Theory and experiment for resource-efficient joint weak-measurement

Aldo C. Martinez-Becerril1, Gabriel Bussières1, Davor Curic2, Lambert Giner1,3, Raphael A. Abrahao1,4, and Jeff S. Lundeen1,4

1Department of Physics and Centre for Research in Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
2Complexity Science Group, Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
3Département de Physique et d’Astronomie, Université de Moncton, 18 Ave. Antonine-Maillet, Moncton, New Brunswick E1A 3E9, Canada
4Joint Centre for Extreme Photonics, University of Ottawa - National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada

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Abstract

Incompatible observables underlie pillars of quantum physics such as contextuality and entanglement. The Heisenberg uncertainty principle is a fundamental limitation on the measurement of the product of incompatible observables, a 'joint' measurement. However, recently a method using weak measurement has experimentally demonstrated joint measurement. This method [Lundeen, J. S., and Bamber, C. Phys. Rev. Lett. 108, 070402, 2012] delivers the standard expectation value of the product of observables, even if they are incompatible. A drawback of this method is that it requires coupling each observable to a distinct degree of freedom (DOF), i.e., a disjoint Hilbert space. Typically, this 'read-out' system is an unused internal DOF of the measured particle. Unfortunately, one quickly runs out of internal DOFs, which limits the number of observables and types of measurements one can make. To address this limitation, we propose and experimentally demonstrate a technique to perform a joint weak-measurement of two incompatible observables using only one DOF as a read-out system. We apply our scheme to directly measure the density matrix of photon polarization states.

Many phenomena in quantum mechanics involve the measurement of the product of two incompatible observables of a quantum system. For example, the fundamental limitation on a joint measurement of two incompatible observables, e.g., position and momentum, is set by the Heisenberg uncertainty principle. Previous work used weak measurement for the realization of joint measurements. Weak measurement is a technique where the disturbance caused by the measurement process is reduced. These joint weak-measurements require a read-out system per measured observable. In this work, we theoretically introduce and experimentally demonstrate a joint weak-measurement technique that uses a single read-out system. This frees useful internal degrees of freedom of the measured quantum system for other tasks. We apply our scheme to directly measure the density matrix of photon polarization states.

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► References

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

[2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, 2009. https:/​/​doi.org/​10.1103/​RevModPhys.81.865.
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[3] Yakir Aharonov and Lev Vaidman. Properties of a quantum system during the time interval between two measurements. Phys. Rev. A, 41: 11–20, 1990. https:/​/​doi.org/​10.1103/​PhysRevA.41.11.
https:/​/​doi.org/​10.1103/​PhysRevA.41.11

[4] K. J. Resch and A. M. Steinberg. Extracting Joint Weak Values with Local, Single-Particle Measurements. Phys. Rev. Lett., 92: 130402, 2004. https:/​/​doi.org/​10.1103/​PhysRevLett.92.130402.
https:/​/​doi.org/​10.1103/​PhysRevLett.92.130402

[5] J.S. Lundeen and K.J. Resch. Practical measurement of joint weak values and their connection to the annihilation operator. Physics Letters A, 334 (5): 337–344, 2005. ISSN 0375-9601. https:/​/​doi.org/​10.1016/​j.physleta.2004.11.037.
https:/​/​doi.org/​10.1016/​j.physleta.2004.11.037

[6] G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen. Direct Measurement of the Density Matrix of a Quantum System. Phys. Rev. Lett., 117: 120401, 2016. https:/​/​doi.org/​10.1103/​PhysRevLett.117.120401.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.120401

[7] F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese. Measuring Incompatible Observables by Exploiting Sequential Weak Values. Phys. Rev. Lett., 117: 170402, 2016. https:/​/​doi.org/​10.1103/​PhysRevLett.117.170402.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.170402

[8] M.A. Ochoa, W. Belzig, and A. Nitzan. Simultaneous weak measurement of non-commuting observables: a generalized Arthurs-Kelly protocol. Sci. Rep., 8 (1): 1–8, 2018. https:/​/​doi.org/​10.1038/​s41598-018-33562-0.
https:/​/​doi.org/​10.1038/​s41598-018-33562-0

[9] Y. Kim, YS Kim, SY Lee, Sang-Wook Han, Sung Moon, Yoon-Ho Kim, and Young-Wook Cho. Direct quantum process tomography via measuring sequential weak values of incompatible observables. Nat. Commun., 9 (1): 1–6, 2018. https:/​/​doi.org/​10.1038/​s41467-017-02511-2.
https:/​/​doi.org/​10.1038/​s41467-017-02511-2

[10] Jiang-Shan Chen, Meng-Jun Hu, Xiao-Min Hu, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Can-Guang Guo, and Yong-Sheng Zhang. Experimental realization of sequential weak measurements of non-commuting Pauli observables. Opt. Express, 27 (5): 6089–6097, 2019. https:/​/​doi.org/​10.1364/​OE.27.006089.
https:/​/​doi.org/​10.1364/​OE.27.006089

[11] Yakir Aharonov, David Z. Albert, and Lev Vaidman. How the result of a measurement of a component of the spin of a spin-1/​2 particle can turn out to be 100. Phys. Rev. Lett., 60: 1351–1354, 1988. https:/​/​doi.org/​10.1103/​PhysRevLett.60.1351.
https:/​/​doi.org/​10.1103/​PhysRevLett.60.1351

[12] G S Thekkadath, F Hufnagel, and J S Lundeen. Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance. New Journal of Physics, 20 (11): 113034, 2018. https:/​/​doi.org/​10.1088/​1367-2630/​aaecdf.
https:/​/​doi.org/​10.1088/​1367-2630/​aaecdf

[13] Onur Hosten and Paul Kwiat. Observation of the Spin Hall Effect of Light via Weak Measurements. Science, 319 (5864): 787–790, 2008. ISSN 0036-8075. https:/​/​doi.org/​10.1126/​science.1152697.
https:/​/​doi.org/​10.1126/​science.1152697

[14] David J. Starling, P. Ben Dixon, Andrew N. Jordan, and John C. Howell. Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values. Phys. Rev. A, 80: 041803, 2009. https:/​/​doi.org/​10.1103/​PhysRevA.80.041803.
https:/​/​doi.org/​10.1103/​PhysRevA.80.041803

[15] Nicolas Brunner and Christoph Simon. Measuring Small Longitudinal Phase Shifts: Weak Measurements or Standard Interferometry? Phys. Rev. Lett., 105: 010405, 2010. https:/​/​doi.org/​10.1103/​PhysRevLett.105.010405.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.010405

[16] Jeff S. Lundeen, Brandon Sutherland, Aabid Patel, Corey Stewart, and Charles Bamber. Direct measurement of the quantum wavefunction. Nature, 474 (7350): 188–191, 2011. https:/​/​doi.org/​10.1038/​nature10120.
https:/​/​doi.org/​10.1038/​nature10120

[17] Justin Dressel, Mehul Malik, Filippo M. Miatto, Andrew N. Jordan, and Robert W. Boyd. Colloquium: Understanding quantum weak values: Basics and applications. Rev. Mod. Phys., 86: 307–316, 2014. https:/​/​doi.org/​10.1103/​RevModPhys.86.307.
https:/​/​doi.org/​10.1103/​RevModPhys.86.307

[18] Graeme Mitchison, Richard Jozsa, and Sandu Popescu. Sequential weak measurement. Phys. Rev. A, 76: 062105, 2007. https:/​/​doi.org/​10.1103/​PhysRevA.76.062105.
https:/​/​doi.org/​10.1103/​PhysRevA.76.062105

[19] Jeff S. Lundeen and Charles Bamber. Procedure for Direct Measurement of General Quantum States Using Weak Measurement. Phys. Rev. Lett., 108: 070402, 2012. https:/​/​doi.org/​10.1103/​PhysRevLett.108.070402.
https:/​/​doi.org/​10.1103/​PhysRevLett.108.070402

[20] Yakir Aharonov, Sandu Popescu, Daniel Rohrlich, and Paul Skrzypczyk. Quantum Cheshire Cats. New Journal of Physics, 15 (11): 113015, 2013. https:/​/​doi.org/​10.1088/​1367-2630/​15/​11/​113015.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​11/​113015

[21] T Denkmayr, H Geppert, S Sponar, H Lemmel, A Matzkin, J Tollaksen, and Y Hasegawa. Observation of a quantum Cheshire cat in a matter-wave interferometer experiment. Nat. Commun., 5 (1): 1–7, 2014. https:/​/​doi.org/​10.1038/​ncomms5492.
https:/​/​doi.org/​10.1038/​ncomms5492

[22] J. S. Lundeen and A. M. Steinberg. Experimental Joint Weak Measurement on a Photon Pair as a Probe of Hardy's Paradox. Phys. Rev. Lett., 102: 020404, 2009. https:/​/​doi.org/​10.1103/​PhysRevLett.102.020404.
https:/​/​doi.org/​10.1103/​PhysRevLett.102.020404

[23] Kazuhiro Yokota, Takashi Yamamoto, Masato Koashi, and Nobuyuki Imoto. Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair. New Journal of Physics, 11 (3): 033011, 2009. https:/​/​doi.org/​10.1088/​1367-2630/​11/​3/​033011.
https:/​/​doi.org/​10.1088/​1367-2630/​11/​3/​033011

[24] Justin Dressel, Areeya Chantasri, Andrew N. Jordan, and Alexander N. Korotkov. Arrow of Time for Continuous Quantum Measurement. Phys. Rev. Lett., 119: 220507, 2017. https:/​/​doi.org/​10.1103/​PhysRevLett.119.220507.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.220507

[25] D. Curic, M. C. Richardson, G. S. Thekkadath, J. Flórez, L. Giner, and J. S. Lundeen. Experimental investigation of measurement-induced disturbance and time symmetry in quantum physics. Phys. Rev. A, 97: 042128, 2018. https:/​/​doi.org/​10.1103/​PhysRevA.97.042128.
https:/​/​doi.org/​10.1103/​PhysRevA.97.042128

[26] Holger F. Hofmann. Uncertainty limits for quantum metrology obtained from the statistics of weak measurements. Phys. Rev. A, 83: 022106, 2011. https:/​/​doi.org/​10.1103/​PhysRevA.83.022106.
https:/​/​doi.org/​10.1103/​PhysRevA.83.022106

[27] Lee A. Rozema, Ardavan Darabi, Dylan H. Mahler, Alex Hayat, Yasaman Soudagar, and Aephraim M. Steinberg. Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements. Phys. Rev. Lett., 109: 100404, 2012. https:/​/​doi.org/​10.1103/​PhysRevLett.109.100404.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.100404

[28] Shengshi Pang, Justin Dressel, and Todd A. Brun. Entanglement-Assisted Weak Value Amplification. Phys. Rev. Lett., 113: 030401, 2014. https:/​/​doi.org/​10.1103/​PhysRevLett.113.030401.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.030401

[29] Andrew N. Jordan, Julián Martínez-Rincón, and John C. Howell. Technical Advantages for Weak-Value Amplification: When Less Is More. Phys. Rev. X, 4: 011031, 2014. https:/​/​doi.org/​10.1103/​PhysRevX.4.011031.
https:/​/​doi.org/​10.1103/​PhysRevX.4.011031

[30] Jérémie Harris, Robert W. Boyd, and Jeff S. Lundeen. Weak Value Amplification Can Outperform Conventional Measurement in the Presence of Detector Saturation. Phys. Rev. Lett., 118: 070802, 2017. https:/​/​doi.org/​10.1103/​PhysRevLett.118.070802.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.070802

[31] Binke Xia, Jingzheng Huang, Chen Fang, Hongjing Li, and Guihua Zeng. High-Precision Multiparameter Weak Measurement with Hermite-Gaussian Pointer. Phys. Rev. Applied, 13: 034023, 2020. https:/​/​doi.org/​10.1103/​PhysRevApplied.13.034023.
https:/​/​doi.org/​10.1103/​PhysRevApplied.13.034023

[32] Nicole Yunger Halpern, Brian Swingle, and Justin Dressel. Quasiprobability behind the out-of-time-ordered correlator. Phys. Rev. A, 97: 042105, Apr 2018. https:/​/​doi.org/​10.1103/​PhysRevA.97.042105.
https:/​/​doi.org/​10.1103/​PhysRevA.97.042105

[33] Marco Barbieri. Multiple-measurement Leggett-Garg inequalities. Phys. Rev. A, 80: 034102, Sep 2009. https:/​/​doi.org/​10.1103/​PhysRevA.80.034102.
https:/​/​doi.org/​10.1103/​PhysRevA.80.034102

[34] N. W. M. Ritchie, J. G. Story, and Randall G. Hulet. Realization of a measurement of a ``weak value''. Phys. Rev. Lett., 66: 1107–1110, 1991. https:/​/​doi.org/​10.1103/​PhysRevLett.66.1107.
https:/​/​doi.org/​10.1103/​PhysRevLett.66.1107

[35] S Glancy, E Knill, and M Girard. Gradient-based stopping rules for maximum-likelihood quantum-state tomography. New Journal of Physics, 14 (9): 095017, 2012. https:/​/​doi.org/​10.1088/​1367-2630/​14/​9/​095017.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​9/​095017

[36] Eliot Bolduc, George C Knee, Erik M Gauger, and Jonathan Leach. Projected gradient descent algorithms for quantum state tomography. npj Quantum Information, 3 (1): 1–9, 2017. https:/​/​doi.org/​10.1038/​s41534-017-0043-1.
https:/​/​doi.org/​10.1038/​s41534-017-0043-1

[37] Rajveer Nehra, Miller Eaton, Carlos González-Arciniegas, M. S. Kim, Thomas Gerrits, Adriana Lita, Sae Woo Nam, and Olivier Pfister. Generalized overlap quantum state tomography. Phys. Rev. Research, 2: 042002, 2020. https:/​/​doi.org/​10.1103/​PhysRevResearch.2.042002.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.042002

[38] J Z Salvail, M Agnew, Allan S Johnson, Eliot Bolduc, Jonathan Leach, and Robert W Boyd. Full characterization of polarization states of light via direct measurement. Nat. Photonics, 7 (4): 316–321, 2013. https:/​/​doi.org/​10.1038/​nphoton.2013.24.
https:/​/​doi.org/​10.1038/​nphoton.2013.24

[39] Charles Bamber and Jeff S. Lundeen. Observing Dirac's Classical Phase Space Analog to the Quantum State. Phys. Rev. Lett., 112: 070405, 2014. https:/​/​doi.org/​10.1103/​PhysRevLett.112.070405.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.070405

[40] Howard M. Wiseman and Gerard J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2009. https:/​/​doi.org/​10.1017/​CBO9780511813948.
https:/​/​doi.org/​10.1017/​CBO9780511813948

[41] Victor Namias. The Fractional Order Fourier Transform and its Application to Quantum Mechanics. IMA Journal of Applied Mathematics, 25 (3): 241–265, 1980. ISSN 0272-4960. https:/​/​doi.org/​10.1093/​imamat/​25.3.241.
https:/​/​doi.org/​10.1093/​imamat/​25.3.241

[42] Adolf W. Lohmann. Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A, 10 (10): 2181–2186, 1993. https:/​/​doi.org/​10.1364/​JOSAA.10.002181.
https:/​/​doi.org/​10.1364/​JOSAA.10.002181

[43] Haldun M. Ozaktas and David Mendlovic. Fractional Fourier optics. J. Opt. Soc. Am. A, 12 (4): 743–751, 1995. https:/​/​doi.org/​10.1364/​JOSAA.12.000743.
https:/​/​doi.org/​10.1364/​JOSAA.12.000743

[44] Steffen Weimann, Armando Perez-Leija, Maxime Lebugle, Robert Keil, Malte Tichy, Markus Gräfe, René Heilmann, Stefan Nolte, Hector Moya-Cessa, Gregor Weihs, et al. Implementation of quantum and classical discrete fractional Fourier transforms. Nat. Commun., 7: 11027, 2016. https:/​/​doi.org/​10.1038/​ncomms11027.
https:/​/​doi.org/​10.1038/​ncomms11027

[45] A. Hariri, D. Curic, L. Giner, and J. S. Lundeen. Experimental simultaneous readout of the real and imaginary parts of the weak value. Phys. Rev. A, 100: 032119, 2019. https:/​/​doi.org/​10.1103/​PhysRevA.100.032119.
https:/​/​doi.org/​10.1103/​PhysRevA.100.032119

[46] Gregory D Scholes, Graham R Fleming, Lin X Chen, Alán Aspuru-Guzik, Andreas Buchleitner, David F Coker, Gregory S Engel, Rienk Van Grondelle, Akihito Ishizaki, David M Jonas, et al. Using coherence to enhance function in chemical and biophysical systems. Nature, 543 (7647): 647–656, 2017. https:/​/​doi.org/​10.1038/​nature21425.
https:/​/​doi.org/​10.1038/​nature21425

[47] E. Merzbacher. Quantum Mechanics. Wiley, 1998. ISBN 9780471887027.

[48] Arthur Fine. Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett., 48: 291–295, Feb 1982. https:/​/​doi.org/​10.1103/​PhysRevLett.48.291.
https:/​/​doi.org/​10.1103/​PhysRevLett.48.291

[49] Matthew F. Pusey. Anomalous Weak Values Are Proofs of Contextuality. Phys. Rev. Lett., 113: 200401, Nov 2014. https:/​/​doi.org/​10.1103/​PhysRevLett.113.200401.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.200401

[50] Ravi Kunjwal, Matteo Lostaglio, and Matthew F. Pusey. Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts. Phys. Rev. A, 100: 042116, Oct 2019. https:/​/​doi.org/​10.1103/​PhysRevA.100.042116.
https:/​/​doi.org/​10.1103/​PhysRevA.100.042116

[51] Valeria Cimini, Ilaria Gianani, Fabrizio Piacentini, Ivo Pietro Degiovanni, and Marco Barbieri. Anomalous values, Fisher information, and contextuality, in generalized quantum measurements. Quantum Science and Technology, 5 (2): 025007, mar 2020. https:/​/​doi.org/​10.1088/​2058-9565/​ab7988.
https:/​/​doi.org/​10.1088/​2058-9565/​ab7988

[52] A. J. Leggett and Anupam Garg. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett., 54: 857–860, Mar 1985. https:/​/​doi.org/​10.1103/​PhysRevLett.54.857.
https:/​/​doi.org/​10.1103/​PhysRevLett.54.857

[53] Alessio Avella, Fabrizio Piacentini, Michelangelo Borsarelli, Marco Barbieri, Marco Gramegna, Rudi Lussana, Federica Villa, Alberto Tosi, Ivo Pietro Degiovanni, and Marco Genovese. Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality. Phys. Rev. A, 96: 052123, Nov 2017. https:/​/​doi.org/​10.1103/​PhysRevA.96.052123.
https:/​/​doi.org/​10.1103/​PhysRevA.96.052123

[54] J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan. Experimental violation of Two-Party Leggett-Garg Inequalities with Semiweak Measurements. Phys. Rev. Lett., 106: 040402, Jan 2011. https:/​/​doi.org/​10.1103/​PhysRevLett.106.040402.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.040402

[55] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde. Violation of the Leggett–Garg inequality with weak measurements of photons. Proceedings of the National Academy of Sciences, 108 (4): 1256–1261, 2011. ISSN 0027-8424. https:/​/​doi.org/​10.1073/​pnas.1005774108.
https:/​/​doi.org/​10.1073/​pnas.1005774108

[56] Justin Dressel and Alexander N. Korotkov. Avoiding loopholes with hybrid Bell-Leggett-Garg inequalities. Phys. Rev. A, 89: 012125, Jan 2014. https:/​/​doi.org/​10.1103/​PhysRevA.89.012125.
https:/​/​doi.org/​10.1103/​PhysRevA.89.012125

[57] Theodore C White, JY Mutus, Justin Dressel, J Kelly, R Barends, E Jeffrey, D Sank, A Megrant, B Campbell, Yu Chen, et al. Preserving entanglement during weak measurement demonstrated with a violation of the Bell–Leggett–Garg inequality. npj Quantum Information, 2 (1): 1–5, 2016. https:/​/​doi.org/​10.1038/​npjqi.2015.22.
https:/​/​doi.org/​10.1038/​npjqi.2015.22

[58] Wei-Wei Pan, Xiao-Ye Xu, Yaron Kedem, Qin-Qin Wang, Zhe Chen, Munsif Jan, Kai Sun, Jin-Shi Xu, Yong-Jian Han, Chuan-Feng Li, and Guang-Can Guo. Direct Measurement of a Nonlocal Entangled Quantum State. Phys. Rev. Lett., 123: 150402, Oct 2019. https:/​/​doi.org/​10.1103/​PhysRevLett.123.150402.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.150402

[59] Ming-Cheng Chen, Yuan Li, Run-Ze Liu, Dian Wu, Zu-En Su, Xi-Lin Wang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan. Directly Measuring a Multiparticle Quantum Wave Function via Quantum Teleportation. Phys. Rev. Lett., 127: 030402, Jul 2021. https:/​/​doi.org/​10.1103/​PhysRevLett.127.030402.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.030402

[60] Mohamed Bourennane, Manfred Eibl, Christian Kurtsiefer, Sascha Gaertner, Harald Weinfurter, Otfried Gühne, Philipp Hyllus, Dagmar Bruß, Maciej Lewenstein, and Anna Sanpera. Experimental Detection of Multipartite Entanglement using Witness Operators. Phys. Rev. Lett., 92: 087902, Feb 2004. https:/​/​doi.org/​10.1103/​PhysRevLett.92.087902.
https:/​/​doi.org/​10.1103/​PhysRevLett.92.087902

[61] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber. Entanglement certification from theory to experiment. Nature Reviews Physics, 1 (1): 72–87, 2019. https:/​/​doi.org/​10.1038/​s42254-018-0003-5.
https:/​/​doi.org/​10.1038/​s42254-018-0003-5

[62] EU Condon. Immersion of the Fourier transform in a continuous group of functional transformations. Proceedings of the National academy of Sciences of the United States of America, 23 (3): 158, 1937. https:/​/​doi.org/​10.1073/​pnas.23.3.158.
https:/​/​doi.org/​10.1073/​pnas.23.3.158

[63] Miguel A. Alonso. Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles. Adv. Opt. Photon., 3 (4): 272–365, Dec 2011. https:/​/​doi.org/​10.1364/​AOP.3.000272.
https:/​/​doi.org/​10.1364/​AOP.3.000272

[64] H.M. Ozaktas, D. Mendlovic, M.A. Kutay, and Z. Zalevsky. The Fractional Fourier Transform: With Applications in Optics and Signal Processing. Wiley Series in Pure and Applied Optics. Wiley, 2001. ISBN 9780471963462.

[65] P. Kunche and N. Manikanthababu. Fractional Fourier Transform Techniques for Speech Enhancement. Springer Briefs in Speech Technology. Springer International Publishing, 2020. ISBN 9783030427450. https:/​/​doi.org/​10.1007/​978-3-030-42746-7.
https:/​/​doi.org/​10.1007/​978-3-030-42746-7

[66] Naveen Kumar Nishchal, Joby Joseph, and Kehar Singh. Securing information using fractional Fourier transform in digital holography. Optics Communications, 235 (4): 253–259, 2004. ISSN 0030-4018. https:/​/​doi.org/​10.1016/​j.optcom.2004.02.052.
https:/​/​doi.org/​10.1016/​j.optcom.2004.02.052

[67] Pierre Pellat-Finet. Fresnel diffraction and the fractional-order Fourier transform. Optics Letters, 19 (18): 1388–1390, 1994. https:/​/​doi.org/​10.1364/​OL.19.001388.
https:/​/​doi.org/​10.1364/​OL.19.001388

[68] David Mendlovic and Haldun M Ozaktas. Fractional Fourier transforms and their optical implementation: I. JOSA A, 10 (9): 1875–1881, 1993. https:/​/​doi.org/​10.1364/​JOSAA.10.001875.
https:/​/​doi.org/​10.1364/​JOSAA.10.001875

[69] Haldun M Ozaktas and David Mendlovic. Fractional Fourier transforms and their optical implementation. II. JOSA A, 10 (12): 2522–2531, 1993. https:/​/​doi.org/​10.1364/​JOSAA.10.002522.
https:/​/​doi.org/​10.1364/​JOSAA.10.002522

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