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.
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.
 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).
 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).
 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).
 S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Reviews of Modern Physics 79, 555–609 (2007).
 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).
 L. Loveridge, T. Miyadera, and P. Busch, Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics, Foundations of Physics 48, 135–198 (2018).
 S. T. Pereira and R. M. Angelo, Galilei covariance and Einstein's equivalence principle in quantum reference frames, Phys. Rev. A 91, 022107 (2015).
 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).
 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].
 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).
 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).
 J. Baez, Dimensional analysis and coordinate systems, The n Category Cafe (2006).
 A. Magidin, Does every set have a group structure?, Mathematics Stack Exchange, url: https://math.stackexchange.com/q/105440 (version: 2017-04-13).
 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.
 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).
 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).
 G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New Journal of Physics 10, 033023 (2008).
 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).
 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).
 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).
 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).
 R. Penrose, Angular momentum: an approach to combinatorial spacetime, Bastin, T. (ed.), Quantum Theory and Beyond , 151–180 (1971).
 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, 587083 (2021).
 Chris Overstreet, Joseph Curti, Minjeong Kim, Peter Asenbaum, Mark A. Kasevich, and Flaminia Giacomini, "Inference of gravitational field superposition from quantum measurements", Physical Review D 108 8, 084038 (2023).
 Flaminia Giacomini, "Spacetime Quantum Reference Frames and superpositions of proper times", Quantum 5, 508 (2021).
 Flaminia Giacomini and Časlav Brukner, "Quantum superposition of spacetimes obeys Einstein's equivalence principle", AVS Quantum Science 4 1, 015601 (2022).
 Victor Dyakin, "Fundamental Cause of Bio-Chirality: Space-Time Symmetry—Concept Review", Symmetry 15 1, 79 (2022).
 Shadi Ali Ahmad, Thomas D. Galley, Philipp A. Höhn, Maximilian P. E. Lock, and Alexander R. H. Smith, "Quantum Relativity of Subsystems", Physical Review Letters 128 17, 170401 (2022).
 Marius Krumm, Philipp A. Höhn, and Markus P. Müller, "Quantum reference frame transformations as symmetries and the paradox of the third particle", Quantum 5, 530 (2021).
 Philipp A. Höhn, Marius Krumm, and Markus P. Müller, "Internal quantum reference frames for finite Abelian groups", Journal of Mathematical Physics 63 11, 112207 (2022).
 Emily Adlam, "Watching the Clocks: Interpreting the Page–Wootters Formalism and the Internal Quantum Reference Frame Programme", Foundations of Physics 52 5, 99 (2022).
 Anne-Catherine de la Hamette, Stefan L. Ludescher, and Markus P. Müller, "Entanglement-Asymmetry Correspondence for Internal Quantum Reference Frames", Physical Review Letters 129 26, 260404 (2022).
 M. F. Savi and R. M. Angelo, "Quantum resource covariance", Physical Review A 103 2, 022220 (2021).
 Isha Kotecha, Springer Theses 61 (2022) ISBN:978-3-030-90968-0.
 Sylvain Carrozza and Philipp A. Höhn, "Edge modes as reference frames and boundary actions from post-selection", Journal of High Energy Physics 2022 2, 172 (2022).
 A Capolupo and A Quaranta, "Neutrinos, mixed bosons, quantum reference frames and entanglement", Journal of Physics G: Nuclear and Particle Physics 50 5, 055003 (2023).
 Yuxiang Yang, Yin Mo, Joseph M. Renes, Giulio Chiribella, and Mischa P. Woods, "Optimal universal quantum error correction via bounded reference frames", Physical Review Research 4 2, 023107 (2022).
 Pierre Martin-Dussaud, "Relational Structures of Fundamental Theories", Foundations of Physics 51 1, 24 (2021).
 Anne-Catherine de la Hamette, Viktoria Kabel, Esteban Castro-Ruiz, and Časlav Brukner, "Quantum reference frames for an indefinite metric", Communications Physics 6 1, 231 (2023).
 Joshua Foo, Cemile Senem Arabaci, Magdalena Zych, and Robert B. Mann, "Quantum superpositions of Minkowski spacetime", Physical Review D 107 4, 045014 (2023).
 Leon Loveridge, "A relational perspective on the Wigner-Araki-Yanase theorem", Journal of Physics: Conference Series 1638 1, 012009 (2020).
 Nikola Paunković and Marko Vojinović, "Equivalence Principle in Classical and Quantum Gravity", Universe 8 11, 598 (2022).
 Flaminia Giacomini and Achim Kempf, "Second-quantized Unruh-DeWitt detectors and their quantum reference frame transformations", Physical Review D 105 12, 125001 (2022).
 Marion Mikusch, Luis C. Barbado, and Časlav Brukner, "Transformation of spin in quantum reference frames", Physical Review Research 3 4, 043138 (2021).
 T. Favalli and A. Smerzi, "A model of quantum spacetime", AVS Quantum Science 4 4, 044403 (2022).
 Isha Kotecha, Springer Theses 167 (2022) ISBN:978-3-030-90968-0.
 Pierre Martin-Dussaud, Titouan Carette, Jan Głowacki, Vaclav Zatloukal, and Federico Zalamea, "Fact-nets: Towards a Mathematical Framework for Relational Quantum Mechanics", Foundations of Physics 53 1, 26 (2023).
 Otto C.W. Kong, "Quantum frames of reference and the noncommutative values of observables", Results in Physics 31, 105033 (2021).
 Viktoria Kabel, Časlav Brukner, and Wolfgang Wieland, "Quantum reference frames at the boundary of spacetime", Physical Review D 108 10, 106022 (2023).
 Angel Ballesteros, Flaminia Giacomini, and Giulia Gubitosi, "The group structure of dynamical transformations between quantum reference frames", Quantum 5, 470 (2021).
 Nuriya Nurgalieva and Renato Renner, "Testing quantum theory with thought experiments", Contemporary Physics 61 3, 193 (2020).
 Christophe Goeller, Philipp A. Hoehn, and Josh Kirklin, "Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance", arXiv:2206.01193, (2022).
 Giovanni Amelino-Camelia, Vittorio D'Esposito, Giuseppe Fabiano, Domenico Frattulillo, Philipp A. Hoehn, and Flavio Mercati, "Quantum Euler angles and agency-dependent spacetime", arXiv:2211.11347, (2022).
The above citations are from Crossref's cited-by service (last updated successfully 2023-12-07 10:31:33) and SAO/NASA ADS (last updated successfully 2023-12-07 10:31:34). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.