# Quantum reference frames for general symmetry groups

1Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland
2Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada

### Abstract

A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry group $G$, we define a general operator for reversibly changing between quantum reference frames associated to a group $G$. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations of $G$. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry group $G$ is a semi-direct product $G = N \rtimes P$ or a direct product $G = N \times P$, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.

Reference frames are essential in the description of physical phenomena.
Whenever a physical quantity is measured or a physical event is described, this is done with respect to a specified reference frame.
Treating reference frames themselves as physical objects and submitting them to the laws of quantum mechanics, they become quantum reference frames. Recently, there has been an increased interest in analysing spatial and temporal quantum reference frames and in establishing a formalism that allows to switch between different perspectives. In this work we construct a formalism to change between the descriptions assigned by different quantum reference frames for general symmetry groups. The change of perspective is quantum in the sense that the operator we construct allows to take on the perspective of an observer who is in superposition relative to the initial one. We apply the relational formalism developed in this work to the Wigner’s friend thought experiment providing an explicit change of perspective from Wigner to the Friend, arriving at a similar conclusion to that of relational quantum mechanics.

### ► References

[1] C. Rovelli, Quantum reference systems, Classical and Quantum Gravity 8, 317–331 (1991).
https:/​/​doi.org/​10.1088/​0264-9381/​8/​2/​012

[2] C. Rovelli, Relational quantum mechanics, International Journal of Theoretical Physics 35, 1637–1678 (1996).
https:/​/​doi.org/​10.1007/​BF02302261

[3] 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).
https:/​/​doi.org/​10.22331/​q-2020-01-27-225

[4] P. A. Hoehn and A. Vanrietvelde, How to switch between relational quantum clocks (2020), arXiv:1810.04153 [gr-qc].
arXiv:1810.04153

[5] P. A. Höhn, Switching Internal Times and a New Perspective on the Wave Function of the Universe', Universe 5, 116 (2019).
https:/​/​doi.org/​10.3390/​universe5050116

[6] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames, Nature Communications 10, 494 (2019).
https:/​/​doi.org/​10.1038/​s41467-018-08155-0

[7] E. Castro-Ruiz, F. Giacomini, A. Belenchia, and Č. Brukner, Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems, Nature Communications 11, 2672 (2020).
https:/​/​doi.org/​10.1038/​s41467-020-16013-1

[8] Y. Aharonov and T. Kaufherr, Quantum frames of reference, Physical Review D 30, 368–385 (1984).
https:/​/​doi.org/​10.1103/​PhysRevD.30.368

[9] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Reviews of Modern Physics 79, 555–609 (2007).
https:/​/​doi.org/​10.1103/​revmodphys.79.555

[10] R. M. Angelo and A. D. Ribeiro, Kinematics and dynamics in noninertial quantum frames of reference, Journal of Physics A: Mathematical and Theoretical 45, 465306 (2012).
https:/​/​doi.org/​10.1088/​1751-8113/​45/​46/​465306

[11] L. Loveridge, T. Miyadera, and P. Busch, Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics, Foundations of Physics 48, 135–198 (2018).
https:/​/​doi.org/​10.1007/​s10701-018-0138-3

[12] S. T. Pereira and R. M. Angelo, Galilei covariance and Einstein's equivalence principle in quantum reference frames, Phys. Rev. A 91, 022107 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.022107

[13] A. R. H. Smith, M. Piani, and R. B. Mann, Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass, Physical Review A 94, 012333 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.012333

[14] A. Vanrietvelde, P. A. Hoehn, and F. Giacomini, Switching quantum reference frames in the n-body problem and the absence of global relational perspectives (2018), arXiv:1809.05093 [quant-ph].
arXiv:1809.05093

[15] P. A. Hoehn, A. R. H. Smith, and M. P. E. Lock, The trinity of relational quantum dynamics (2019), arXiv:1912.00033 [quant-ph].
arXiv:1912.00033

[16] G. Bene and D. Dieks, A Perspectival Version of the Modal Interpretation of Quantum mechanics and the Origin of Macroscopic Behavior, Foundations of Physics 32, 645–671 (2001).
https:/​/​doi.org/​10.1023/​A:1016014008418

[17] M. L. Dalla Chiara, Logical self reference, set theoretical paradoxes and the measurement problem in quantum mechanics, Journal of Philosophical Logic 6, 331–347 (1977).
https:/​/​doi.org/​10.1007/​BF00262066

[18] T. Breuer, The Impossibility of Accurate State Self-Measurements, Philosophy of Science 62, 197–214 (1995).
http:/​/​www.jstor.org/​stable/​188430

[19] J. Baez, Dimensional analysis and coordinate systems, The n Category Cafe (2006).
https:/​/​golem.ph.utexas.edu/​category/​2006/​09/​dimensional_analysis_and_coord.html

[20] T. Tao, Compactness and Contradiction, Miscellaneous Books (American Mathematical Society, 2013).

[21] A. Kitaev, D. Mayers, and J. Preskill, Superselection rules and quantum protocols, Physical Review A 69, 052326 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.052326

[22] A. Magidin, Does every set have a group structure?, Mathematics Stack Exchange, url: https:/​/​math.stackexchange.com/​q/​105440 (version: 2017-04-13).
https:/​/​math.stackexchange.com/​q/​105440

[23] E. P. Wigner, Remarks on the mind-body question, in Philosophical Reflections and Syntheses, edited by J. Mehra (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995) pp. 247–260.
https:/​/​doi.org/​10.1007/​978-3-642-78374-6_20

[24] D. Frauchiger and R. Renner, Quantum theory cannot consistently describe the use of itself, Nature Communications 9, 3711 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-05739-8

[25] D. Poulin, Toy Model for a Relational Formulation of Quantum Theory, International Journal of Theoretical Physics 45, 1189–1215 (2006).
https:/​/​doi.org/​10.1007/​s10773-006-9052-0

[26] L. Loveridge, P. Busch, and T. Miyadera, Relativity of quantum states and observables, EPL (Europhysics Letters) 117, 40004 (2017).
https:/​/​doi.org/​10.1209/​0295-5075/​117/​40004

[27] N. D. Mermin, What is quantum mechanics trying to tell us?, American Journal of Physics 66, 753–767 (1998).
https:/​/​doi.org/​10.1119/​1.18955

[28] Y. Aharonov and L. Susskind, Charge superselection rule, Phys. Rev. 155, 1428–1431 (1967).
https:/​/​doi.org/​10.1103/​PhysRev.155.1428

[29] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Dialogue Concerning Two Views on Quantum Coherence: Factist and Fictionist, International Journal of Quantum Information 04, 17–43 (2006).
https:/​/​doi.org/​10.1142/​S0219749906001591

[30] R. M. Angelo, N. Brunner, S. Popescu, A. J. Short, and P. Skrzypczyk, Physics within a quantum reference frame, Journal of Physics A: Mathematical and Theoretical 44, 145304 (2011).
https:/​/​doi.org/​10.1088/​1751-8113/​44/​14/​145304

[31] M. Zych, F. Costa, and T. C. Ralph, Relativity of quantum superpositions (2018), arXiv:1809.04999 [quant-ph].
arXiv:1809.04999

[32] M. C. Palmer, F. Girelli, and S. D. Bartlett, Changing quantum reference frames, Physical Review A 89, 052121 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.052121

[33] G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New Journal of Physics 10, 033023 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​3/​033023

[34] S. Popescu, A. B. Sainz, A. J. Short, and A. Winter, Quantum Reference Frames and Their Applications to Thermodynamics, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20180111 (2018).
https:/​/​doi.org/​10.1098/​rsta.2018.0111

[35] D. Poulin and J. Yard, Dynamics of a quantum reference frame, New Journal of Physics 9, 156–156 (2007).
https:/​/​doi.org/​10.1088/​1367-2630/​9/​5/​156

[36] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner, Quantum communication using a bounded-size quantum reference frame, New Journal of Physics 11, 063013 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​6/​063013

[37] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges, Nature Communications 7, 12051 (2016).
https:/​/​doi.org/​10.1038/​ncomms12051

[38] T. Miyadera, L. Loveridge, and P. Busch, Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off, Journal of Physics A Mathematical General 49, 185301 (2016).
https:/​/​doi.org/​10.1088/​1751-8113/​49/​18/​185301

[39] M. Skotiniotis, W. Dür, and P. Sekatski, Macroscopic superpositions require tremendous measurement devices, Quantum 1, 34 (2017).
https:/​/​doi.org/​10.22331/​q-2017-11-21-34

[40] R. Penrose, Angular momentum: an approach to combinatorial spacetime, Bastin, T. (ed.), Quantum Theory and Beyond , 151–180 (1971).
https:/​/​math.ucr.edu/​home/​baez/​penrose/​Penrose-AngularMomentum.pdf

[41] J. Baez, Torsors made easy, John Baez Stuff (2009).
http:/​/​math.ucr.edu/​home/​baez/​torsors.html

### Cited by

[1] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock, "Equivalence of approaches to relational quantum dynamics in relativistic settings", arXiv:2007.00580.

[2] Marius Krumm, Philipp A. Hoehn, and Markus P. Mueller, "Quantum reference frame transformations as symmetries and the paradox of the third particle", arXiv:2011.01951.

[3] Leon Loveridge, "A relational perspective on the Wigner-Araki-Yanase theorem", Journal of Physics Conference Series 1638 1, 012009 (2020).

[4] Pierre Martin-Dussaud, "Relational structures of fundamental theories", arXiv:2012.05584.

The above citations are from SAO/NASA ADS (last updated successfully 2021-01-18 20:43:16). The list may be incomplete as not all publishers provide suitable and complete citation data.

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