupta, K., Srivastava, S.: “Exact relaxation dynamics and quantum information scrambling in multiply quenched harmonic chains’’. Phys. Rev. E 100, 012215 (2019)
Park, D.: “Dynamics of entanglement in three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants’’. Quantum Inf. Process. 18, 282 (2019)
Park, D., Jung, E.: “Sum rule of quantum uncertainties: coupled harmonic oscillator system with time-dependent parameters’’. Quantum Inf. Process. 19, 259 (2020)
Hab-Arrih, R., Jellal, A., Merdaci, A.: “Dynamics and redistribution of entanglement and coherence in three time-dependent coupled harmonic oscillators’’. Int J. Geom. Methods Mod. Phys. 18, 2150120 (2021)
Kim, S., Santana, A., Khanna, F.: Decoherence of quantum damped oscillators. J. Korean Phys. Soc. 43, 452 (2003)
Husimi, K.: “Miscellanea in elementary quantum mechanics, II’’. Prog. Theor. Phys. 9, 381 (1953)
O’Connell, R., Zuo, J.: Effect of an external field on decoherence: Part II. J. Mod. Opt. 51, 821 (2004)
Das, S.: “Old and new scaling laws in quantum quench” PTEP (2016) 12C107
López-Ruiz, R., Mancini, H., Calbet, X.: “A statistical measure of complexity’’. Phys. Lett. A 209, 321 (1995)
Ebert, M., Volosniev, A., Hammer, H.-W.: “Two cold atoms in a time-dependent harmonic trap in one dimension’’. Ann. Phys. 528, 693 (2016)
Dinc, C., Oktay, O.: “Entanglement dynamics of coupled oscillators from Gaussian states” arXiv:2104.12332 (2021)
Chandran, S. Mahesh, Shankaranarayanan, S.: “Divergence of entanglement entropy in quantum systems: Zero-modes” Phys. Rev. D 99 (2019) 045010
Das, S., Hampton, S., Liu, S.: Quantum quench in c=1 matrix model and emergent space-times. JHEP 04, 107 (2020)
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Andrzejewski, K. Dynamics of entropy and information of time-dependent quantum systems: exact results. Quantum Inf Process 21, 117 (2022). https://doi.org/10.1007/s11128-022-03440-w
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DOI: https://doi.org/10.1007/s11128-022-03440-w