1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
3International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
4MU-NECTEC Collaborative Research Unit on Quantum Information, Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand.
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes — classical, quantum or beyond — that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to $all$ theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.
![](https://xlera8.com/wp-content/uploads/2020/06/kolmogorov-extension-theorem-for-quantum-causal-modelling-and-general-probabilistic-theories.png)
Popular summary
► BibTeX data
► References
[1] Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach (Springer, New York, 2009).
[2] M. Liao, Applied Stochastic Processes (Chapman and Hall/CRC, Boca Raton, 2013).
[3] A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Berlin, 1933) [Foundations of the Theory of Probability (Chelsea, New York, 1956)].
[4] W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971).
[5] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
[6] T. Tao, An Introduction to Measure Theory (American Mathematical Society, 2011).
[7] A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985).
https://doi.org/10.1103/PhysRevLett.54.857
[8] A. J. Leggett, Rep. Prog. Phys. 71, 022001 (2008).
https://doi.org/10.1088/0034-4885/71/2/022001
[9] C. Emary, N. Lambert, and F. Nori, Rep. Prog. Phys. 77, 016001 (2014).
https://doi.org/10.1088/0034-4885/77/1/016001
[10] N. Gilbert, Agent-Based Models (SAGE Publications Inc., Los Angeles, 2007).
[11] J. Pearl, Causality: Models, Reasoning and Inference (Cambridge University Press, Cambridge, U.K.; New York, 2009).
[12] S. Milz, F. A. Pollock, and K. Modi, Open Sys. Info. Dyn. , 1740016 (2017).
https://doi.org/10.1142/S1230161217400169
[13] F. Costa and S. Shrapnel, , 063032 (2016).
https://doi.org/10.1088%20%20New%20J.%20Phys.%20textbf%2018
[14] O. Oreshkov and C. Giarmatzi, New J. Phys. 18, 093020 (2016).
https://doi.org/10.1088/1367-2630/18/9/093020
[15] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, Phys. Rev. X 7, 031021 (2017).
https://doi.org/10.1103/PhysRevX.7.031021
[16] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016).
https://doi.org/10.1103/RevModPhys.88.021002
[17] A. Smirne, D. Egloff, M. G. Díaz, M. B. Plenio, and S. F. Huelga, Quantum Sci. Technol. 4, 01LT01 (2018).
https://doi.org/10.1088/2058-9565/aaebd5
[18] L. Accardi, A. Frigerio, and J. T. Lewis, Publ. Rest. Inst. Math. Sci. 18, 97 (1982).
https://doi.org/10.2977/prims/1195184017
[19] N. Wiener, A. Siegel, B. Rankin, and W. T. Martin, eds., Differential Space, Quantum Systems, and Prediction (The MIT Press, Cambridge (MA), 1966).
[20] M. P. Lévy, Am. J. Math. 62, 487 (1940).
https://doi.org/10.2307/2371467
[21] Z. Ciesielski, Lectures on Brownian motion, heat conduction and potential theory (Aarhus Universitet, Mathematisk Institutt, 1966).
[22] R. Bhattacharya and E. C. Waymire, A Basic Course in Probability Theory (Springer, New York, NY, 2017).
[23] S. Shrapnel and F. Costa, Quantum 2, 63 (2018).
https://doi.org/10.22331/q-2018-05-18-63
[24] G. Lindblad, Comm. Math. Phys. 65, 281 (1979).
https://doi.org/10.1007/BF01197883
[25] E. B. Davis, Quantum Theory of Open Systems (Academic Press Inc, London; New York, 1976).
[26] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. A 81, 062348 (2010).
https://doi.org/10.1103/PhysRevA.81.062348
[27] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 180501 (2008a).
https://doi.org/10.1103/PhysRevLett.101.180501
[28] S. Shrapnel, F. Costa, and G. Milburn, New J. Phys. 20, 053010 (2018).
https://doi.org/10.1088/1367-2630/aabe12
[29] K. Modi, Sci. Rep. 2, 581 (2012).
https://doi.org/10.1038/srep00581
[30] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. A 97, 012127 (2018a).
https://doi.org/10.1103/PhysRevA.97.012127
[31] S. Milz, F. A. Pollock, and K. Modi, Phys. Rev. A 98, 012108 (2018a).
https://doi.org/10.1103/PhysRevA.98.012108
[32] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. A 80, 022339 (2009).
https://doi.org/10.1103/PhysRevA.80.022339
[33] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, Phys. Rev. A 88, 022318 (2013).
https://doi.org/10.1103/PhysRevA.88.022318
[34] O. Oreshkov, F. Costa, and Č. Brukner, Nat. Commun. 3, 1092 (2012).
https://doi.org/10.1038/ncomms2076
[35] G. M. D’Ariano, G. Chiribella, and P. Perinotti, Quantum Theory from First Principles: An Informational Approach, 1st ed. (Cambridge University Press, Cambridge, United Kingdom ; New York, NY, 2017).
[36] G. Chiribella and D. Ebler, New J. Phys. 18, 093053 (2016).
https://doi.org/10.1088/1367-2630/18/9/093053
[37] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Europhys. Lett. 83, 30004 (2008b).
https://doi.org/10.1209/0295-5075/83/30004
[38] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 060401 (2008c).
https://doi.org/10.1103/PhysRevLett.101.060401
[39] T. Tyc and J. Vlach, Eur. Phys. J. D 69, 209 (2015).
https://doi.org/10.1140/epjd/e2015-60191-7
[40] J. v. Neumann, Comp. Math. 6, 1 (1939).
https://eudml.org/doc/88704
[41] E. Kreyszig, Introductory Functional Analysis with Applications, 1st ed. (Wiley, New York, 1989).
[42] R. Haag and D. Kastler, J. Math. Phys. 5, 848 (1964).
https://doi.org/10.1063/1.1704187
[43] D. Kretschmann and R. F. Werner, Phys. Rev. A 72, 062323 (2005).
https://doi.org/10.1103/PhysRevA.72.062323
[44] C. M. Caves, C. A. Fuchs, K. K. Manne, and J. M. Renes, Found. Phys. 34, 193 (2004).
https://doi.org/10.1023/B:FOOP.0000019581.00318.a5
[45] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, New J. Phys. 17, 102001 (2015).
https://doi.org/10.1088/1367-2630/17/10/102001
[46] R. W. Spekkens, Phys. Rev. A 71, 052108 (2005).
https://doi.org/10.1103/PhysRevA.71.052108
[47] E. G. Cavalcanti, Phys. Rev. X 8, 021018 (2018).
https://doi.org/10.1103/PhysRevX.8.021018
[48] P. Strasberg and M. G. Díaz, Phys. Rev. A 100, 022120 (2019).
https://doi.org/10.1103/PhysRevA.100.022120
[49] S. Milz, D. Egloff, P. Taranto, T. Theurer, M. B. Plenio, A. Smirne, and S. F. Huelga, arXiv:1907.05807 (2019).
arXiv:1907.05807
[50] M.-D. Choi, Linear Algebra Appl. 10, 285 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0
[51] A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972).
https://doi.org/10.1016/0034-4877(72)90011-0
[52] F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, Rev. Mod. Phys. 86, 1203 (2014).
https://doi.org/10.1103/RevModPhys.86.1203
[53] C. Portmann, C. Matt, U. Maurer, R. Renner, and B. Tackmann, IEEE Trans. Inf. Theory 63, 3277 (2017).
https://doi.org/10.1109/TIT.2017.2676805
[54] L. Hardy, arXiv:1608.06940 (2016).
arXiv:1608.06940
[55] L. Hardy, Phil. Trans. R. Soc. A 370, 3385 (2012).
https://doi.org/10.1098/rsta.2011.0326
[56] J. Cotler, C.-M. Jian, X.-L. Qi, and F. Wilczek, J. High Energy Phys. 2018, 93 (2018).
https://doi.org/10.1007/JHEP09(2018)093
[57] N. Barnett and J. P. Crutchfield, J. Stat. Phys. 161, 404 (2015).
https://doi.org/10.1007/s10955-015-1327-5
[58] J. Thompson, A. J. P. Garner, V. Vedral, and M. Gu, npj Quantum Inf. 3, 6 (2017).
https://doi.org/10.1038/s41534-016-0001-3
[59] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. Lett. 120, 040405 (2018b).
https://doi.org/10.1103/PhysRevLett.120.040405
[60] R. Tumulka, Lett. Math. Phys. 84, 41 (2008).
https://doi.org/10.1007/s11005-008-0229-8
[61] E. Haapasalo, T. Heinosaari, and Y. Kuramochi, J. Phys. A 49, 33LT01 (2016).
https://doi.org/10.1088/1751-8113/49/33/33LT01
[62] S. Milz, F. A. Pollock, T. P. Le, G. Chiribella, and K. Modi, New J. Phys. 20, 033033 (2018b).
https://doi.org/10.1088/1367-2630/aaafee
Cited by
[1] Philipp Strasberg, “Operational approach to quantum stochastic thermodynamics”, Physical Review E 100 2, 022127 (2019).
[2] A. Smirne, D. Egloff, M. G. Díaz, M. B. Plenio, and S. F. Huelga, “Coherence and non-classicality of quantum Markov processes”, Quantum Science and Technology 4 1, 01LT01 (2019).
[3] Jacques Pienaar, “Quantum causal models via QBism”, arXiv:1806.00895.
[4] Philipp Strasberg and Andreas Winter, “Stochastic thermodynamics with arbitrary interventions”, Physical Review E 100 2, 022135 (2019).
[5] Philip Taranto, Felix A. Pollock, Simon Milz, Marco Tomamichel, and Kavan Modi, “Quantum Markov Order”, Physical Review Letters 122 14, 140401 (2019).
[6] 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).
[7] Philipp Strasberg and María García Díaz, “Classical quantum stochastic processes”, Physical Review A 100 2, 022120 (2019).
[8] 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).
[9] Joshua Morris, Felix A. Pollock, and Kavan Modi, “Non-Markovian memory in IBMQX4”, arXiv:1902.07980.
[10] 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).
[11] Simon Milz, Felix A. Pollock, and Kavan Modi, “Reconstructing non-Markovian quantum dynamics with limited control”, Physical Review A 98 1, 012108 (2018).
[12] Hong-Bin Chen, Ping-Yuan Lo, Clemens Gneiting, Joonwoo Bae, Yueh-Nan Chen, and Franco Nori, “Quantifying the nonclassicality of pure dephasing”, Nature Communications 10, 3794 (2019).
[13] Simon Milz, Felix A. Pollock, and Kavan Modi, “Reconstructing open quantum system dynamics with limited control”, arXiv:1610.02152.
[14] Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, “Equilibration on average of temporally non-local observables in quantum systems”, arXiv:1905.08469.
[15] Philipp Strasberg, “Repeated Interactions and Quantum Stochastic Thermodynamics at Strong Coupling”, Physical Review Letters 123 18, 180604 (2019).
[16] Simon Milz, Dario Egloff, Philip Taranto, Thomas Theurer, Martin B. Plenio, Andrea Smirne, and Susana F. Huelga, “When is a non-Markovian quantum process classical?”, arXiv:1907.05807.
[17] Fattah Sakuldee, Simon Milz, Felix A. Pollock, and Kavan Modi, “Non-Markovian quantum control as coherent stochastic trajectories”, Journal of Physics A Mathematical General 51 41, 414014 (2018).
[18] Philip Taranto, Felix A. Pollock, and Kavan Modi, “Memory Strength and Recoverability of Non-Markovian Quantum Stochastic Processes”, arXiv:1907.12583.
[19] 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”, arXiv:1907.07003.
[20] Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, “Almost Markovian processes from closed dynamics”, arXiv:1802.10344.
[21] Andrea Smirne, Thomas Nitsche, Dario Egloff, Sonja Barkhofen, Syamsundar De, Ish Dhand, Christine Silberhorn, Susana F. Huelga, and Martin B. Plenio, “Experimental Control of the Degree of Non-Classicality via Quantum Coherence”, arXiv:1910.11830.
[22] Matheus Capela, Lucas C. Céleri, Kavan Modi, and Rafael Chaves, “Monogamy of temporal correlations: Witnessing non-Markovianity beyond data processing”, Physical Review Research 2 1, 013350 (2020).
[23] Jacques Pienaar, “Quantum causal models via quantum Bayesianism”, Physical Review A 101 1, 012104 (2020).
[24] Gregory A. L. White, Charles D. Hill, Felix A. Pollock, Lloyd C. L. Hollenberg, and Kavan Modi, “Experimental non-Markovian process characterisation and control on a quantum processor”, arXiv:2004.14018.
[25] Philip Taranto, “Memory effects in quantum processes”, International Journal of Quantum Information 18 2, 1941002-574 (2020).
[26] Kavan Modi, “George Sudarshan and Quantum Dynamics”, Open Systems and Information Dynamics 26 3, 1950013 (2019).
[27] Pedro Figueroa-Romero, Felix A. Pollock, and Kavan Modi, “Markovianization by design”, arXiv:2004.07620.
[28] Philipp Strasberg, “Thermodynamics of Quantum Causal Models: An Inclusive, Hamiltonian Approach”, Quantum 4, 240 (2020).
The above citations are from SAO/NASA ADS (last updated successfully 2020-06-03 19:21:59). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref’s cited-by service no data on citing works was found (last attempt 2020-06-03 19:21:57).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
Source: https://quantum-journal.org/papers/q-2020-04-20-255/