Quantum reference frame transformations as symmetries and the paradox of the third particle

Marius Krumm1,2, Philipp A. Höhn3,4, and Markus P. Müller1,2,5

1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Vienna, Austria
3Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904 0495, Japan
4Department of Physics and Astronomy, University College London, London, United Kingdom
5Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo ON N2L 2Y5, Canada

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

Abstract

In a quantum world, reference frames are ultimately quantum systems too – but what does it mean to "jump into the perspective of a quantum particle"? In this work, we show that quantum reference frame (QRF) transformations appear naturally as symmetries of simple physical systems. This allows us to rederive and generalize known QRF transformations within an alternative, operationally transparent framework, and to shed new light on their structure and interpretation. We give an explicit description of the observables that are measurable by agents constrained by such quantum symmetries, and apply our results to a puzzle known as the `paradox of the third particle'. We argue that it can be reduced to the question of how to relationally embed fewer into more particles, and give a thorough physical and algebraic analysis of this question. This leads us to a generalization of the partial trace (`relational trace') which arguably resolves the paradox, and it uncovers important structures of constraint quantization within a simple quantum information setting, such as relational observables which are key in this resolution. While we restrict our attention to finite Abelian groups for transparency and mathematical rigor, the intuitive physical appeal of our results makes us expect that they remain valid in more general situations.

► BibTeX data

► References

[1] Y. Aharonov and L. Susskind, Charge Superselection Rule, Phys. Rev. 155, 1428 (1967).
https:/​/​doi.org/​10.1103/​PhysRev.155.1428

[2] Y. Aharonov and L. Susskind, Observability of the Sign Change of Spinors under $2\pi$ Rotations, Phys. Rev. 158, 1237 (1967).
https:/​/​doi.org/​10.1103/​PhysRev.158.1237

[3] Y. Aharonov and T. Kaufherr, Quantum frames of reference, Phys. Rev. D 30, 368 (1984).
https:/​/​doi.org/​10.1103/​PhysRevD.30.368

[4] E. Wigner, Die Messung quantenmechanischer Operatoren, Z. Physik 133, 101–108 (1952).
https:/​/​doi.org/​10.1007/​BF01948686

[5] H. Araki and M. M. Yanase, Measurement of Quantum Mechanical Operators, Phys. Rev. 120, 622 (1960).
https:/​/​doi.org/​10.1103/​PhysRev.120.622

[6] M. M. Yanase, Optimal Measuring Apparatus, Phys. Rev. 123, 666 (1961).
https:/​/​doi.org/​10.1103/​PhysRev.123.666

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

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

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

[10] L. Loveridge, A relational perspective on the Wigner-Araki-Yanase theorem, J. Phys.: Conf. Ser. 1638, 012009 (2020).
https:/​/​doi.org/​10.1088/​1742-6596/​1638/​1/​012009

[11] P. A. Höhn and M. P. Müller, An operational approach to spacetime symmetries: Lorentz transformations from quantum communication, New J. Phys. 18, 063026 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​6/​063026

[12] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Rev. Mod. Phys. 79, 555 (2007).
https:/​/​doi.org/​10.1103/​RevModPhys.79.555

[13] A. R. H. Smith, Communicating without shared reference frames, Phys. Rev. A 99, 052315 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.052315

[14] I. Marvian, Symmetry, Asymmetry and Quantum Information, PhD thesis, University of Waterloo, 2012.
https:/​/​uwspace.uwaterloo.ca/​handle/​10012/​7088

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

[16] G. Gour, I. Marvian, and R. W. Spekkens, Measuring the quality of a quantum reference frame: The relative entropy of frameness, Phys. Rev. A 80, 012307 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.012307

[17] I. Marvian and R. W. Spekkens, Modes of asymmetry: The application of harmonic analysis to symmetric quantum dynamics and quantum reference frames, Phys. Rev. A 90, 062110 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.062110

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

[19] 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, Phys. Rev. A 94, 012333 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.012333

[20] J. Åberg, Catalytic Coherence, Phys. Rev. Lett. 113, 150402 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.150402

[21] M. Lostaglio, D. Jennings, and T. Rudolph, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy, Nat. Commun. 6, 6383 (2015).
https:/​/​doi.org/​10.1038/​ncomms7383

[22] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Quantum Coherence, Time-Translation Symmetry, and Thermodynamics, Phys. Rev. X 5, 021001 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.021001

[23] M. Lostaglio and M. P. Müller, Coherence and Asymmetry Cannot be Broadcast, Phys. Rev. Lett. 123, 020403 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.020403

[24] I. Marvian and R. W. Spekkens, No-Broadcasting Theorem for Quantum Asymmetry and Coherence and a Trade-off Relation for Approximate Broadcasting, Phys. Rev. Lett. 123, 020404 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.020404

[25] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?, Phys. Rev. X 7, 031022 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.031022

[26] P. Ć wikliński, M. Studziński, M. Horodecki, and J. Oppenheim, Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics, Phys. Rev. Lett. 115, 210403 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.210403

[27] M. P. Woods, R. Silva, and J. Oppenheim, Autonomous Quantum Machines and Finite-Sized Clocks, Ann. Henri Poincaré 20, 125 (2019).
https:/​/​doi.org/​10.1007/​s00023-018-0736-9

[28] M. P. Woods and M. Horodecki, The resource theoretic paradigm of quantum thermodynamics with control, arXiv:1912.05562 [quant-ph].
arXiv:1912.05562

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

[30] T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press, 2007.
https:/​/​doi.org/​10.1017/​CBO9780511755682

[31] J. Tambornino, Relational Observables in Gravity: a Review, SIGMA 8, 017 (2012).
https:/​/​doi.org/​10.3842/​SIGMA.2012.017

[32] C. Rovelli, What is observable in classical and quantum gravity?, Class. Quant. Grav. 8, 297 (1991).
https:/​/​doi.org/​10.1088/​0264-9381/​8/​2/​011

[33] C. Rovelli, Quantum reference systems, Class. Quant. Grav. 8, 317 (1991).
https:/​/​doi.org/​10.1088/​0264-9381/​8/​2/​012

[34] C. Rovelli, Time in quantum gravity: An hypothesis, Phys. Rev. D 43, 442-456 (1991).
https:/​/​doi.org/​10.1103/​PhysRevD.43.442

[35] B. Dittrich, Partial and complete observables for Hamiltonian constrained systems, Gen. Rel. Grav. 39, 1891 (2007).
https:/​/​doi.org/​10.1007/​s10714-007-0495-2

[36] B. Dittrich, Partial and complete observables for canonical general relativity, Class. Quant. Grav. 23, 6155 (2006).
https:/​/​doi.org/​10.1088/​0264-9381/​23/​22/​006

[37] L. Chataignier, Construction of quantum Dirac observables and the emergence of WKB time, Phys. Rev. D 101, 086001 (2020).
https:/​/​doi.org/​10.1103/​PhysRevD.101.086001

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

[39] A. Vanrietvelde, P. A. Höhn, 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

[40] A. de la Hamette and T. Galley, Quantum reference frames for general symmetry groups, Quantum 4, 367 (2020).
https:/​/​doi.org/​10.22331/​q-2020-11-30-367

[41] A. Vanrietvelde, P. A. Höhn, and F. Giacomini, Switching quantum reference frames in the N-body problem and the absence of global relational perspectives, arXiv:1809.05093 [quant-ph].
arXiv:1809.05093

[42] P. A. Höhn and A. Vanrietvelde, How to switch between relational quantum clocks, New J. Phys. 22, 123048 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​abd1ac

[43] 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

[44] P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, The Trinity of Relational Quantum Dynamics, Phys. Rev. D (in press), arXiv:1912.00033 [quant-ph].
arXiv:1912.00033

[45] P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Equivalence of approaches to relational quantum dynamics in relativistic settings, Front. Phys. 9, 587083 (2021).
https:/​/​doi.org/​10.3389/​fphy.2021.587083

[46] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, Relativistic Quantum Reference Frames: The Operational Meaning of Spin, Phys. Rev. Lett. 123, 090404 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.090404

[47] L. F. Streiter, F. Giacomini, and Č. Brukner, A Relativistic Bell Test within Quantum Reference Frames, Phys. Rev. Lett. 126, 230403 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.230403

[48] 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, Nat. Commun. 11, 2672 (2020).
https:/​/​doi.org/​10.1038/​s41467-020-16013-1

[49] J. M. Yang, Switching Quantum Reference Frames for Quantum Measurement, Quantum 4, 283 (2020).
https:/​/​doi.org/​10.22331/​q-2020-06-18-283

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

[51] T. P. Le, P. Mironowicz, and P. Horodecki, Blurred quantum Darwinism across quantum reference frames, Phys. Rev. A 102, 062420 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.062420

[52] J. Tuziemski, Decoherence and information encoding in quantum reference frames, arXiv:2006.07298 [quant-ph].
arXiv:2006.07298

[53] M. F. Savi, and R. M. Angelo, Quantum Resources Covariance, Phys. Rev. A 103, 022220 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.022220

[54] P. A. Guérin and Č. Brukner, Observer-dependent locality of quantum events, New J. Phys. 20, 103031 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aae742

[55] L. Hardy, Implementation of the Quantum Equivalence Principle, arXiv:1903.01289 [quant-ph].
arXiv:1903.01289

[56] S. Gielen and L. Menéndez-Pidal, Singularity resolution depends on the clock, Class. Quant. Grav. 37, 205018 (2020).
https:/​/​doi.org/​10.1088/​1361-6382/​abb14f

[57] K. Giesel, L. Herold, B. F. Li, and P. Singh, Mukhanov-Sasaki equation in a manifestly gauge-invariant linearized cosmological perturbation theory with dust reference fields, Phys. Rev. D 102, 023524 (2020).
https:/​/​doi.org/​10.1103/​PhysRevD.102.023524

[58] K. Giesel, B. F. Li, and P. Singh, Towards a reduced phase space quantization in loop quantum cosmology with an inflationary potential, Phys. Rev. D 102, 126024 (2020).
https:/​/​doi.org/​10.1103/​PhysRevD.102.126024

[59] P. A. Dirac, Lectures on Quantum Mechanics, Yeshiva University Press, 1964.

[60] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, 1992.
https:/​/​doi.org/​10.1515/​9780691213866

[61] D. Giulini and D. Marolf, A Uniqueness theorem for constraint quantization, Class. Quant. Grav. 16, 2489 (1999).
https:/​/​doi.org/​10.1088/​0264-9381/​16/​7/​322

[62] D. Marolf, Group averaging and refined algebraic quantization: Where are we now?, arXiv:gr-qc/​0011112 [gr-qc].
arXiv:gr-qc/0011112

[63] C. Rovelli, Why Gauge?, Found. Phys. 44, 91–104 (2014).
https:/​/​doi.org/​10.1007/​s10701-013-9768-7

[64] W. Donnelly and A. C. Wall, Entanglement Entropy of Electromagnetic Edge Modes, Phys. Rev. Lett. 114, 111603 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.111603

[65] W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09, 102 (2016).
https:/​/​doi.org/​10.1007/​JHEP09(2016)102

[66] M. Geiller and P. Jai-akson, Extended actions, dynamics of edge modes, and entanglement entropy, JHEP 20, 134 (2020).
https:/​/​doi.org/​10.1007/​JHEP09(2020)134

[67] L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 2020, 26 (2020).
https:/​/​doi.org/​10.1007/​JHEP11(2020)026

[68] H. Gomes and A. Riello, Unified geometric framework for boundary charges and particle dressings, Phys. Rev. D 98, 025013 (2018).
https:/​/​doi.org/​10.1103/​PhysRevD.98.025013

[69] A. Riello, Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back, SciPost Phys. 10, 125 (2021).
https:/​/​doi.org/​10.21468/​SciPostPhys.10.6.125

[70] W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34, no.21, 215008 (2017).
https:/​/​doi.org/​10.1088/​1361-6382/​aa8d06

[71] W. Wieland, Fock representation of gravitational boundary modes and the discreteness of the area spectrum, Annales Henri Poincare 18, no.11, 3695 (2017).
https:/​/​doi.org/​10.1007/​s00023-017-0598-6

[72] P. A. Höhn, M. Krumm, and M. P. Müller, Internal quantum reference frames for finite Abelian groups, arXiv:2107.07545 [quant-ph].
arXiv:2107.07545

[73] B. Simon, Representations of Finite and Compact Groups, American Mathematical Society, 1996.

[74] K. R. Davidson, C$^*$-Algebras by Example, American Mathematical Society, 1996.

[75] A. Savage, Modern Group Theory, lecture notes, University of Ottawa, 2017. Available at https:/​/​alistairsavage.ca/​mat5145/​notes/​MAT5145-Modern_group_theory.pdf.
https:/​/​alistairsavage.ca/​mat5145/​notes/​MAT5145-Modern_group_theory.pdf

[76] P. A. Höhn, Reflections on the information paradigm in quantum and gravitational physics, J. Phys. Conf. Ser. 880, 012014 (2017).
https:/​/​doi.org/​10.1088/​1742-6596/​880/​1/​012014

[77] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, New York, 2010.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[78] M. Tinkham, Group Theory and Quantum Mechanics, Dover Publications, 1992.

[79] S. Khandelwal, M. P. E. Lock, and M. P. Woods, Universal quantum modifications to general relativistic time dilation in delocalised clocks, Quantum 4, 309 (2020).
https:/​/​doi.org/​10.22331/​q-2020-08-14-309

[80] A. R. H. Smith and M. Ahmadi, Quantum clocks observe classical and quantum time dilation, Nat. Commun. 11, 5360 (2020).
https:/​/​doi.org/​10.1038/​s41467-020-18264-4

[81] P. T. Grochowski, A. R. H. Smith, A. Dragan, and K. Debski, Quantum time dilation in atomic spectra, Phys. Rev. Research 3, 023053 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.023053

[82] R. Gambini and J. Pullin, The Montevideo Interpretation: How the inclusion of a Quantum Gravitational Notion of Time Solves the Measurement Problem, Universe 6, 236 (2020).
https:/​/​doi.org/​10.3390/​universe6120236

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

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

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

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

[87] E. Moreva, G. Brida, M. Gramegna, V. Giovannetti, L. Maccone, and M. Genovese, Time from quantum entanglement: An experimental illustration, Phys. Rev. A 89, 052122 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.052122

Cited by

[1] Wolfgang Wieland, "Null infinity as an open Hamiltonian system", Journal of High Energy Physics 2021 4, 95 (2021).

[2] Flaminia Giacomini and Časlav Brukner, "Einstein's Equivalence principle for superpositions of gravitational fields", arXiv:2012.13754.

[3] Philipp A. Hoehn, Maximilian P. E. Lock, Shadi Ali Ahmad, Alexander R. H. Smith, and Thomas D. Galley, "Quantum Relativity of Subsystems", arXiv:2103.01232.

[4] Angel Ballesteros, Flaminia Giacomini, and Giulia Gubitosi, "The group structure of dynamical transformations between quantum reference frames", arXiv:2012.15769.

[5] Flaminia Giacomini, "Spacetime Quantum Reference Frames and superpositions of proper times", arXiv:2101.11628.

[6] Marion Mikusch, Luis C. Barbado, and Časlav Brukner, "Transformation of Spin in Quantum Reference Frames", arXiv:2103.05022.

[7] Bharath Ron, "Emergence of Time in a Participatory Universe", arXiv:1704.01416.

[8] Philipp A. Hoehn, Marius Krumm, and Markus P. Mueller, "Internal quantum reference frames for finite Abelian groups", arXiv:2107.07545.

[9] Wolfgang Wieland, "Barnich-Troessaert Bracket as a Dirac Bracket on the Covariant Phase Space", arXiv:2104.08377.

[10] Isha Kotecha, "On Generalised Statistical Equilibrium and Discrete Quantum Gravity", arXiv:2010.15445.

[11] Sylvain Carrozza and Philipp A. Hoehn, "Edge modes as reference frames and boundary actions from post-selection", arXiv:2109.06184.

[12] Flaminia Giacomini and Časlav Brukner, "Quantum superposition of spacetimes obeys Einstein's Equivalence Principle", arXiv:2109.01405.

The above citations are from SAO/NASA ADS (last updated successfully 2021-09-17 09:12:04). 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 2021-09-17 09:12:02).