The arrow of time in operational formulations of quantum theory

Andrea Di Biagio1, Pietro Donà2, and Carlo Rovelli2,3,4

1Dipartimento di Fisica, Sapienza Università di Roma, 00185 Roma, Italy
2Aix-Marseille Université, Université de Toulon, CNRS, CPT, 13288 Marseille, France
3Perimeter Institute, 31 Caroline Street North, Waterloo Ontario N2L2Y5, Canada
4The Rotman Institute of Philosophy, 1151 Richmond St. N, London Ontario N6A5B7, Canada

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


The operational formulations of quantum theory are drastically time oriented. However, to the best of our knowledge, microscopic physics is time-symmetric. We address this tension by showing that the asymmetry of the operational formulations does not reflect a fundamental time-orientation of physics. Instead, it stems from built-in assumptions about the $users$ of the theory. In particular, these formalisms are designed for predicting the future based on information about the past, and the main mathematical objects contain implicit assumption about the past, but not about the future. The main asymmetry in quantum theory is the difference between knowns and unknowns.

► BibTeX data

► References

[1] Huw Price. ``Time's arrow & Archimedes' point: New directions for the physics of time''. Oxford Paperbacks. Oxford University Press (1997). DOI: 10.1093/​acprof:oso/​9780195117981.001.0001.

[2] Carlo Rovelli. ``Memory and entropy'' (2020) eprint: arXiv:2003.06687.

[3] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge University Press (2000). DOI: 10.1017/​CBO9780511976667.

[4] Giacomo Mauro D'Ariano, Giulio Chiribella, and Paolo Perinotti. ``Quantum Theory from First Principles: An Informational Approach''. Cambridge University PressCambridge (2017). DOI: 10.1017/​9781107338340.

[5] Bob Coecke and Aleks Kissinger. ``Picturing quantum processes''. Cambridge University Press (2017). DOI: 10.1017/​9781316219317.

[6] J. S. Bell. ``On the Einstein Podolsky Rosen paradox''. Physics Physique Fizika 1, 195–200 (1964). DOI: 10.1103/​physicsphysiquefizika.1.195.

[7] John S. Bell. ``On the Problem of Hidden Variables in Quantum Mechanics''. Reviews of Modern Physics 38, 447–452 (1966). DOI: 10.1103/​revmodphys.38.447.

[8] W K Wootters and W H Zurek. ``A single quantum cannot be cloned''. Nature 299, 802–803 (1982). DOI: 10.1038/​299802a0.

[9] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Journal on Computing 26, 1484–1509 (1997). DOI: 10.1137/​s0097539795293172. eprint: arXiv:quant-ph/​9508027.

[10] Artur K Ekert. ``Quantum cryptography based on Bell’s theorem''. Physical Review Letters 67, 661–663 (1991). DOI: 10.1103/​PhysRevLett.67.661.

[11] Lov K. Grover. ``A fast quantum mechanical algorithm for database search''. Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC ’96 (1996). DOI: 10.1145/​237814.237866. eprint: arXiv:quant-ph/​9605043.

[12] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. Theoretical Computer Science 560, 7–11 (2014). DOI: 10.1016/​j.tcs.2014.05.025. eprint: arXiv:2003.06557.

[13] D Dieks. ``Communication by EPR devices''. Physics Letters A 92, 271–272 (1982). DOI: 10.1016/​0375-9601(82)90084-6.

[14] Charles H Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K Wootters. ``Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels''. Physical Review Letters 70, 1895–1899 (1993). DOI: 10.1103/​PhysRevLett.70.1895.

[15] A.S. Holevo. ``The capacity of the quantum channel with general signal states''. IEEE Transactions on Information Theory 44, 269–273 (1998). DOI: 10.1109/​18.651037. eprint: arXiv:quant-ph/​9611023.

[16] G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron. ``Quantum computations without definite causal structure''. Physical Review A 88, 022318 (2013). DOI: 10.1103/​physreva.88.022318. eprint: arXiv:0912.0195.

[17] Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner. ``Quantum mechanics and the covariance of physical laws in quantum reference frames''. Nature Communications 10, 494 (2019). DOI: 10.1038/​s41467-018-08155-0. eprint: arXiv:1712.07207.

[18] Sougato Bose, Anupam Mazumdar, Gavin W. Morley, Hendrik Ulbricht, Marko Toroš, Mauro Paternostro, Andrew Geraci, Peter Barker, M. S. Kim, and Gerard Milburn. ``A Spin Entanglement Witness for Quantum Gravity''. Physical Review Letters 119, 240401 (2017). DOI: 10.1103/​physrevlett.119.240401. eprint: arXiv:1707.06050.

[19] Chiara Marletto and Vlatko Vedral. ``Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity''. Physical Review Letters 119, 240402 (2017). DOI: 10.1103/​physrevlett.119.240402. eprint: arXiv:1707.06036.

[20] Kok-Wei Bong, Aníbal Utreras-Alarcón, Farzad Ghafari, Yeong-Cherng Liang, Nora Tischler, Eric G. Cavalcanti, Geoff J. Pryde, and Howard M. Wiseman. ``A strong no-go theorem on the Wigner's friend paradox''. Nature PhysicsPages 1–7 (2020). DOI: 10.1038/​s41567-020-0990-x.

[21] Lucien Hardy. ``Quantum Theory From Five Reasonable Axioms'' (2001) eprint: arXiv:quant-ph/​0101012.

[22] Borivoje Dakic and Časlav Brukner. ``Quantum theory and beyond: Is entanglement special?'' (2009) eprint: arXiv:0911.0695.

[23] Lluís Masanes and Markus P. Müller. ``A derivation of quantum theory from physical requirements''. New Journal of Physics 13, 063001 (2011). DOI: 10.1088/​1367-2630/​13/​6/​063001.

[24] G. Chiribella, G. M. D'Ariano, and P. Perinotti. ``Informational derivation of Quantum Theory''. Physical Review A 84, 012311 (2011). DOI: 10.1103/​physreva.84.012311. eprint: arXiv:1011.6451.

[25] Lucien Hardy. ``Reconstructing quantum theory'' (2013) eprint: arXiv:1303.1538.

[26] Philipp A. Höhn. ``Toolbox for reconstructing quantum theory from rules on information acquisition''. Quantum 1, 38 (2017). DOI: 10.22331/​q-2017-12-14-38. eprint: arXiv:1412.8323.

[27] Philipp A. Höhn and Christopher Wever. ``Quantum theory from questions''. Physical Review A 95, 012102 (2017). DOI: 10.1103/​PhysRevA.95.012102. eprint: arXiv:1511.01130.

[28] John H. Selby, Carlo Maria Scandolo, and Bob Coecke. ``Reconstructing quantum theory from diagrammatic postulates''. Quantum 5, 445 (2021). DOI: 10.22331/​q-2021-04-28-445. eprint: arXiv:1802.00367.

[29] Ding Jia. ``Quantum from principles without assuming definite causal structure''. Physical Review A 98, 032112 (2018). DOI: 10.1103/​physreva.98.032112. eprint: arXiv:1808.00898.

[30] Robert Oeckl. ``A local and operational framework for the foundations of physics''. Advances in Theoretical and Mathematical Physics 23, 437–592 (2019). DOI: 10.4310/​ATMP.2019.v23.n2.a4. eprint: arXiv:1610.09052.

[31] Carlo Rovelli. ``Relational Quantum Mechanics''. International Journal of Theoretical Physics 35, 1637–1678 (1996). DOI: 10.1007/​BF02302261. eprint: arXiv:quant-ph/​9609002.

[32] Christopher A. Fuchs. ``Quantum Mechanics as Quantum Information (and only a little more)''. eprint: arXiv:quant-ph/​0205039.

[33] ``The Quantum Information Structure of Spacetime''. http:/​/​​.

[34] Andrea Di Biagio. ``Can we think time-symmetrically about causation?''. http:/​/​​20120022 (2020). PIRSA:20120022.

[35] David Schmid, John H. Selby, and Robert W. Spekkens. ``Unscrambling the omelette of causation and inference: The framework of causal-inferential theories'' (2020) eprint: arXiv:2009.03297.

[36] Lucien Hardy. ``Time Symmetry in Operational Theories'' (2021) eprint: arXiv:2104.00071.

[37] Albert Einstein, Richard C Tolman, and Boris Podolsky. ``Knowledge of Past and Future in Quantum Mechanics''. Physical Review 37, 780–781 (1931). DOI: 10.1103/​PhysRev.37.780.

[38] Yakir Aharonov, Peter G Bergmann, and Joel L Lebowitz. ``Time Symmetry in the Quantum Process of Measurement''. Physical Review 134, B1410–B1416 (1964). DOI: 10.1103/​PhysRev.134.B1410.

[39] Satosi Watanabe. ``Symmetry of physical laws. Part III. Prediction and retrodiction''. Reviews of Modern Physics 27, 179–186 (1955). DOI: 10.1103/​RevModPhys.27.179.

[40] Stephen M. Barnett, David T. Pegg, and John Jeffers. ``Bayes' theorem and quantum retrodiction''. Journal of Modern Optics 47, 1779–1789 (2000). DOI: 10.1080/​09500340008232431. eprint: arXiv:quant-ph/​0106139.

[41] David T. Pegg, Stephen M. Barnett, and John Jeffers. ``Quantum retrodiction in open systems''. Physical Review A 66, 022106 (2002). DOI: 10.1103/​physreva.66.022106. eprint: arXiv:quant-ph/​0208082.

[42] Fiona C. Speirits, Matthias Sonnleitner, and Stephen M. Barnett. ``From retrodiction to Bayesian quantum imaging''. Journal of Optics 19, 044001 (2017). DOI: 10.1088/​2040-8986/​aa5ccf.

[43] Matthew Leifer and Matthew Pusey. ``Is a time symmetric interpretation of quantum theory possible without retrocausality?''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, 20160607 (2017). DOI: 10.1103/​physreva.88.052130. eprint: arXiv:1607.07871.

[44] Ognyan Oreshkov and Nicolas J Cerf. ``Operational formulation of time reversal in quantum theory''. Nature PhysicsPages 853–858 (2015). DOI: 10.1038/​nphys3414. eprint: arXiv:1507.07745.

[45] Robert Oeckl. ``What is Quantum Theory?, QISS Virtual Seminar''. https:/​/​​watch?v=_Fwkh9cBMZw&t=5095s (2020).

[46] Edwin Thompson Jaynes. ``Probability in Quantum Theory''. In Wojciech Hubert Zurek, editor, Complexity, Entropy And The Physics Of Information Page 381. Addison-Wesley Pub. Co (1990).

[47] Christopher A. Fuchs and Blake C. Stacey. ``QBism: Quantum theory as a hero's handbook'' (2016) eprint: arXiv:1612.07308.

[48] M. S. Leifer and R. W. Spekkens. ``Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference''. Physical Review A 88, 052130 (2013). DOI: 10.1103/​physreva.88.052130. eprint: arXiv:1107.5849.

[49] Stephen M. Barnett. ``Quantum Retrodiction''. In Erika Andersson and Patrik Öhberg, editors, Quantum Information and Coherence Pages 1–30. Springer International Publishing (2014).

[50] Dov Fields, Abdelali Sajia, and János A. Bergou. ``Quantum retrodiction made fully symmetric'' (2020) eprint: arXiv:2006.15692.

[51] Yuan Yuan, Zhibo Hou, Kang-Da Wu, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo. ``Experimental retrodiction of trajectories of single photons in double interferometers''. Physical Review A 97, 062115 (2018). DOI: 10.1103/​PhysRevA.97.062115.

[52] Massimiliano Rossi, David Mason, Junxin Chen, and Albert Schliesser. ``Observing and Verifying the Quantum Trajectory of a Mechanical Resonator''. Physical Review Letters 123, 163601 (2019). DOI: 10.1103/​PhysRevLett.123.163601.

[53] Carlo Rovelli. ``Agency in physics''. eprint: arXiv:2007.05300.

[54] W. Forrest Stinespring. ``Positive functions on C$^*$-algebras''. Proceedings of the American Mathematical Society 6, 211–216 (1955). DOI: 10.2307/​2032342.

[55] Masanao Ozawa. ``Quantum measuring processes of continuous observables''. Journal of Mathematical Physics 25 (1984). DOI: 10.1063/​1.526000.

[56] L. J. Landau and R. F. Streater. ``On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras''. Linear Algebra and its Applications 193, 107–127 (1993). DOI: 10.1016/​0024-3795(93)90274-r.

[57] Christian B. Mendl and Michael M. Wolf. ``Unital Quantum Channels – Convex Structure and Revivals of Birkhoff's Theorem''. Communications in Mathematical Physics 289, 1057–1086 (2009). DOI: 10.1007/​s00220-009-0824-2. eprint: arXiv:0806.2820.

[58] Giulio Chiribella and Carlo Maria Scandolo. ``Microcanonical thermodynamics in general physical theories''. New Journal of Physics 19, 123043 (2017). DOI: 10.1088/​1367-2630/​aa91c7. eprint: arXiv:1608.04460.

[59] Alexander Streltsov, Hermann Kampermann, Sabine Wölk, Manuel Gessner, and Dagmar Bruß. ``Maximal Coherence and the Resource Theory of Purity''. New Journal of Physics 20, 053058 (2018). DOI: 10.1088/​1367-2630/​aac484. eprint: arXiv:1612.07570.

[60] Peter Shor. ``The structure of unital maps and the asymptotic quantum Birkhoff conjecture''. https:/​/​​bA9Pz0AKycY?t=824 (2010).

[61] Giulio Chiribella, Erik Aurell, and Karol Życzkowski. ``Symmetries of quantum evolutions''. Physical Review Research 3, 033028 (2021). DOI: 10.1103/​PhysRevResearch.3.033028. eprint: arXiv:2101.04962.

[62] Giulio Chiribella and Zixuan Liu. ``Quantum operations with indefinite time direction''. eprint: arXiv:2012.03859.

[63] Jonathan Barrett. ``Information processing in generalized probabilistic theories''. Physical Review A 75, 032304 (2007). DOI: 10.1103/​physreva.75.032304. eprint: arXiv:quant-ph/​0508211.

[64] Bob Coecke, Stefano Gogioso, and John H. Selby. ``The time-reverse of any causal theory is eternal noise'' (2017) eprint: arXiv:1711.05511.

[65] Asher Peres. ``Quantum theory: concepts and methods''. Volume 57. Springer Science & Business Media (2002). DOI: 10.1007/​0-306-47120-5.

[66] G. J. Milburn and S. Shrapnel. ``Physical grounds for causal perspectivalism'' (2020) eprint: arXiv:2009.04121.

[67] Rolf Landauer. ``Irreversibility and Heat Generation in the Computing Process''. IBM Journal of Research and Development 5, 261–269 (1961). DOI: 10.1147/​rd.53.0183.

[68] Wayne C. Myrvold. ``Shakin' All Over: Proving Landauer's Principle without neglect of fluctuations'' (2020) eprint: arXiv:2007.11748.

[69] David Bohm. ``A Suggested Interpretation of the Quantum Theory in Terms of ``Hidden'' Variables. II''. Physical Review 85, 180–193 (1952). DOI: 10.1103/​physrev.85.180.

[70] Sheldon Goldstein. ``Bohmian Mechanics''. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford Universitysummer 2017 edition (2017).

[71] Simon Saunders, Jonathan Barrett, Adrian Kent, and David Wallace, editors. ``Many Worlds? Everett, Quantum Theory, and Reality''. Oxford University PressOxford, England (2010). DOI: 10.1093/​acprof:oso/​9780199560561.001.0001.

[72] Lev Vaidman. ``Many-Worlds Interpretation of Quantum Mechanics''. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford Universityfall 2018 edition (2018).

[73] Carlo Rovelli. ``An argument against the realistic interpretation of the wave function''. Foundations of Physics 46, 1229–1237 (2016). DOI: 10.1007/​s10701-016-0032-9. eprint: arXiv:1508.05533.

[74] Christopher A. Fuchs. ``On Participatory Realism'' (2016) eprint: arXiv:1601.04360. to appear in ``Information & Interaction: Eddington, Wheeler, and the Limits of Knowledge'', edited by Ian T. Durham and Dean Rickles.

[75] Carlo Rovelli. ``Space is blue and birds fly through it''. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170312 (2018). DOI: 10.1098/​rsta.2017.0312. eprint: arXiv:1712.02894.

[76] Andrea Di Biagio and Carlo Rovelli. ``Stable Facts, Relative Facts''. Foundations of Physics 51, 30 (2021). DOI: 10.1007/​s10701-021-00429-w. eprint: arXiv:2006.15543.

[77] Wayne C. Myrvold. ``The Science of $\Theta\Delta^{cs}$''. Foundations of Physics 50, 1219–1251 (2020). DOI: 10.1007/​s10701-020-00371-3. eprint: arXiv:2007.11729.

[78] Leonard Mlodinow and Todd A. Brun. ``Relation between the psychological and thermodynamic arrows of time''. Physical Review E 89, 052102 (2014). DOI: 10.1103/​physreve.89.052102.

Cited by

[1] Giuseppe Giglia, Dimitri Ognibene, Nadia Bolognini, Marina De Tommaso, Francesco Cappello, Pierangelo Sardo, Giuseppe Ferraro, and Filippo Brighina, "Editorial: Timing the Brain: From Basic Sciences to Clinical Implications", Frontiers in Human Neuroscience 16, 880443 (2022).

[2] Porter Williams, "Entanglement, Complexity, and Causal Asymmetry in Quantum Theories", Foundations of Physics 52 2, 47 (2022).

[3] Giulio Chiribella and Zixuan Liu, "Quantum operations with indefinite time direction", Communications Physics 5 1, 190 (2022).

[4] V. Vilasini and Roger Colbeck, "General framework for cyclic and fine-tuned causal models and their compatibility with space-time", Physical Review A 106 3, 032204 (2022).

[5] Anthony Sudbery, "Histories Without Collapse", International Journal of Theoretical Physics 61 2, 39 (2022).

[6] Andrea Di Biagio and Carlo Rovelli, "Relational Quantum Mechanics is About Facts, Not States: A Reply to Pienaar and Brukner", Foundations of Physics 52 3, 62 (2022).

[7] Alessandro Capurso, "The Potential of a Thick Present through Undefined Causality and Non-Locality", Entropy 24 3, 410 (2022).

[8] Robin Lorenz, "Quantum causal models: the merits of the spirit of Reichenbach’s principle for understanding quantum causal structure", Synthese 200 5, 424 (2022).

[9] Chris Fields, Karl Friston, James F. Glazebrook, and Michael Levin, "A free energy principle for generic quantum systems", Progress in Biophysics and Molecular Biology 173, 36 (2022).

[10] Lucien Hardy, "Time Symmetry in Operational Theories", arXiv:2104.00071.

[11] Stephen M. Barnett, John Jeffers, and David T. Pegg, "Quantum Retrodiction: Foundations and Controversies", Symmetry 13 4, 586 (2021).

[12] Giulio Chiribella, Erik Aurell, and Karol Życzkowski, "Symmetries of quantum evolutions", Physical Review Research 3 3, 033028 (2021).

[13] Arthur J. Parzygnat and Francesco Buscemi, "Axioms for retrodiction: achieving time-reversal symmetry with a prior", arXiv:2210.13531.

[14] John H. Selby, Maria E. Stasinou, Stefano Gogioso, and Bob Coecke, "Time symmetry in quantum theories and beyond", arXiv:2209.07867.

[15] Sergey G. Rubin, "Can a particle moves zigzag in time?", arXiv:2203.04200.

The above citations are from Crossref's cited-by service (last updated successfully 2022-12-08 06:29:48) and SAO/NASA ADS (last updated successfully 2022-12-08 06:29:49). The list may be incomplete as not all publishers provide suitable and complete citation data.