Abstract
We show the star-product quantization procedure for spin-1/2 particles (qubits) employing the construction of a pair of operators – dequantizers and quantizers. We present an explicit description of all minimal systems of such dequantizers and quantizers and discuss their relation to the probability representation of spin states where the fair probability distribution is identified with the spin states. We give some examples and discuss the possibility of constructing a symplectic structure in the finite-dimensional phase space.
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F. Lizzi and P. Vitale, SIGMA, 10, 086 (2014).
O. Man’ko, V. Man’ko, and G. Marmo, J. Phys. A: Math. Gen., 35, 699 (2002).
S. L. Stratonovich, Sov. Phys. JETP, 4, 891 (1957).
K. E. Cahill and R. J. Glauber, Phys. Rev., 176, 1857 (1968).
K. E. Cahill and R. J. Glauber, Phys. Rev., 177, 1882 (1969).
M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep., 106, 121 (1984).
V. I. Tatarskii, Sov. Phys. Uspekhi, 26, 331 (1983).
A. B. Klimov and S. M. Chumakov, A Group-Theoretical Approach to Quantum Optics, Wiley, Wenheim (2009).
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press (1995).
M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
W. P. Schleich, Quantum Optics in Phase Space, Wiley (2001).
E. P. Wigner, Phys. Rev., 40, 749 (1932).
K. Husimi, Proc. Phys. Math. Soc. Jpn., 22, 264 (1940).
Y. J. Kano, J. Math. Phys., 6, 1913 (1965).
R. J. Glauber, Phys. Rev. Lett., 10, 84 (1963).
E. C. G. Sudarshan, Phys. Rev. Lett., 10, 277 (1963).
S. Mancini, V. I. Man’ko, and P. Tombesi, Quantum Semiclass. Opt., 7, 615 (1995).
S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A, 213, 1 (1996).
S. Mancini, V. I. Man’ko, and P. Tombesi, Found. Phys., 27, 801 (1997).
A. Ibort, V. I. Man’ko, G. Marmo, et al, Phys. Scr., 79, 065013 (2009).
V. A. Andreev, D. M. Davidović, M. G. Davidović, et al., Theor. Math. Phys., 166, 356 (2011).
J. Schwinger, Proc. Natl. Acad. Sci. USA, 46, 570 (1960).
W. K. Wootters, IBM J. Res. Dev., 48, 99 (2004) [arXiv:quant-ph/0306135v4 9 Aug 2003].
A. Vourdas, Acta Applic. Math., 93, 197 (2006).
K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A, 70, 062101 (2004) [arXiv:quantph/0401155v6 2 Apr 2005].
A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez-Soto, J. Phys. A: Math. Gen., 40, 3987 (2007).
M. O Scully, Phys. Rev. D, 28, 2477 (1983).
P. Stóvicék and J. Tolar, Rep. Math. Phys., 20, 157 (1984).
S. Chaturvedi, E. Ercolessi, G. Marmo, et al., J. Phys. A: Math. Gen., 39, 1405 (2006).
S. N. Filippov and V. I. Man’ko, Phys. Scr., T143, 014010 (2011).
S. N. Filippov and V. I. Man’ko, Phys. Scr., 83, 058101 (2011).
S. N. Filippov and V. I. Man’ko, J. Phys. A: Math. Gen., 32, 56 (2011).
M. O. Terra-Cuñha, V. I. Man’ko, and M. O. Scully, Found. Phys. Lett., 14, 103 (2001).
V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 229, 335 (1997).
V. I. Man’ko and O. V. Man’ko, J. Exp. Theor. Phys., 85, 430 (1997).
V. A. Andreev and V. I. Man’ko, Sov. Phys. JETP, 87, 239 (1998).
V. A. Andreev, O. V. Man’ko, V. I. Man’ko, and S. S. Safonov, J. Russ. Laser Res., 19, 340 (1998).
V. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, Phys. Lett. A, 372, 6490 (2008).
F. Bayen, M. Flato, C. Fronsdal, et al., Lett. Math. Phys., 1, 521 (1977).
P. Adam, V. A. Andreev, A. Isar, et al., Theor. Math. Phys., 186, 346 (2016).
V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 24, 507 (2003).
V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Lett. A, 327, 353 (2004).
A. A. Kirillov, Elements of the Theory of Representations, Grundlehren der Mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, Springer-Verlag (1976), Vol. 220.
B. Kostant, Amer. J. Math., 85, 327 (1963).
P. Adam, V. A. Andreev, I. Ghiu, et al., J. Russ. Laser Res., 35, 3 (2014).
P. Adam, V. A. Andreev, I. Ghiu, et al., J. Russ. Laser Res., 35, 427 (2014).
S. Weigert, Int. J. Mod. Phys., 20, 1942 (2006).
J.-P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 31, L543 (1998).
J.-P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 32, 2777 (1999)
J.-P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 32, L269 (1999).
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*Based on the Poster presented at the 23nd Central European Workshop on Quantum Optics (27 June – 1 July 2016, Kolymbari, Crete, Greece).
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Adam, P., Andreev, V.A., Isar, A. et al. Continuous Sets of Dequantizers and Quantizers for One-Qubit States* . J Russ Laser Res 37, 544–555 (2016). https://doi.org/10.1007/s10946-016-9606-4
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DOI: https://doi.org/10.1007/s10946-016-9606-4