Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories

Simon Milz1,2, Fattah Sakuldee3,4, Felix A. Pollock2, and Kavan Modi2

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.

While theories of general (quantum) processes with interventions have attracted considerable interest over the last decades, their axiomatic underpinnings are still opaque. In the classical case, the Kolmogorov extension theorem (KET) establishes the basic properties of stochastic processes; however, this theorem breaks down when interventions are allowed. For quantum processes, interventions are unavoidable. The present work closes this conceptual gap by providing a generalised version of the KET. Our generalised theorem lays the theoretical foundation for the description of all processes with interventions, be they classical, quantum or beyond. Two prominent and timely examples are the theories of quantum stochastic processes and quantum causal modelling. Our results have direct consequences for the characterisation and modelling of causal structure in stochastic processes throughout the quantitative sciences, and, notably, allows for a complete representation of arbitrary controlled quantum systems.

► 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).

[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] 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).

[2] Philipp Strasberg, Andreas Winter, Jochen Gemmer, and Jiaozi Wang, "Classicality, Markovianity, and local detailed balance from pure-state dynamics", Physical Review A 108 1, 012225 (2023).

[3] 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).

[4] Pedro Figueroa-Romero, Kavan Modi, Robert J. Harris, Thomas M. Stace, and Min-Hsiu Hsieh, "Randomized Benchmarking for Non-Markovian Noise", PRX Quantum 2 4, 040351 (2021).

[5] Pedro Figueroa-Romero, Kavan Modi, and Min-Hsiu Hsieh, "Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise", Quantum 6, 868 (2022).

[6] Christina Giarmatzi and Fabio Costa, "Witnessing quantum memory in non-Markovian processes", Quantum 5, 440 (2021).

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

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

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

[11] Yu Guo, Philip Taranto, Bi-Heng Liu, Xiao-Min Hu, Yun-Feng Huang, Chuan-Feng Li, and Guang-Can Guo, "Experimental Demonstration of Instrument-Specific Quantum Memory Effects and Non-Markovian Process Recovery for Common-Cause Processes", Physical Review Letters 126 23, 230401 (2021).

[12] G. A. L. White, C. D. Hill, F. A. Pollock, L. C. L. Hollenberg, and K. Modi, "Demonstration of non-Markovian process characterisation and control on a quantum processor", Nature Communications 11 1, 6301 (2020).

[13] Simon Milz, Cornelia Spee, Zhen-Peng Xu, Felix Pollock, Kavan Modi, and Otfried Gühne, "Genuine multipartite entanglement in time", SciPost Physics 10 6, 141 (2021).

[14] Neil Dowling, Pedro Figueroa-Romero, Felix A. Pollock, Philipp Strasberg, and Kavan Modi, "Relaxation of Multitime Statistics in Quantum Systems", Quantum 7, 1027 (2023).

[15] Neil Dowling, Pedro Figueroa-Romero, Felix A. Pollock, Philipp Strasberg, and Kavan Modi, "Equilibration of multitime quantum processes in finite time intervals", SciPost Physics Core 6 2, 043 (2023).

[16] Fattah Sakuldee and Łukasz Cywiński, "Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system", Quantum Information Processing 21 7, 244 (2022).

[17] Moritz F. Richter, Andrea Smirne, Walter T. Strunz, and Dario Egloff, "Classical Invasive Description of Informationally‐Complete Quantum Processes", Annalen der Physik 2300304 (2023).

[18] Cristhiano Duarte, Lorenzo Catani, and Raphael C. Drumond, "Relating Compatibility and Divisibility of Quantum Channels", International Journal of Theoretical Physics 61 7, 189 (2022).

[19] Paolo Perinotti, "Causal influence in operational probabilistic theories", Quantum 5, 515 (2021).

[20] Matheus Capela, Harshit Verma, Fabio Costa, and Lucas C. Céleri, "Reassessing thermodynamic advantage from indefinite causal order", Physical Review A 107 6, 062208 (2023).

[21] 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).

[22] Fattah Sakuldee, Philip Taranto, and Simon Milz, "Connecting commutativity and classicality for multitime quantum processes", Physical Review A 106 2, 022416 (2022).

[23] Yun-Hua Kuo and Hong-Bin Chen, "Adaptively partitioned analog quantum simulation on near-term quantum computers: The nonclassical free-induction decay of NV centers in diamond", Physical Review Research 5 4, 043139 (2023).

[24] Neil Dowling and Kavan Modi, "Operational Metric for Quantum Chaos and the Corresponding Spatiotemporal-Entanglement Structure", PRX Quantum 5 1, 010314 (2024).

[25] Philipp Strasberg, "Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach", SciPost Physics 15 1, 024 (2023).

[26] Andrea Smirne, Nina Megier, and Bassano Vacchini, "On the connection between microscopic description and memory effects in open quantum system dynamics", Quantum 5, 439 (2021).

[27] Joshua Morris, Felix A. Pollock, and Kavan Modi, "Quantifying non-Markovian Memory in a Superconducting Quantum Computer", Open Systems & Information Dynamics 29 02, 2250007 (2022).

[28] Philip Taranto, Thomas J. Elliott, and Simon Milz, "Hidden Quantum Memory: Is Memory There When Somebody Looks?", Quantum 7, 991 (2023).

[29] A Smirne, T Nitsche, D Egloff, S Barkhofen, S De, I Dhand, C Silberhorn, S F Huelga, and M B Plenio, "Experimental control of the degree of non-classicality via quantum coherence", Quantum Science and Technology 5 4, 04LT01 (2020).

[30] Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, "Equilibration on average in quantum processes with finite temporal resolution", Physical Review E 102 3, 032144 (2020).

[31] Daniel Burgarth, Paolo Facchi, Davide Lonigro, and Kavan Modi, "Quantum non-Markovianity elusive to interventions", Physical Review A 104 5, L050404 (2021).

[32] Andreas Albrecht, Rose Baunach, and Andrew Arrasmith, "Einselection, equilibrium, and cosmology", Physical Review D 106 12, 123507 (2022).

[33] 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).

[34] 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?", Physical Review X 10 4, 041049 (2020).

[35] Matheus Capela, Lucas C. Céleri, Rafael Chaves, and Kavan Modi, "Quantum Markov monogamy inequalities", Physical Review A 106 2, 022218 (2022).

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

[37] Hendra I. Nurdin and John Gough, 2021 60th IEEE Conference on Decision and Control (CDC) 4164 (2021) ISBN:978-1-6654-3659-5.

[38] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Quantum Causal Models", arXiv:1906.10726, (2019).

[39] 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).

[40] 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).

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

[42] Philipp Strasberg, "Operational approach to quantum stochastic thermodynamics", Physical Review E 100 2, 022127 (2019).

[43] Philipp Strasberg, "Repeated Interactions and Quantum Stochastic Thermodynamics at Strong Coupling", Physical Review Letters 123 18, 180604 (2019).

[44] 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).

[45] 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).

[46] Philipp Strasberg and María García Díaz, "Classical quantum stochastic processes", Physical Review A 100 2, 022120 (2019).

[47] Gregory A. L. White, Felix A. Pollock, Lloyd C. L. Hollenberg, Charles D. Hill, and Kavan Modi, "From many-body to many-time physics", arXiv:2107.13934, (2021).

[48] 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).

[49] Simon Milz, Felix A. Pollock, and Kavan Modi, "Reconstructing non-Markovian quantum dynamics with limited control", Physical Review A 98 1, 012108 (2018).

[50] Philipp Strasberg and Andreas Winter, "Stochastic thermodynamics with arbitrary interventions", Physical Review E 100 2, 022135 (2019).

[51] 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, (2019).

[52] 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).

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

[54] Philipp Strasberg, "Thermodynamics of Quantum Causal Models: An Inclusive, Hamiltonian Approach", Quantum 4, 240 (2020).

[55] Philip Taranto, "Memory effects in quantum processes", International Journal of Quantum Information 18 2, 1941002-574 (2020).

[56] Jacques Pienaar, "Quantum causal models via quantum Bayesianism", Physical Review A 101 1, 012104 (2020).

[57] Simon Milz, Felix A. Pollock, and Kavan Modi, "Reconstructing open quantum system dynamics with limited control", arXiv:1610.02152, (2016).

[58] Gregory A. L. White, Petar Jurcevic, Charles D. Hill, and Kavan Modi, "Unifying non-Markovian characterisation with an efficient and self-consistent framework", arXiv:2312.08454, (2023).

[59] Jacques Pienaar, "Quantum causal models via QBism", arXiv:1806.00895, (2018).

[60] Kavan Modi, "George Sudarshan and Quantum Dynamics", Open Systems and Information Dynamics 26 3, 1950013 (2019).

[61] 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).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-29 04:35:56) and SAO/NASA ADS (last updated successfully 2024-03-29 04:35:57). The list may be incomplete as not all publishers provide suitable and complete citation data.