# Switching Quantum Reference Frames for Quantum Measurement

Jianhao M. Yang

Qualcomm, San Diego, CA 92121, USA

### Abstract

Physical observation is made relative to a reference frame. A reference frame is essentially a quantum system given the universal validity of quantum mechanics. Thus, a quantum system must be described relative to a quantum reference frame (QRF). Further requirements on QRF include using only relational observables and not assuming the existence of external reference frame. To address these requirements, two approaches are proposed in the literature. The first one is an operational approach (F. Giacomini, et al, Nat. Comm. 10:494, 2019) which focuses on the quantization of transformation between QRFs. The second approach attempts to derive the quantum transformation between QRFs from first principles (A. Vanrietvelde, et al, $\textit{Quantum}$ 4:225, 2020). Such first principle approach describes physical systems as symmetry induced constrained Hamiltonian systems. The Dirac quantization of such systems before removing redundancy is interpreted as perspective-neutral description. Then, a systematic redundancy reduction procedure is introduced to derive description from perspective of a QRF. The first principle approach recovers some of the results from the operational approach, but not yet include an important part of a quantum theory - the measurement theory. This paper is intended to bridge the gap. We show that the von Neumann quantum measurement theory can be embedded into the perspective-neutral framework. This allows us to successfully recover the results found in the operational approach, with the advantage that the transformation operator can be derived from the first principle. In addition, the formulation presented here reveals several interesting conceptual insights. For instance, the projection operation in measurement needs to be performed after redundancy reduction, and the projection operator must be transformed accordingly when switching QRFs. These results represent one step forward in understanding how quantum measurement should be formulated when the reference frame is also a quantum system.

The paper is a step forward in the investigation of quantum measurement with quantum reference frames (QRF). Utilizing the recently published first-principle approach to the formulation of quantum mechanics when switching QRFs, the paper addresses the following issue of quantum measurement: How should the same measurement process be described from the points of view of different QRFs? How do the descriptions relate to one another? In the context of the first-principle approach, the answer is this: the unitary evolution of the measurement process can be embedded in a perspective-neutral framework, but the measurement projection is perspectival, and thus shall be implemented after the QRF is specified.

### ► References

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

[2] Aharonov, Y. and Susskind, L. 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] Aharonov, Y. and Kaufherr, T. Quantum frames of reference. Phys. Rev. D. 30.2,368 (1984).
https:/​/​doi.org/​10.1103/​PhysRevD.30.368

[4] Palmer, M. C., Girelli, F. and Bartlett, S. D. Changing quantum reference frames. Phys. Rev. A. 89.5, 052121 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.052121

[5] Bartlett, S. D., Rudolph, T., Spekkens, R. W. and Turner, P. S. Degradation of a quantum reference frame. N. J. Phys. 8.4, 58 (2006).
https:/​/​doi.org/​10.1088/​1367-2630/​8/​4/​058

[6] Poulin, D. and Yard, J. Dynamics of a quantum reference frame. N. J. Phys. 9.5, 156 (2007).
https:/​/​doi.org/​10.1088/​1367-2630/​9/​5/​156

[7] Rovelli, C. Quantum reference systems. Class. Quantum Gravity 8.2, 317 (1991).
https:/​/​doi.org/​10.1088/​0264-9381/​8/​2/​012

[8] Poulin, D. Toy model for a relational formulation of quantum theory. Int. J. Theor. Phys. 45.7, 1189–1215 (2006).
https:/​/​doi.org/​10.1007/​s10773-006-9052-0

[9] Girelli, F. and Poulin, D. Quantum reference frames and deformed symmetries. Phys. Rev. D 77.10, 104012 (2008).
https:/​/​doi.org/​10.1103/​PhysRevD.77.104012

[10] Loveridge, L., Miyadera, T. and Busch, P. Symmetry, reference frames, and relational quantities in quantum mechanics. Found. Phys. 48, 135–198 (2018).
https:/​/​doi.org/​10.1007/​s10701-018-0138-3

[11] J. Pienaar, A relational approach to quantum reference frames for spins. arXiv preprint at arXiv:1601.07320 (2016).
arXiv:1601.07320

[12] Angelo, R. M., Brunner, N., Popescu, S., Short, A. and Skrzypczyk, P. Physics within a quantum reference frame. J. Phys. A 44.14, 145304 (2011).
https:/​/​doi.org/​10.1088/​1751-8113/​44/​14/​145304

[13] Angelo, R. M. and Ribeiro, A. D. Kinematics and dynamics in noninertial quantum frames of reference. J. Phys. A 45.46, 465306 (2012).
https:/​/​doi.org/​10.1088/​1751-8113/​45/​46/​465306

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

[15] Gour, G., and Spekkens, R. W. The resource theory of quantum reference frames: manipulations and monotones. N. J. Phys. 10.3, 033023 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​3/​033023

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

[17] Rovelli, C.: Relational Quantum Mechanics, Int. J. Theor. Phys., 35, 1637-1678 (1996).
https:/​/​doi.org/​10.1007/​BF02302261

[18] Smerlak M., and Rovelli, C.: Relational EPR, Found. Phys., 37, 427-445 (2007).
https:/​/​doi.org/​10.1007/​s10701-007-9105-0

[19] Transsinelli, M.: Relational Quantum Mechanics and Probability, Found. Phys., 48, 1092-1111 (2018).
https:/​/​doi.org/​10.1007/​s10701-018-0207-7

[20] Rovelli, C.: Space is blue and birds fly through it", Phil. Trans. R. Soc. A 376.
https:/​/​doi.org/​10.1098/​rsta.2017.0312

[21] Yang, J. M.: A Relational Formulation of Quantum Mechanics, Sci. Rep. 8:13305 (2018), arXiv:1706.01317.
https:/​/​doi.org/​10.1038/​s41598-018-31481-8
arXiv:1706.01317

[22] Yang, J. M.: Relational Formulation of Quantum Measurement, Int. J. Theor. Phys. 58 (3), 757-785 (2019), arXiv:1803.04843.
https:/​/​doi.org/​10.1007/​s10773-018-3973-2
arXiv:1803.04843

[23] P. A. Höhn, Toolbox for reconstructing quantum theory from rules on information acquisition, Quantum 1, 38 (2017).
https:/​/​doi.org/​10.22331/​q-2017-12-14-38

[24] P. A. Höhn, Quantum theory from questions, Phys. Rev. A 95, 012102 (2017), arXiv:1511.01130v7.
https:/​/​doi.org/​10.1103/​PhysRevA.95.012102
arXiv:1511.01130v7

[25] F. Giacomini, E. Castro-Ruiz, C. Brukner, Quantum Mechanics and the Covariance of Physical Laws in Quantum Reference Frame, Nat. Comm. 10:494 (2019).
https:/​/​doi.org/​10.1038/​s41467-018-08155-0

[26] A. Vanrietvelde, P. Höhn, F. Giacomini, and E. Castro-Ruiz, A Change of Perspective: Switching Quantum Reference Frames via a Perspective-neutral Framework, Quantum 4:225 (2020), arXiv:1809.00556.
https:/​/​doi.org/​10.22331/​q-2020-01-27-225
arXiv:1809.00556

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

[28] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992.

[29] C. Rovelli, Quantum Gravity. Cambridge University Press, 2004.

[30] T. Thiemann, Modern Canonical Quantum General Relativity. Cambridge University Press, 2007.

[31] A. Vanrietvelde, P. Hoehn, and F. Giacomini, Switching Quantum Reference Frames in the N-body Problem and the Absence of Global Relational Perspectives, arXiv:1809.05093.
arXiv:1809.05093

[32] Von Neumann, J.: Mathematical Foundations of Quantum Mechanics, Chap. VI. Princeton University Press, Princeton Translated by Robert T. Beyer (1932/​1955).

[33] Wigner, E.H.: Remarks on the mind-body question, in Symmetries and Reflections, pp 171-184 (Indiana University, 1967).

[34] E. Wigner, The Scientist Speculates, edited by I. Good, pp. 284–302 (1961).

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

[36] Nielsen, M. A., and Chuang, I. L.: Quantum computation and quantum information. page 94-95, Cambridge University Press, Cambridge (2000).

[37] E. P. Wigner, Die Messun Quantenmechanischer Operatoren, Z. Phys. 133, 101 (1952).

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

[39] M. Ahmadi1, D. Jennings and T. Rudolph, The Wigner–Araki–Yanase theorem and the quantum resource theory of asymmetry, New J. of Phys. 15 (2013) 013057.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​1/​013057

[40] M. Proietti, et al., Experimental rejection of observer-independence in the quantum world, Sci. Adv., Vol. 5, No. 9 (2019), arXiv:1902.05080 (2019).
arXiv:1902.05080

[41] Einstein, A., Podolsky, B., and Rosen, N.: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777-780 (1935).
https:/​/​doi.org/​10.1103/​PhysRev.47.777

[42] D. Frauchiger, R. Renner, Quantum Theory Cannot Consistently Describe The Use of Itself, Nature Comm. 3711, 9 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-05739-8

[43] Yang, J. M.: Consistent Descriptions of Quantum Measurement, Found. Phys. 49 (11): 1306-1324 (2019), arXiv:1812.00985.
https:/​/​doi.org/​10.1007/​s10701-019-00305-8
arXiv:1812.00985

[44] Brukner, C.: A no-go theorem for observer-independent facts, Entropy 20, 350, (2018).
https:/​/​doi.org/​10.3390/​e20050350

### Cited by

[1] Pierre Martin-Dussaud, "Perspective on: Switching Quantum Reference Frames for Quantum Measurement", Quantum Views 4, 40 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2020-07-14 03:14:59). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2020-07-14 03:14:59).