Tomographically reconstructed master equations for any open quantum dynamics

Felix A. Pollock and Kavan Modi

School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Memory effects in open quantum dynamics are often incorporated in the equation of motion through a superoperator known as the memory kernel, which encodes how past states affect future dynamics. However, the usual prescription for determining the memory kernel requires information about the underlying system-environment dynamics. Here, by deriving the transfer tensor method from first principles, we show how a memory kernel master equation, for any quantum process, can be entirely expressed in terms of a family of completely positive dynamical maps. These can be reconstructed through quantum process tomography on the system alone, either experimentally or numerically, and the resulting equation of motion is equivalent to a generalised Nakajima-Zwanzig equation. For experimental settings, we give a full prescription for the reconstruction procedure, rendering the memory kernel operational. When simulation of an open system is the goal, we show how our procedure yields a considerable advantage for numerically calculating dynamics, even when the system is arbitrarily periodically (or transiently) driven or initially correlated with its environment. Namely, we show that the long time dynamics can be efficiently obtained from a set of reconstructed maps over a much shorter time.

► BibTeX data

► References

[1] I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455 (1997).

[2] K. Modi, Sci. Rep. 2, 581 (2012).

[3] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. A 97, 012127 (2018a).

[4] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. Lett. 120, 040405 (2018b).

[5] S. Milz, F. A. Pollock, and K. Modi, accepted in Phys. Rev. A (2018).

[6] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).

[7] A. Fruchtman, N. Lambert, and E. M. Gauger, Sci. Rep. 6, 28204 (2016).

[8] J. Iles-Smith, A. G. Dijkstra, N. Lambert, and A. Nazir, J. Chem. Phys. 144, 044110 (2016).

[9] Y. Tanimura, J. Phys. Soc. Jpn. 75, 082001 (2006).

[10] J. Strümpfer and K. Schulten, J. Chem. Theory Comput. 8, 2808 (2012).

[11] A. W. Chin, S. F. Huelga, and M. B. Plenio, in Semiconductors and Semimetals, Vol. 85, edited by U. Würfel, M. Thorwart, and E. R. Weber (Elsevier, 2011) pp. 115 – 143.

[12] N. Makri and D. E. Makarov, J. Chem. Phys. 102, 4611 (1995).

[13] I. de Vega and D. Alonso, Rev. Mod. Phys. 89, 015001 (2017).

[14] J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014).

[15] P. Nalbach, A. Ishizaki, G. R. Fleming, and M. Thorwart, New J. Phys. 13, 063040 (2011).

[16] A. Strathearn, B. W. Lovett, and P. Kirton, New J. Phys. 19, 093009 (2017a).

[17] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, arXiv:1711.09641 (2017b).

[18] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).

[19] V. Gorini, A. Kossakokowski, and E. C. G. Sudarshan, J. Math. Phys 17, 821 (1976).

[20] K. Modi, Open Systems & Information Dynamics 18, 253 (2011).

[21] M. Ringbauer, C. J. Wood, K. Modi, A. Gilchrist, A. G. White, and A. Fedrizzi, Phys. Rev. Lett. 114, 090402 (2015).

[22] D. Kretschmann, D. Schlingemann, and R. F. Werner, J. Funct. Anal. 255, 1889 (2008).

[23] B. Dive, F. Mintert, and D. Burgarth, Phys. Rev. A 92, 032111 (2015).

[24] L. M. Norris, G. A. Paz-Silva, and L. Viola, Phys. Rev. Lett. 116, 150503 (2016).

[25] F. A. Pollock, A. Chęcińska, S. Pascazio, and K. Modi, Phys. Rev. A 94, 032112 (2016).

[26] J. Jeske, J. H. Cole, C. Müller, M. Marthaler, and G. Schön, New J. Phys. 14, 023013 (2012).

[27] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, Phys. Rev. A 80, 052108 (2009).

[28] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, Phys. Rev. A 82, 062104 (2010a).

[29] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, J. Phys. A 43, 395303 (2010b).

[30] R. Rosenbach, J. Cerrillo, S. F. Huelga, J. Cao, and M. B. Plenio, New J. Phys. 18, 023035 (2016).

[31] A. A. Kananenka, C.-Y. Hsieh, J. Cao, and E. Geva, J. Phys. Chem. Lett. 7, 4809 (2016).

[32] S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).

[33] S. Daffer, K. Wódkiewicz, J. D. Cresser, and J. K. McIver, Phys. Rev. A 70, 010304 (2004).

[34] H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008).

[35] D. Chruściński and A. Kossakowski, Phys. Rev. A 94, 020103 (2016).

[36] B. Vacchini, Phys. Rev. Lett. 117, 230401 (2016).

[37] Q. Shi and E. Geva, J. Chem. Phys. 119, 12063 (2003).

[38] G. Cohen and E. Rabani, Phys. Rev. B 84, 075150 (2011).

[39] G. Cohen, E. Gull, D. R. Reichman, A. J. Millis, and E. Rabani, Phys. Rev. B 87, 195108 (2013).

[40] M. Buser, J. Cerrillo, G. Schaller, and J. Cao, Phys. Rev. A 96, 062122 (2017).

[41] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 120, 030402 (2018).

[42] C. R. Willis and R. H. Picard, Phys. Rev. A 9, 1343 (1974).

[43] R. H. Picard and C. R. Willis, Phys. Rev. A 16, 1625 (1977).

[44] P. Degenfeld-Schonburg and M. J. Hartmann, Phys. Rev. B 89, 245108 (2014).

[45] P. Degenfeld-Schonburg, C. Navarrete-Benlloch, and M. J. Hartmann, Phys. Rev. A 91, 053850 (2015).

[46] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).

[47] P. Facchi and S. Pascazio, J. Phys. A 41, 493001 (2008).

[48] T. M. Stace, A. C. Doherty, and D. J. Reilly, Phys. Rev. Lett. 111, 180602 (2013).

[49] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).

[50] P.-Y. Yang and W.-M. Zhang, arXiv:1605.08521 (2016).

[51] S. Kitajima, M. Ban, and F. Shibata, J. Phys. A 50, 125303 (2017).

[52] V. Prepeliţă, M. Doroftei, and T. Vasilache, Balkan J. Geom. Appl. 3, 111 (1998).

Cited by

[1] S. Alipour, A. T. Rezakhani, A. P. Babu, K. Mølmer, M. Möttönen, and T. Ala-Nissila, "Correlation-Picture Approach to Open-Quantum-System Dynamics", Physical Review X 10 4, 041024 (2020).

[2] Simon Milz and Kavan Modi, "Quantum Stochastic Processes and Quantum non-Markovian Phenomena", PRX Quantum 2 3, 030201 (2021).

[3] Mathias R. Jørgensen and Felix A. Pollock, "Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals", Physical Review Letters 123 24, 240602 (2019).

[4] Philip Taranto, Felix A. Pollock, and Kavan Modi, "Non-Markovian memory strength bounds quantum process recoverability", npj Quantum Information 7 1, 149 (2021).

[5] Alexander F. Kemper, Chao Yang, and Emanuel Gull, "Denoising and Extension of Response Functions in the Time Domain", Physical Review Letters 132 16, 160403 (2024).

[6] Leonardo Banchi, Edward Grant, Andrea Rocchetto, and Simone Severini, "Modelling non-markovian quantum processes with recurrent neural networks", New Journal of Physics 20 12, 123030 (2018).

[7] Yu-Qin Chen, Kai-Li Ma, Yi-Cong Zheng, Jonathan Allcock, Shengyu Zhang, and Chang-Yu Hsieh, "Non-Markovian Noise Characterization with the Transfer Tensor Method", Physical Review Applied 13 3, 034045 (2020).

[8] Philip Taranto, Felix A. Pollock, Simon Milz, Marco Tomamichel, and Kavan Modi, "Quantum Markov Order", Physical Review Letters 122 14, 140401 (2019).

[9] Stefano Gherardini, Andrea Smirne, Susana F Huelga, and Filippo Caruso, "Transfer-tensor description of memory effects in open-system dynamics and multi-time statistics", Quantum Science and Technology 7 2, 025005 (2022).

[10] Stefano Gherardini, Lorenzo Buffoni, Guido Giachetti, Andrea Trombettoni, and Stefano Ruffo, "Energy fluctuation relations and repeated quantum measurements", Chaos, Solitons & Fractals 156, 111890 (2022).

[11] Robin Blume-Kohout, Marcus P. da Silva, Erik Nielsen, Timothy Proctor, Kenneth Rudinger, Mohan Sarovar, and Kevin Young, "A Taxonomy of Small Markovian Errors", PRX Quantum 3 2, 020335 (2022).

[12] Francesco Campaioli, Felix A. Pollock, and Kavan Modi, "Tight, robust, and feasible quantum speed limits for open dynamics", Quantum 3, 168 (2019).

[13] Pedro Figueroa–Romero, Felix A. Pollock, and Kavan Modi, "Markovianization with approximate unitary designs", Communications Physics 4 1, 127 (2021).

[14] Thomas Sayer and Andrés Montoya-Castillo, "Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra", The Journal of Chemical Physics 160 4, 044108 (2024).

[15] Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, "Almost Markovian processes from closed dynamics", Quantum 3, 136 (2019).

[16] I. A. Luchnikov, S. V. Vintskevich, D. A. Grigoriev, and S. N. Filippov, "Machine Learning Non-Markovian Quantum Dynamics", Physical Review Letters 124 14, 140502 (2020).

[17] I.A. Aloisio, G.A.L. White, C.D. Hill, and K. Modi, "Sampling Complexity of Open Quantum Systems", PRX Quantum 4 2, 020310 (2023).

[18] Mathias R. Jørgensen and Felix A. Pollock, "Discrete memory kernel for multitime correlations in non-Markovian quantum processes", Physical Review A 102 5, 052206 (2020).

[19] Felix Pollock, Emanuel Gull, Kavan Modi, and Guy Cohen, "Reduced Dynamics of Full Counting Statistics", SciPost Physics 13 2, 027 (2022).

[20] G.A.L. White, F.A. Pollock, L.C.L. Hollenberg, K. Modi, and C.D. Hill, "Non-Markovian Quantum Process Tomography", PRX Quantum 3 2, 020344 (2022).

[21] Chu Guo, Kavan Modi, and Dario Poletti, "Tensor-network-based machine learning of non-Markovian quantum processes", Physical Review A 102 6, 062414 (2020).

[22] Dominikus Brian and Xiang Sun, "Generalized quantum master equation: A tutorial review and recent advances", Chinese Journal of Chemical Physics 34 5, 497 (2021).

[23] I. A. Luchnikov, E. O. Kiktenko, M. A. Gavreev, H. Ouerdane, S. N. Filippov, and A. K. Fedorov, "Probing non-Markovian quantum dynamics with data-driven analysis: Beyond “black-box” machine-learning models", Physical Review Research 4 4, 043002 (2022).

[24] Kavan Modi, "George Sudarshan and Quantum Dynamics", Open Systems & Information Dynamics 26 03, 1950013 (2019).

[25] Rolando Ramirez Camasca and Gabriel T. Landi, "Memory kernel and divisibility of Gaussian collisional models", Physical Review A 103 2, 022202 (2021).

[26] Philip Taranto, Simon Milz, Felix A. Pollock, and Kavan Modi, "Structure of quantum stochastic processes with finite Markov order", Physical Review A 99 4, 042108 (2019).

[27] Simon Milz, M. S. Kim, Felix A. Pollock, and Kavan Modi, "Completely Positive Divisibility Does Not Mean Markovianity", Physical Review Letters 123 4, 040401 (2019).

[28] Shlok Nahar and Sai Vinjanampathy, "Preparations and weak-field phase control can witness initial correlations", Physical Review A 100 6, 062120 (2019).

[29] Michael te Vrugt and Raphael Wittkowski, "Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians", Physical Review E 99 6, 062118 (2019).

[30] Anthony J. Dominic, Siqin Cao, Andrés Montoya-Castillo, and Xuhui Huang, "Memory Unlocks the Future of Biomolecular Dynamics: Transformative Tools to Uncover Physical Insights Accurately and Efficiently", Journal of the American Chemical Society 145 18, 9916 (2023).

[31] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Universal construction of decoders from encoding black boxes", Quantum 7, 957 (2023).

[32] A. P. Babu, S. Alipour, A. T. Rezakhani, and T. Ala-Nissila, "Unfolding system-environment correlation in open quantum systems: Revisiting master equations and the Born approximation", Physical Review Research 6 1, 013243 (2024).

[33] Frederik vom Ende, "Finite-Dimensional Stinespring Curves Can Approximate Any Dynamics", Open Systems & Information Dynamics 31 01, 2450004 (2024).

[34] Philip Taranto, "Memory effects in quantum processes", International Journal of Quantum Information 18 02, 1941002 (2020).

[35] Andrew Wu, Javier Cerrillo, and Jianshu Cao, "Extracting kinetic information from short-time trajectories: relaxation and disorder of lossy cavity polaritons", Nanophotonics (2024).

[36] Giovanni Cemin, Francesco Carnazza, Sabine Andergassen, Georg Martius, Federico Carollo, and Igor Lesanovsky, "Inferring interpretable dynamical generators of local quantum observables from projective measurements through machine learning", Physical Review Applied 21 4, L041001 (2024).

[37] Graeme D. Berk, Andrew J. P. Garner, Benjamin Yadin, Kavan Modi, and Felix A. Pollock, "Resource theories of multi-time processes: A window into quantum non-Markovianity", Quantum 5, 435 (2021).

[38] Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi, "Non-Markovian quantum processes: Complete framework and efficient characterization", Physical Review A 97 1, 012127 (2018).

[39] Simon Milz, Felix A. Pollock, and Kavan Modi, "An Introduction to Operational Quantum Dynamics", Open Systems and Information Dynamics 24 4, 1740016 (2017).

[40] Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi, "Non-Markovian quantum processes: complete framework and efficient characterisation", arXiv:1512.00589, (2015).

[41] Maximilian Buser, Javier Cerrillo, Gernot Schaller, and Jianshu Cao, "Initial system-environment correlations via the transfer-tensor method", Physical Review A 96 6, 062122 (2017).

[42] Leonardo Banchi, Edward Grant, Andrea Rocchetto, and Simone Severini, "Modelling Non-Markovian Quantum Processes with Recurrent Neural Networks", arXiv:1808.01374, (2018).

[43] Tanmay Neema, Susmit Jha, and Tuhin Sahai, "Non-Markovian Quantum Control via Model Maximum Likelihood Estimation and Reinforcement Learning", arXiv:2402.05084, (2024).

[44] Shlok Nahar and Sai Vinjanampathy, "Preparations and Weak Quantum Control can Witness non-Markovianity", arXiv:1803.08443, (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 07:26:52) and SAO/NASA ADS (last updated successfully 2024-05-26 07:26:53). The list may be incomplete as not all publishers provide suitable and complete citation data.