# Generalized probability rules from a timeless formulation of Wigner’s friend scenarios

Veronika Baumann1,2,3, Flavio Del Santo1,2, Alexander R. H. Smith4, Flaminia Giacomini1,2,5, Esteban Castro-Ruiz1,2,6, and Caslav Brukner1,2

1Institute for Quantum Optics and Quantum Information (IQOQI-Vienna) of the Austrian Academy of Sciences, Boltzmanngasse 3, A-1090, Vienna, Austria
2Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
3Faculty of Informatics, Università della Svizzera italiana, Via G. Buffi 13, CH-6900 Lugano, Switzerland
4Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
5Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
6QuIC, Ecole polytechnique de Bruxelles, C.P. 165, Université libre de Bruxelles, 1050 Brussels, Belgium

### Abstract

The quantum measurement problem can be regarded as the tension between the two alternative dynamics prescribed by quantum mechanics: the unitary evolution of the wave function and the state-update rule (or "collapse") at the instant a measurement takes place. The notorious Wigner's friend gedankenexperiment constitutes the paradoxical scenario in which different observers (one of whom is observed by the other) describe one and the same interaction differently, one –the Friend– via state-update and the other –Wigner– unitarily. This can lead to Wigner and his friend assigning different probabilities to the outcome of the same subsequent measurement. In this paper, we apply the Page-Wootters mechanism (PWM) as a timeless description of Wigner's friend-like scenarios. We show that the standard rules to assign two-time conditional probabilities within the PWM need to be modified to deal with the Wigner's friend gedankenexperiment. We identify three main definitions of such modified rules to assign two-time conditional probabilities, all of which reduce to standard quantum theory for non-Wigner's friend scenarios. However, when applied to the Wigner's friend setup each rule assigns different conditional probabilities, potentially resolving the probability-assignment paradox in a different manner. Moreover, one rule imposes strict limits on when a joint probability distribution for the measurement outcomes of Wigner and his Friend is well-defined, which single out those cases where Wigner's measurement does not disturb the Friend's memory and such a probability has an operational meaning in terms of collectible statistics. Interestingly, the same limits guarantee that said measurement outcomes fulfill the consistency condition of the consistent histories framework.

The measurement problem plagues a common sense interpretation of quantum theory. It arises from the tension between two different dynamics describing the evolution of a quantum system: the first (unitary evolution) resembles the deterministic evolution of classical physics, while the second one ("collapse") applies whenever a measurement takes place and is a discontinuous, probabilistic process. Despite an ambiguity between which of these two dynamics to apply when, physicists know well which description to use in their daily practice. However, since there is today no strong evidence that quantum mechanics has a limit of applicability—in principle it holds at any scale, from subatomic particles to chairs, from human bodies to planets, etc.—one can think that a whole laboratory including the physicist carrying out her experiments can be regarded as a quantum system, and therefore observed by another physicist outside the laboratory. This scenario is called a Wigner's friend experiment, and it shows that the physicist inside the lab (the friend) and the one measuring them from outside (Wigner) both using the same textbook of quantum mechanics arrive at different conclusions about the probabilities of certain measurements, because they both apply one of the two different dynamics.

In our work, we have argued that a way to clarify the conceptual issues of the measurement problem in a Wigner's friend scenario is to eliminate time (in a classical sense, that is, a parameter which always increases) from the description. We have shown that in this way, Wigner and the friend remove the which-dynamics ambiguity because they always agree on a single timeless state of the whole physical situation. However, even if they always agree, they can use different rules to predict probabilities for measurement results. We have shown that some probability rules give the probabilities provided by unitary dynamics, while others lead to probabilities equal to those given by "collapse“ dynamics. This means, depending on which rule Wigner and his friend use, they will either agree on the probabilities for unitary dynamics or those for collapse dynamics. In our framework, the ambiguity between the two dynamics is thus pushed back to the choice of the probability-assignment rule, but once Wigner and the friend agree on a rule (ideally based on some physical arguments or evidence), there cannot be disagreement on the probabilities.

### ► References

[1] P. Busch, P. J. Lahti, and P. Mittelstaedt. The quantum theory of measurement. Springer Berlin Heidelberg, 1996. doi:10.1007/​978-3-540-49239-9.
https:/​/​doi.org/​10.1007/​978-3-540-49239-9

[2] T. Maudlin. Three measurement problems. Topoi, 14(1):7–15, 1995. doi:https:/​/​doi.org/​10.1007/​BF00763473.
https:/​/​doi.org/​10.1007/​BF00763473

[3] J. Bub and I. Pitowsky. Many worlds, chapter Two dogmas about quantum mechanics, pages 433–459. 2010. doi:https:/​/​doi.org/​10.1093/​acprof:oso/​9780199560561.003.0016.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199560561.003.0016

[4] E. P. Wigner. Remarks on the Mind-Body Question. In Jagdish Mehra, editor, Philosophical Reflections and Syntheses, pages 247–260. Springer Berlin Heidelberg, 1995. doi:https:/​/​doi.org/​10.1007/​978-3-642-78374-6_20.
https:/​/​doi.org/​10.1007/​978-3-642-78374-6_20

[5] Č. Brukner. On the quantum measurement problem. In Quantum [Un] Speakables II, pages 95–117. Springer Cham, 2017. doi:https:/​/​doi.org/​10.1007/​978-3-319-38987-5_5.
https:/​/​doi.org/​10.1007/​978-3-319-38987-5_5

[6] D. Frauchiger and R. Renner. Quantum theory cannot consistently describe the use of itself. Nat. Commun., 9(1), 2018. doi:https:/​/​doi.org/​10.1038/​s41467-018-05739-8.
https:/​/​doi.org/​10.1038/​s41467-018-05739-8

[7] Č. Brukner. A No-Go Theorem for Observer-Independent Facts. Entropy, 20(5):350, 2018. doi:https:/​/​doi.org/​10.3390/​e20050350.
https:/​/​doi.org/​10.3390/​e20050350

[8] D. N. Page and W. K. Wootters. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D, 27:2885, 1983. doi:https:/​/​doi.org/​10.1103/​PhysRevD.27.2885.
https:/​/​doi.org/​10.1103/​PhysRevD.27.2885

[9] W. K. Wootters. Time'' replaced by quantum correlations. Int. J. Theor. Phys., 23:701, 1984. doi:https:/​/​doi.org/​10.1007/​BF02214098.
https:/​/​doi.org/​10.1007/​BF02214098

[10] K. V. Kuchař. Time and interpretations of quantum gravity. Int. J. Mod. Phys. D, 20(supp01):3–86, 2011. Originally published in Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, eds. G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992). doi:https:/​/​doi.org/​10.1142/​S0218271811019347.
https:/​/​doi.org/​10.1142/​S0218271811019347

[11] C. J. Isham. Canonical Quantum Gravity and the Problem of Time. In L. A. Ibort and M. A. Rodríguez, editors, Integrable Systems, Quantum Groups, and Quantum Field Theories, pages 157–287. Springer Netherlands, 1993. doi:https:/​/​doi.org/​10.1007/​978-94-011-1980-1_6.
https:/​/​doi.org/​10.1007/​978-94-011-1980-1_6

[12] C. Rovelli. Quantum Gravity. Cambridge University Press, 2004. doi:https:/​/​doi.org/​10.1017/​CBO9780511755804.
https:/​/​doi.org/​10.1017/​CBO9780511755804

[13] C. Kiefer. Quantum Gravity. Oxford University Press, 3rd edition, 2012. doi:https:/​/​doi.org/​10.1093/​acprof:oso/​9780199585205.001.0001.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199585205.001.0001

[14] A. S. Holevo. Probabilistic and Statistical Aspects of Quantum Theory, volume 1 of Statistics and Probability. North-Holland, Amsterdam, 1982. doi:https:/​/​doi.org/​10.1007/​978-88-7642-378-9.
https:/​/​doi.org/​10.1007/​978-88-7642-378-9

[15] P. Busch, M. Grabowski, and P. J. Lahti. Operational Quantum Physics, volume 31 of Lecture Notes in Physics Monographs. Springer Berlin Heidelberg, 1995. doi:https:/​/​doi.org/​10.1007/​978-3-540-49239-9.
https:/​/​doi.org/​10.1007/​978-3-540-49239-9

[16] A. R. H. Smith and M. Ahmadi. Quantizing time: Interacting clocks and systems. Quantum, 3:160, 2019. doi:https:/​/​doi.org/​10.22331/​q-2019-07-08-160.
https:/​/​doi.org/​10.22331/​q-2019-07-08-160

[17] F. Hellmann, M. Mondragon, A. Perez, and C. Rovelli. Multiple-event probability in general-relativistic quantum mechanics. Phys. Rev. D, 75(8):084033, 2007. doi:https:/​/​doi.org/​10.1103/​PhysRevD.75.084033.
https:/​/​doi.org/​10.1103/​PhysRevD.75.084033

[18] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum time. Phys. Rev. D, 92(4):045033, 2015. doi:https:/​/​doi.org/​10.1103/​PhysRevD.92.045033.
https:/​/​doi.org/​10.1103/​PhysRevD.92.045033

[19] C. E. Dolby. The conditional probability interpretation of the hamiltonian constraint. arXiv preprint gr-qc/​0406034, 2004. URL: https:/​/​arxiv.org/​abs/​gr-qc/​0406034.
https:/​/​arxiv.org/​abs/​gr-qc/​0406034

[20] V. Baumann and S. Wolf. On formalisms and interpretations. Quantum, 2:99, 2018. doi:https:/​/​doi.org/​10.22331/​q-2018-10-15-99.
https:/​/​doi.org/​10.22331/​q-2018-10-15-99

[21] V. Baumann and Č. Brukner. Quantum, Probability, Logic, chapter Wigner's Friend as a Rational Agent. 2020. doi:https:/​/​doi.org/​10.1007/​978-3-030-34316-3_4.
https:/​/​doi.org/​10.1007/​978-3-030-34316-3_4

[22] R. B. Griffiths. Consistent quantum theory. Cambridge University Press, 2003. doi:https:/​/​doi.org/​10.1017/​CBO9780511606052.
https:/​/​doi.org/​10.1017/​CBO9780511606052

[23] R. Gambini, R. A. Porto, Sebastian Torterolo, and J. Pullin. Conditional probabilities with Dirac observables and the problem of time in quantum gravity. Phys. Rev. D, 79:041501, 2009. doi:https:/​/​doi.org/​10.1103/​PhysRevD.79.041501.
https:/​/​doi.org/​10.1103/​PhysRevD.79.041501

[24] M. Losada, R. Laura, and O. Lombardi. Frauchiger-renner argument and quantum histories. Phys. Rev. A, 100:052114, 2019. doi:https:/​/​doi.org/​10.1103/​PhysRevA.100.052114.
https:/​/​doi.org/​10.1103/​PhysRevA.100.052114

[25] A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz. A change of perspective: switching quantum reference frames via a perspective-neutral framework. Quantum, 4:225, 2020. doi:10.22331/​q-2020-01-27-225.
https:/​/​doi.org/​10.22331/​q-2020-01-27-225

[26] L. Diosi. Anomalies of weakened decoherence criteria for quantum histories. Phys. Rev. Lett., 92(17):170401, 2004. doi:https:/​/​doi.org/​10.1103/​PhysRevLett.92.170401.
https:/​/​doi.org/​10.1103/​PhysRevLett.92.170401

### Cited by

[1] Tommaso Favalli and Augusto Smerzi, "Peaceful coexistence of thermal equilibrium and the emergence of time", Physical Review D 105 2, 023525 (2022).

[2] Tommaso Favalli and Augusto Smerzi, "Time Observables in a Timeless Universe", arXiv:2003.09042, Quantum 4, 354 (2020).

[3] Flavio Del Santo and Nicolas Gisin, "The Relativity of Indeterminacy", Entropy 23 10, 1326 (2021).

[4] I. L. Paiva, M. Nowakowski, and E. Cohen, "Dynamical nonlocality in quantum time via modular operators", Physical Review A 105 4, 042207 (2022).

[5] M. Trassinelli, "Conditional probabilities of measurements, quantum time, and the Wigner's-friend case", Physical Review A 105 3, 032213 (2022).

[6] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock, "The Trinity of Relational Quantum Dynamics", arXiv:1912.00033, Physical Review D 104 6, 066001 (2019).

[7] Philipp A. Höhn, Alexander R. H. Smith, and Maximilian P. E. Lock, "Equivalence of approaches to relational quantum dynamics in relativistic settings", Frontiers in Physics 9, 181 (2021).

[8] Marek Żukowski and Marcin Markiewicz, "Physics and Metaphysics of Wigner's Friends: Even Performed Premeasurements Have No Results", Physical Review Letters 126 13, 130402 (2021).

[9] Philippe Allard Guérin, Veronika Baumann, Flavio Del Santo, and Časlav Brukner, "A no-go theorem for the persistent reality of Wigner's friend's perception", Communications Physics 4 1, 93 (2021).

[10] Vinicius P. Rossi and Diogo O. Soares-Pinto, "Wigner's friend and the quasi-ideal clock", Physical Review A 103 5, 052206 (2021).

[11] Eric G. Cavalcanti, "The View from a Wigner Bubble", Foundations of Physics 51 2, 39 (2021).

[12] Caterina Foti, Alessandro Coppo, Giulio Barni, Alessandro Cuccoli, and Paola Verrucchi, "Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states", Nature Communications 12, 1787 (2021).

[13] Matteo Lostaglio and Joseph Bowles, "The original Wigner's friend paradox within a realist toy model", Proceedings of the Royal Society of London Series A 477 2254, 20210273 (2021).

[14] Anthony Sudbery, "Histories without collapse", arXiv:2012.13430.

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