Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States

John Martin1, Stefan Weigert2, and Olivier Giraud3

1Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège, B-4000 Liège, Belgium
2Department of Mathematics, University of York, UK-York YO10 5DD, United Kingdom
3Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France

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Abstract

Coherent and anticoherent states of spin systems up to spin $j=2$ are known to be optimal in order to detect rotations by a known angle but unknown rotation axis. These optimal quantum rotosensors are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. We calculate a closed-form expression for the average fidelity in terms of anticoherent measures, valid for arbitrary values of the quantum number $j$. We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to $j=7/2$ and (ii) for small rotation angles in the case of spin quantum numbers up to $j=5$. The closed-form expression we derive allows us to explain the central role of anticoherence measures in the problem of optimal detection of rotation angles for arbitrary values of $j$.

Advances in measurement techniques have often led to progress in physics. Over time, metrology developed as a subject of its own, especially in the context of defining standard units for physical quantities. Quantum theory provides new perspectives on measurements but also new challenges. The currently emerging quantum technologies require ever better control of microscopic systems and, hence, measurements which are as accurate as possible. In this paper, we are interested to determine whether a quantum system has undergone a rotation by a known angle about an unknown axis. The states optimally suited for this task are called "optimal quantum rotosensors". They are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. The closed-form expression we derive for the fidelity and numerical computations allow us to explain the central role of a specific class of quantum states in this problem which are known as "anticoherent" states.

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[1] W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43, 172 (1927).
https:/​/​doi.org/​10.1007/​BF01397280

[2] V. Giovannetti, S. Lloyd, L. Maccone, Advances in quantum metrology, Nature Photonics 5, 222 (2011).
https:/​/​doi.org/​10.1038/​nphoton.2011.35

[3] P. Busch, P. Lahti, J.-P. Pellonpää, K. Ylinen: Quantum Measurement, Springer 2016.
https:/​/​doi.org/​10.1007/​978-3-319-43389-9

[4] W. Nawrocki, Introduction to Quantum Metrology, Springer Nature 2019.
https:/​/​doi.org/​10.1007/​978-3-319-15669-9

[5] A. Peres and P. F. Scudo, Transmission of a Cartesian Frame by a Quantum System, Phys. Rev. Lett. 87, 167901 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.87.167901

[6] H. Hakoshima and Y. Matsuzaki, Efficient detection of inhomogeneous magnetic fields from the single spin with Dicke states, arXiv:2003.12524.
arXiv:2003.12524

[7] C. R. Rao, Information and the accuracy attainable in the estimation of statistical parameters, Bulletin of the Calcutta Mathematical Society 37, 81 (1945).
https:/​/​doi.org/​10.1007/​978-1-4612-0919-5_16

[8] H. Cramér, Mathematical Methods of Statistics (PMS-9), Princeton University Press 1946.
https:/​/​press.princeton.edu/​books/​paperback/​9780691005478/​mathematical-methods-of-statistics-pms-9-volume-9

[9] C. W. Helstrom, Quantum Detection and Estimation Theory, volume 123 of Mathematics in Science and Engineering, Elsevier (1976).
https:/​/​doi.org/​10.1007/​BF01007479

[10] M. Hübner, Explicit computation of the Bures distance for density matrices, Phys. Lett. A 163, 239 (1992).
https:/​/​doi.org/​10.1016/​0375-9601(92)91004-B

[11] M. Hübner, Computation of Uhlmann’s parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space, Phys. Lett. A 179, 226 (1993).
https:/​/​doi.org/​10.1016/​0375-9601(93)90668-P

[12] P. Kolenderski and R. Demkowicz-Dobrzanski, Optimal state for keeping reference frames aligned and the Platonic solids, Phys. Rev. A 78, 052333 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.052333

[13] A. Z. Goldberg and D. F. V. James, Quantum-limited Euler angle measurements using anticoherent states, Phys. Rev. A 98, 032113 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.032113

[14] Y. Mo and G. Chiribella, Quantum-enhanced learning of rotations about an unknown direction, New J. Phys. 21, 113003 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​ab4d9a

[15] C. Chryssomalakos and H. Hernández-Coronado, Optimal quantum rotosensors, Phys. Rev. A 95, 052125 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.052125

[16] F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Atomic Coherent States in Quantum Optics, Phys. Rev. A 6, 2211 (1972).
https:/​/​doi.org/​10.1103/​PhysRevA.6.2211

[17] J. Zimba, ``Anticoherent” Spin States via the Majorana Representation, Electr. J. Theor. Phys. 3, 143 (2006).
http:/​/​www.ejtp.com/​articles/​ejtpv3i10p143.pdf

[18] F. Bouchard, P. de la Hoz, G. Björk, R. W. Boyd, M. Grassl, Z. Hradil, E. Karimi, A. B. Klimov, G. Leuchs, J. Rehacek, and L. L. Sánchez-Soto, Quantum metrology at the limit with extremal Majorana constellations, Optica 4, 1429 (2017).
https:/​/​doi.org/​10.1364/​OPTICA.4.001429

[19] T. Chalopin, C. Bouazza, A. Evrard, V. Makhalov, D. Dreon, J. Dalibard, L. A. Sidorenkov, and S. Nascimbene, Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom, Nature Communications 9, 4955 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-07433-1

[20] D. Baguette and J. Martin, Anticoherence measures for pure spin states, Phys. Rev. A 96, 032304 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.032304

[21] L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Cambridge University Press 1984.
https:/​/​doi.org/​10.1017/​CBO9780511759888

[22] I. Bengtsson and K. Życzkowski, Geometry of Quantum States : An Introduction to Quantum Entanglement, 2nd ed. Cambridge University Press 2017.
https:/​/​doi.org/​10.1017/​9781139207010

[23] B. Coecke, A Representation for a Spin-S Entity as a Compound System in $\mathbb{R}^3$ Consisting of 2S Individual Spin-1/​2 Entities, Foundations of Physics 28, 1347 (1998).
https:/​/​doi.org/​10.1023/​A:1018878927186

[24] O. Giraud, D. Braun, D. Baguette, T. Bastin, and J. Martin, Tensor representation of spin states, Phys. Rev. Lett. 114, 080401 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.080401

[25] S. Weinberg, Feynman Rules for Any Spin, Phys. Rev. 133, B1318 (1964).
https:/​/​doi.org/​10.1103/​PhysRev.133.B1318

[26] D. Baguette, T. Bastin, and J. Martin, Multiqubit symmetric states with maximally mixed one-qubit reductions, Phys. Rev. A 90, 032314 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.032314

[27] O. Giraud, P. Braun, and D. Braun, Quantifying Quantumness and the Quest for Queens of Quantum, New J. Phys. 12, 063005 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​6/​063005

[28] G. Björk, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, L. L. Sánchez-Soto, Extremal quantum states and their Majorana constellations, Phys. Rev. A 92, 031801(R) (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.031801

[29] G. Björk, M. Grassl, P. de la Hoz, G. Leuchs and L. L. Sánchez-Soto, Stars of the quantum Universe: extremal constellations on the Poincaré sphere, Phys. Scr. 90, 108008 (2015).
https:/​/​doi.org/​10.1088/​0031-8949/​90/​10/​108008

[30] P. Delsarte, J. M. Goethals, J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6, 363 (1977).
https:/​/​doi.org/​10.1007/​BF03187604

[31] R. H. Hardin and N. J. A. Sloane, McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions, Discrete and Computational Geometry 15, 429 (1996).
https:/​/​doi.org/​10.1007/​BF02711518

[32] R. E. Schwartz, The Five-Electron Case of Thomson’s Problem, Experimental Mathematics 22, 157 (2013).
https:/​/​doi.org/​10.1080/​10586458.2013.766570

[33] D. Baguette, F. Damanet, O. Giraud, and J. Martin, Anticoherence of spin states with point-group symmetries, Phys. Rev. A 92, 052333 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.052333

[34] See e.g. http:/​/​polarization.markus-grassl.de/​, http:/​/​www.oq.ulg.ac.be and Refs. Bag17,Bjo15.
http:/​/​polarization.markus-grassl.de/​

[35] J. S. Sidhu and P. Kok, A Geometric Perspective on Quantum Parameter Estimation, AVS Quantum Science 2, 014701 (2020).
https:/​/​doi.org/​10.1116/​1.5119961

[36] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory Of Angular Momentum, World Scientific (1988).
https:/​/​doi.org/​10.1142/​0270

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