We develop the quantizer–dequantizer formalism for the probability and mean-value representations of qubits and qutrits. In these representations, the elements of symbols of operators are measurable physical quantities. We derive the star-product kernels, which lead to explicit expressions of the associative product of the symbols of the density operators and quantum observables for qubits. The extension of the quantizer–dequantizer formalism associated with the probability and mean-value descriptions of qudit states is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
05 January 2021
In section Acknowledgments, the Russian Science Foundation Grant No. 1871-10091 should read No. 1971-10091.
References
C. Ferrie, Rep. Prog. Phys., 74, 116001 (2011).
S. N. Filippov and V. I. Man’ko, J. Phys. A: Math. Theor., 45, 15305 (2012).
J. Messamah, F. E. Schroeck, M. Hachemane, et al., Quantum Inf. Process., 14, 1035 (2015).
A. Garon, R. Zeier, S. J. Glaser, Phys. Rev. A, 91, 42122 (2015).
J. L. Romero, A. B. Klimov, and S. Wallentowitz, New J. Phys., 17, 43015 (2015).
P. Campos, M. G. R. Martins, M. C. B. Fernandes, and J. D. M. Vianna, Ann. Phys., 390, 60 (2018).
E. Wigner, Phys. Rev., 40, 749 (1932).
K. Husimi, Proc. Phys.-Math. Soc. Jpn., 22, 264 (1940).
Y. 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).
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland Series in Statistics and Probability, North-Holland Pub. Co. (1982).
M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep., 106, 121 (1984).
A. B. Klimov and S. M. Chumakov, A Group-Theoretical Approach to Quantum Optics, Wiley (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-VCH Verlag GmbH & Co. KGaA (2005).
S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A, 213, 1 (1996).
A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 79, 65013 (2009).
G. G. Amosov, Y. A. Korennoy, and V. I. Man’ko, Phys. Rev. A, 85, 52119 (2012).
M. Asorey, A. Ibort, G. Marmo, and F. Ventriglia, Phys. Scr., 90, 74031 (2015).
J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Phys. Rev. A, 49, 4101 (1994).
H. A. Kastrup, Phys. Rev. A, 94, 62113 (2016).
T. Tilma, M. J. Everitt, J. H. Samson, et al., Phys. Rev. Lett., 117, 180401 (2016).
C. Muñoz, A. B. Klimov, and L. Sánchez-Soto, Quantum Inform. Process., 16, 158 (2017).
W. K. Wootters, Ann. Phys., 176, 1 (1987).
U. Leonhardt, Phys. Rev. A, 53, 2998 (1996).
R. Asplund and G. Bjõrk, Phys. Rev. A, 64, 12106 (2001).
K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A, 70, 62101 (2004).
S. Chaturvedi, E. Ercolessi, G. Marmo, et al., J. Phys. A: Math. Gen., 39, 1405 (2006).
A. B. Klimov, J. L. Romero, G. Bjõrk, and L. Sánchez-Soto, J. Phys. A: Math. Theor., 40, 3987 (2007).
E. R. Livine, J. Phys. A: Math. Theor., 43, 75303 (2010).
S. N. Filippov and V. I. Man’ko, Phys. Scr., 83, 58101 (2011).
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, J. Exp. Theor. Phys., 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. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, Phys. Lett. A, 372, 6490 (2008).
R. L. Stratonovich, Sov. Phys. JETP, 4, 891 (1957).
F. Bayen, M. Flato, C. Fronsdal, et al., Lett. Math. Phys., 1, 521 (1977).
C. Brif and A. Mann, Phys. Rev. A, 59, 971 (1999).
O. V. Man’ko, V. I. Man’ko, and G. Marmo, J. Phys. A: Math. Gen., 35, 699 (2002).
F. Lizzi and P. Vitale, SIGMA, 10, 86 (2014).
V. I. Man’ko, G. Marmo, F. Ventriglia, and P. Vitale, J. Phys. A: Math. Theor., 50, 335302 (2017).
F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo, Ann. Phys., 385, 769 (2017).
M. A. Man’ko, V. I. Man’ko, and G. Marmo, Il Nuovo Cimento C, 38, 167 (2015)
A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 87, 38107 (2013).
A. Vourdas, J. Phys. A: Math. Gen., 39, R65 (2006).
V. A. Andreev, D. M. Davidovic, L. D. Davidovic, et al., Theor. Math. Phys., 166, 356 (2011).
V. A. Andreev, L. D. Davidovic, M. D. Davidovic, et al., Theor. Math. Phys., 179, 559 (2014).
P. Adam, V. A. Andreev, A. Isar, et al., Theor. Math. Phys., 186, 346 (2016).
S. N. Filippov and V. I. Man’ko, J. Russ. Laser Res., 32, 56 (2011).
P. Adam, V. A. Andreev, A. Isar, et al., Phys. Lett. A, 381, 2778 (2017).
P. Adam, V. A. Andreev, M. A. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 491 (2017).
P. Adam, V. A. Andreev, M. A. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 39, 360 (2018).
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).
V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 141 (2017).
V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 324 (2017).
V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 38, 416 (2017).
M. O. Terra-Cunha, V. I. Man’ko, and M. O. Scully, Found. Phys. Lett., 14, 103 (2001).
A. Shatskikh and M. Schwartz, Black Square: Malevich and the Origin of Suprematism, Yale University Press (2012).
S. Mancini, O. V. Man’ko, V. I. Man’ko, and P. Tombesi, J. Phys. A: Math. Gen., 34, 3461 (2001).
M. H. Devoret and R. J. Schoelkopf, Science, 339, 1169 (2013).
Y. Shalibo, R. Resh, O. Fogel, et al., Phys. Rev. Lett., 110, 100404 (2013).
V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Phys. Conf. Ser., 1071, 12008 (2018).
V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Russ. Laser Res., 39, 128 (2018).
M. A. Man’ko, J. Phys. Conf. Ser., 1071, 12015 (2018).
M. A. Man’ko and V. I. Man’ko, Phys. Scr., 93, 84002 (2018).
M. A. Man’ko and V. I. Man’ko, Entropy, 20, 692 (2018).
J. A. López-Saldívar, O. Castaños, E. Nahmad-Achar, et al., Entropy, 20, 630 (2018).
J. A. López-Saldívar, O. Castaños, M. A. Man’ko, and V. I. Man’ko, Quantum Inform. Process., 18, 210 (2019).
I. Ya. Doskoch and M. A. Manko, J. Phys. Conf. Ser., 1612, 012003 (2020).
I. Ya. Doskoch and M. A. Man’ko, J. Russ. Laser Res., 39, 499 (2018).
I. Ya. Doskoch and M. A. Man’ko, J. Russ. Laser Res., 40, 293 (2019).
I. Ya. Doskoch and M. A. Man’ko, Quantum Rep., 1(2), 130 (2019).
V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, J. Phys. Conf. Ser., 1612, 012009 (2020).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Adam, P., Andreev, V.A., Man’ko, M.A. et al. Star-Product Formalism for the Probability and Mean-Value Representations of Qudits. J Russ Laser Res 41, 470–483 (2020). https://doi.org/10.1007/s10946-020-09900-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-020-09900-x