Publication: Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures
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2015-10-15
Authors
Bent, N.
Qassim, H.
Tahir, A. A.
Sych, D.
Leuchs, Gerd
Karimi, E.
Boyd, R. W.
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American Physical Society
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Abstract
Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime. Fidelities of reconstruction within the range of 0.81-0.96 are obtained for both pure and mixed states. These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.
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© American Physical Society.
The authors thank Markus Grassl and Joseph M. Renes for fruitful discussions. The authors acknowledge the support of the Canada Excellence Research Chairs (CERC) Program. R.W. B. was supported by the DARPA InPho program. L. L. S.-S. acknowledges support from the Spanish MINECO (Grant No. FIS2011-26786) and the Program UCM-BSCH (Grant No. GR3/14).
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1. W. K. Wootters and B. D. Fields, Optimal State-Determination by Mutually Unbiased Measurements, Ann. Phys. (N.Y.) 191, 363 (1989).
2. R. Derka, V. Bužek, and A. Ekert, Universal Algorithm for Optimal Estimation of Quantum States from Finite Ensembles via Realizable Generalized Measurement, Phys. Rev. Lett. 80, 1571 (1998).
3. A. J. Scott, Tight Informationally Complete Quantum Measurements, J. Phys. A 39, 13507 (2006).
4. H. Zhu and B.-G. Englert, Quantum State Tomography with Fully Symmetric Measurements and Product Measurements, Phys. Rev. A 84, 022327 (2011).
5. J. Nunn, B. J. Smith, G. Puentes, I. A. Walmsley, and J. S. Lundeen, Optimal Experiment Design for Quantum State Tomography: Fair, Precise, and Minimal Tomography, Phys. Rev. A 81, 042109 (2010).
6. D. McNulty and S. Weigert, On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six, Int. J. Quantum. Inform. 10, 1250056 (2012).
7. V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, Test of Mutually Unbiased Bases for Six-Dimensional Photonic Quantum Systems, Sci. Rep. 3, 2726 (2013).
8. D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, Characterization of High-Dimensional Entangled Systems via Mutually Unbiased Measurements, Phys. Rev. Lett. 110, 143601 (2013).
9. T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, On Mutually Unbiased Bases, Int. J. Quantum. Inform. 08, 535 (2010).
10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).
11. D. Sych, J. Řeháček, Z. Hradil, G. Leuchs, and L. L. Sánchez-Soto, Informational Completeness of Continuous-Variable Measurements, Phys. Rev. A 86, 052123 (2012).
12. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric Informationally Complete Quantum Measurements, J. Math. Phys. (N.Y.) 45, 2171 (2004).
13. G. Zauner, Quantum Designs: Foundations of a Noncommutative Design Theory, Int. J. Quantum. Inform. 09, 445 (2011).
14. A. J. Scott and M. Grassl, Symmetric Informationally Complete Positive-Operator-Valued Measures: A New Computer Study, J. Math. Phys. (N.Y.) 51, 042203 (2010).
15. B.-G. Englert, W. K. Chua, J. Anders, D. Kaszlikowski, and H. K. Ng, Highly Efficient Quantum Key Distribution with Minimal State Tomography, 2004.
16. T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, Wigner Tomography of Two-Qubit States and Quantum Cryptography, Phys. Rev. A 78, 042338 (2008).
17. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear Optical Quantum Computing with Photonic Qubits, Rev. Mod. Phys. 79, 135 (2007).
18. Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, Experimental Characterization of Qutrits Using Symmetric Informationally Complete Positive Operator-Valued Measurements, Phys. Rev. A 83, 051801 (2011).
19. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP Publishing, Bristol, England, 2003).
20. G. Molina-Terriza, J. P. Torres, and L. Torner, Twisted Photons, Nat. Phys. 3, 305 (2007).
21. S. Franke-Arnold, L. Allen, and M. Padgett, Advances in Optical Angular Momentum, Laser Photonics Rev. 2, 299 (2008).
22. L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, Spin-to-Orbital Conversion of the Angular Momentum of Light and Its Classical and Quantum Applications, J. Opt. 13, 064001 (2011).
23. M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, Higher-Dimensional Orbital-Angular-Momentum-Based Quantum Key Distribution with Mutually Unbiased Bases, Phys. Rev. A 88, 032305 (2013).
24. J. Schwinger, Unitary Operator Basis, Proc. Natl. Acad. Sci. U.S.A. 46, 570 (1960).
25. I. D. Ivanovic, Geometrical Description of Quantal State Determination, J. Phys. A 14, 3241 (1981).
26. S. T. Flammia, A. Silberfarb, and C. M. Caves, Minimal Informationally Complete Measurements, Found. Phys. 35, 1985 (2005).
27. J. Romero, J. Leach, B. Jack, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, Violation of Leggett Inequalities in Orbital Angular Momentum Subspaces, New J. Phys. 12, 123007 (2010).
28. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, Violation of Leggett-Type Inequalities in the Spin-Orbit Degrees of Freedom of a Single Photon, Phys. Rev. A 88, 032101 (2013).
29. D. Petz and L. Ruppert, Optimal Quantum-State Tomography with Known Parameters, J. Phys. A 45, 085306 (2012).
30. B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G. Guo, Quantum State Tomography via Linear Regression Estimation, Sci. Rep. 3, 03496 (2013).
31. J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Quantum State Estimation (Springer, Berlin, 2004), pp. 113–145.
32. M. S. Kaznady and D. F. V. James, Numerical Strategies for Quantum Tomography: Alternatives to Full Optimization, Phys. Rev. A 79, 022109 (2009).
33. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, Exact Solution to Simultaneous Intensity and Phase Encryption with a Single Phase-Only Hologram, Opt. Lett. 38, 3546 (2013).
34. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of the Orbital Angular Momentum States of Photons, Nature (London) 412, 313 (2001).
35. E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, Exploring the Quantum Nature of the Radial Degree of Freedom of a Photon via Hong-Ou-Mandel Interference, Phys. Rev. A 89, 013829 (2014).
36. H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, Limitations to the Determination of a Laguerre-Gauss Spectrum via Projective, Phase-Flattening Measurement, J. Opt. Soc. Am. B 31, A20 (2014).
37. Fidelity is defined as F=(Trρ^√ρ^rρ^√−−−−−−−−√)2, where ρr and ρ are the reconstructed and expected density matrices, respectively.
38. K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Maximum-Likelihood Estimation of the Density Matrix, Phys. Rev. A 61, 010304 (1999).
39. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Qudit Quantum-State Tomography, Phys. Rev. A 66, 012303 (2002).
40. C. H. Bennett and G. Brassard, Quantum Cryptography: Public Key Distribution and Coin Tossing, in Proceedings of the International Conference on Computers, Systems and Signal Processing, Bangalore, 1984 (IEEE, Bangalore, 1984), pp. 175–179.
41. A. K. Ekert, Quantum Cryptography Based on Bell’s Theorem, Phys. Rev. Lett. 67, 661 (1991).
42. D. V. Sych, B. A. Grishanin, and V. N. Zadkov, Analysis of Limiting Information Characteristics of Quantum-Cryptography Protocols, Quantum Electron. 35, 80 (2005).
43. J. M. Renes, Spherical-Code Key-Distribution Protocols for Qubits, Phys. Rev. A 70, 052314 (2004).
44. B. G. Englert, D. Kaszlikowski, H. K. Ng, W. K. Chua, J. Řeháček, and J. Anders, Efficient and Robust Quantum Key Distribution with Minimal State Tomography, arXiv:quant-ph/0412075.