Almost Markovian processes from closed dynamics

Pedro Figueroa-Romero; Kavan Modi; Felix A. Pollock

doi:10.22331/q-2019-04-30-136

Almost Markovian processes from closed dynamics

Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock

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

Published:	2019-04-30, volume 3, page 136
Eprint:	arXiv:1802.10344v5
Doi:	https://doi.org/10.22331/q-2019-04-30-136
Citation:	Quantum 3, 136 (2019).

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Abstract

It is common, when dealing with quantum processes involving a subsystem of a much larger composite closed system, to treat them as effectively memory-less (Markovian). While open systems theory tells us that non-Markovian processes should be the norm, the ubiquity of Markovian processes is undeniable. Here, without resorting to the Born-Markov assumption of weak coupling or making any approximations, we formally prove that processes are close to Markovian ones, when the subsystem is sufficiently small compared to the remainder of the composite, with a probability that tends to unity exponentially in the size of the latter. We also show that, for a fixed global system size, it may not be possible to neglect non-Markovian effects when the process is allowed to continue for long enough. However, detecting non-Markovianity for such processes would usually require non-trivial entangling resources. Our results have foundational importance, as they give birth to $\textit{almost}$ Markovian processes from composite closed dynamics, and to obtain them we introduce a new notion of equilibration that is far stronger than the conventional one and show that this stronger equilibration is attained.

Featured image: In the space of all processes $\Upsilon$, the probability that the non-Markovianity $\mathcal{N}$ is larger than $\mathcal{B}_k\geq\mathbb{E}[\mathcal{N}]$ by some $\epsilon > 0$ decreases exponentially in $\epsilon^2$. When the environment dimension is much larger than that of the system, $\mathcal{B}_k$ becomes small and quantum processes concentrate around the Markovian ones $\Upsilon^{\scriptstyle{(\mathrm{M})}}$.

► BibTeX data

@article{FigueroaRomero2019almostmarkovian,
  doi = {10.22331/q-2019-04-30-136},
  url = {https://doi.org/10.22331/q-2019-04-30-136},
  title = {Almost {M}arkovian processes from closed dynamics},
  author = {Figueroa-Romero, Pedro and Modi, Kavan and Pollock, Felix A.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {136},
  month = apr,
  year = {2019}
}

► References

[1] C. Gogolin and J. Eisert, ``Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems,'' Rep. Prog. Phys. 79, 056001 (2016).
https://doi.org/10.1088/0034-4885/79/5/056001

[2] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, ``The role of quantum information in thermodynamics–a topical review,'' J. Phys. A: Math. Theor. 49, 143001 (2016).
https://doi.org/10.1088/1751-8113/49/14/143001

[3] L. P. García-Pintos, N. Linden, A. S. L. Malabarba, A. J. Short, and A. Winter, ``Equilibration time scales of physically relevant observables,'' Phys. Rev. X 7, 031027 (2017).
https://doi.org/10.1103/PhysRevX.7.031027

[4] M. Srednicki, ``The approach to thermal equilibrium in quantized chaotic systems,'' J. Phys. A: Math. Gen. 32, 1163 (1999).
https://doi.org/10.1088/0305-4470/32/7/007

[5] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Operational markov condition for quantum processes,'' Phys. Rev. Lett. 120, 040405 (2018a).
https://doi.org/10.1103/PhysRevLett.120.040405

[6] C. Arenz, R. Hillier, M. Fraas, and D. Burgarth, ``Distinguishing decoherence from alternative quantum theories by dynamical decoupling,'' Phys. Rev. A 92, 022102 (2015).
https://doi.org/10.1103/PhysRevA.92.022102

[7] M. A. Schlosshauer, Decoherence and the Quantum-To-Classical Transition (Springer Berlin Heidelberg, 2007).
https://doi.org/10.1007/978-3-540-35775-9

[8] I. de Vega and D. Alonso, ``Dynamics of non-Markovian open quantum systems,'' Rev. Mod. Phys. 89, 015001 (2017).
https://doi.org/10.1103/RevModPhys.89.015001

[9] F. Ciccarello, ``Collision models in quantum optics,'' Quantum Meas. Quantum Metrol. 4, 53 (2017).
https://doi.org/10.1515/qmetro-2017-0007

[10] H. Spohn and J. L. Lebowitz, ``Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs,'' in Advances in Chemical Physics, edited by S. A. Rice (John Wiley & Sons, Ltd, 2007) pp. 109–142.
https://doi.org/10.1002/9780470142578.ch2

[11] M. Ledoux, The Concentration of Measure Phenomenon, Mathematical surveys and monographs (American Mathematical Society, 2001).
https://bookstore.ams.org/surv-89-s

[12] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics No. 1200 (Springer-Verlag, 1986).
https://doi.org/10.1007/978-3-540-38822-7

[13] S. Boucheron, G. Lugosi, and P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence (OUP Oxford, 2013).
https://doi.org/10.1093/acprof:oso/9780199535255.001.0001

[14] S. Popescu, A. J. Short, and A. Winter, ``Entanglement and the foundations of statistical mechanics,'' Nat. Phys. 2, 754 (2006).
https://doi.org/10.1038/nphys444

[15] Y. Li, ``Seminar 6 of selected topics in mathematical physics: Quantum information theory,'' http://www.mpmueller.net/seminar/talk6.pdf (2013).
http://www.mpmueller.net/seminar/talk6.pdf

[16] L. Masanes, A. J. Roncaglia, and A. Acín, ``Complexity of energy eigenstates as a mechanism for equilibration,'' Phys. Rev. E 87, 032137 (2013).
https://doi.org/10.1103/PhysRevE.87.032137

[17] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum circuit architecture,'' Phys. Rev. Lett. 101, 060401 (2008).
https://doi.org/10.1103/PhysRevLett.101.060401

[18] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Phys. Rev. A 80, 022339 (2009).
https://doi.org/10.1103/PhysRevA.80.022339

[19] F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New J. Phys. 18, 063032 (2016).
https://doi.org/10.1088/1367-2630/18/6/063032

[20] K. Modi, ``Operational approach to open dynamics and quantifying initial correlations,'' Sci. Rep. 2, 581 (2012).
https://doi.org/10.1038/srep00581

[21] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Non-Markovian quantum processes: Complete framework and efficient characterization,'' Phys. Rev. A 97, 012127 (2018b).
https://doi.org/10.1103/PhysRevA.97.012127

[22] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, ``Colloquium: Non-Markovian dynamics in open quantum systems,'' Rev. Mod. Phys. 88, 021002 (2016).
https://doi.org/10.1103/RevModPhys.88.021002

[23] M.-D. Choi, ``Completely positive linear maps on complex matrices,'' Linear Algebra Its Appl. 10, 285 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0

[24] A. Jamiołkowski, ``Linear transformations which preserve trace and positive semidefiniteness of operators,'' Rep. Math. Phys. 3, 275 (1972).
https://doi.org/10.1016/0034-4877(72)90011-0

[25] S. Milz, F. A. Pollock, and K. Modi, ``An introduction to operational quantum dynamics,'' Open Syst. Inf. Dyn. 24, 1740016 (2017a).
https://doi.org/10.1142/S1230161217400169

[26] A. J. Short and T. C. Farrelly, ``Quantum equilibration in finite time,'' New J. Phys. 14, 013063 (2012).
https://doi.org/10.1088/1367-2630/14/1/013063

[27] C. Gogolin, M. P. Müller, and J. Eisert, ``Absence of thermalization in nonintegrable systems,'' Phys. Rev. Lett. 106, 040401 (2011).
https://doi.org/10.1103/PhysRevLett.106.040401

[28] A. Gilchrist, N. K. Langford, and M. A. Nielsen, ``Distance measures to compare real and ideal quantum processes,'' Phys. Rev. A 71, 062310 (2005).
https://doi.org/10.1103/PhysRevA.71.062310

[29] L. Zhang, ``Matrix integrals over unitary groups: An application of Schur-Weyl duality,'' arXiv:1408.3782 (2014).
arXiv:1408.3782

[30] Y. Gu, Moments of Random Matrices and. Weingarten Functions, Master's thesis, Queen's University, Ontario, Canada (2013).
http://hdl.handle.net/1974/8241

[31] B. Collins and P. Śniady, ``Integration with respect to the Haar measure on unitary, orthogonal and symplectic group,'' Commun. Math. Phys. 264, 773 (2006).
https://doi.org/10.1007/s00220-006-1554-3

[32] Z. Puchała and J. A. Miszczak, ``Symbolic integration with respect to the Haar measure on the unitary groups,'' Bull. Pol. Acad. Sci. Tech. Sci. 65, 21 (2017).
https://doi.org/10.1515/bpasts-2017-0003

[33] D. A. Roberts and B. Yoshida, ``Chaos and complexity by design,'' J. High Energy Phys. 2017, 121 (2017).
https://doi.org/10.1007/JHEP04(2017)121

[34] F. Brandão, A. W. Harrow, and M. Horodecki, ``Local random quantum circuits are approximate polynomial-designs,'' Commun. Math. Phys. 346, 397 (2016).
https://doi.org/10.1007/s00220-016-2706-8

[35] T. Gläßle, ``Seminar 8 of selected topics in mathematical physics: Quantum information theory,'' http://www.mpmueller.net/seminar/talk8.pdf (2013).
http://www.mpmueller.net/seminar/talk8.pdf

[36] E. Lubkin, ``Entropy of an n-system from its correlation with a k-reservoir,'' J. Math. Phys. 19, 1028 (1978).
https://doi.org/10.1063/1.523763

[37] D. N. Page, ``Average entropy of a subsystem,'' Phys. Rev. Lett. 71, 1291 (1993).
https://doi.org/10.1103/PhysRevLett.71.1291

[38] S. Lloyd and H. Pagels, ``Complexity as thermodynamic depth,'' Ann. Phys. New York 188, 186 (1988).
https://doi.org/10.1016/0003-4916(88)90094-2

[39] A. J. Scott and C. M. Caves, ``Entangling power of the quantum baker's map,'' J. Phys. A: Math. Gen. 36, 9553 (2003).
https://doi.org/10.1088/0305-4470/36/36/308

[40] O. Giraud, ``Purity distribution for bipartite random pure states,'' J. Phys. A: Math. Theor. 40, F1053 (2007).
https://doi.org/10.1088/1751-8113/40/49/f03

[41] A. D. Pasquale, P. Facchi, V. Giovannetti, G. Parisi, S. Pascazio, and A. Scardicchio, ``Statistical distribution of the local purity in a large quantum system,'' J. Phys. A: Math. Theor. 45, 015308 (2012).
https://doi.org/10.1088/1751-8113/45/1/015308

[42] F. Mezzadri, ``How to generate random matrices from the classical compact groups,'' Notices of the AMS 54, 592 (2007).
https://www.ams.org/journals/notices/200705/fea-mezzadri-web.pdf

[43] J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, ``Chaos, complexity, and random matrices,'' J. High Energy Phys. 2017, 48 (2017).
https://doi.org/10.1007/JHEP11(2017)048

[44] R. A. Low, ``Large deviation bounds for k-designs,'' Proc. R. Soc. A 465, 3289 (2009).
https://doi.org/10.1098/rspa.2009.0232

[45] Y. Nakata, C. Hirche, M. Koashi, and A. Winter, ``Efficient quantum pseudorandomness with nearly time-independent Hamiltonian dynamics,'' Phys. Rev. X 7, 021006 (2017).
https://doi.org/10.1103/PhysRevX.7.021006

[46] S. Milz, F. Sakuldee, F. A. Pollock, and K. Modi, ``Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories,'' arXiv:1712.02589 (2017b).
arXiv:1712.02589

[47] F. Anza, C. Gogolin, and M. Huber, ``Eigenstate thermalization for degenerate observables,'' Phys. Rev. Lett. 120, 150603 (2018).
https://doi.org/10.1103/PhysRevLett.120.150603

[48] F. A. Pollock and K. Modi, ``Tomographically reconstructed master equations for any open quantum dynamics,'' Quantum 2, 76 (2018).
https://doi.org/10.22331/q-2018-07-11-76

[49] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, ``Nonperturbative treatment of non-Markovian dynamics of open quantum systems,'' Phys. Rev. Lett. 120, 030402 (2018).
https://doi.org/10.1103/PhysRevLett.120.030402

[50] I. A. Luchnikov, S. V. Vintskevich, H. Ouerdane, and S. N. Filippov, ``Simulation complexity of open quantum dynamics: Connection with tensor networks,'' Phys. Rev. Lett. 122, 160401 (2019).
https://doi.org/10.1103/PhysRevLett.122.160401

[51] A. Arias, A. Gheondea, and S. Gudder, ``Fixed points of quantum operations,'' J. Math. Phys. 43, 5872 (2002).
https://doi.org/10.1063/1.1519669

[52] M. Cramer, ``Thermalization under randomized local Hamiltonians,'' New J. Phys. 14, 053051 (2012).
https://doi.org/10.1088/1367-2630/14/5/053051

[53] D. Weingarten, ``Asymptotic behavior of group integrals in the limit of infinite rank,'' J. Math. Phys. 19, 999 (1978).
https://doi.org/10.1063/1.523807

[54] A. Ginory and J. Kim, ``Weingarten calculus and the IntHaar package for integrals over compact matrix groups,'' arXiv:1612.07641 (2016).
arXiv:1612.07641

[55] P. Taranto, K. Modi, and F. A. Pollock, ``Emergence of a fluctuation relation for heat in nonequilibrium Landauer processes,'' Phys. Rev. E 97, 052111 (2018).
https://doi.org/10.1103/PhysRevE.97.052111

[56] J. M. Epstein and K. B. Whaley, ``Quantum speed limits for quantum-information-processing tasks,'' Phys. Rev. A 95, 042314 (2017).
https://doi.org/10.1103/PhysRevA.95.042314

[57] M. Deza and E. Deza, Encyclopedia of Distances, Encyclopedia of Distances (Springer Berlin Heidelberg, 2009).
https://doi.org/10.1007/978-3-642-00234-2

[58] L. Zhang and H. Xiang, ``Average entropy of a subsystem over a global unitary orbit of a mixed bipartite state,'' Quantum Inf. Process. 16, 112 (2017).
https://doi.org/10.1007/s11128-017-1570-6

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