Abstract
We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Shrödinger, Naturwissenschaften, 23, 807 (1935).
J. S. Bell, Physica, 1, 195 (1965).
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (1987).
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett., 23, 880 (1969).
A. A. Klyachko, H. Ali Can, S. Binicioglu, and A. S. Shumovsky, Phys. Rev. Lett., 101, 020403 (2008).
R. Lapkiewicz, P. Li, C. Schaeff, et al., Nature, 460, 490 (2011).
H. Olliver and W. Zurek, Phys. Rev. Lett., 88, 017901 (2002).
V. I. Man’ko and O. V. Man’ko, J. Exp. Theor. Phys., 85, 430 (1997).
V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 239, 335 (1997).
V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Lett. A, 327, 353 (2004).
A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 79, 065013 (2009).
C. E. Shannon, Bell Syst. Tech. J., 27, 379 (1948).
A. Rényi, Probability Theory, North-Holland, Amsterdam (1970).
C. Tsallis, “Nonextensive statistical mechanics and thermodynamics: historical background and present status,” in: S. Abe and Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and Its Applications, Lecture Notes in Physics, Springer, Berlin (2001), Vol. 560, p. 3.
M. A. Man’ko, V. I. Man’ko, S. De Nicola, and R. Fedele, Acta Phys. Hung. B, 26/1-2, 71 (2006).
S. De Nicola, R. Fedele, M. A. Man’ko, and V. I. Man’ko, Eur. J. Phys. B, 52, 191 (2006).
M. A. Man’ko, J. Russ. Laser Res., 27, 405 (2006).
S. De Nicola, R. Fedele, M. A. Man’ko, and V. I. Man’ko, J. Phys. Conf. Ser., 70, 012007 (2007).
S. De Nicola, R. Fedele, M. A. Man’ko, and V. I. Man’ko, Theor. Math. Phys., 152, 1081 (2007) [tr. Teor. Mat. Fiz., 152, 241 (2007)].
M. A. Man’ko, “Tomographic entropy and new entropic uncertainty relations,” in: G. Adenier, A. Yu. Khrennikov, P. Lahti, V. I. Man’ko, and T. Nieuwenhuizen (Eds.), Quantum Theory: Reconsideration of Foundations-4, AIP Conference Proceedings, New York (2007), Vol. 962, p. 132.
V. N. Chernega and V. I. Man’ko, J. Russ. Laser Res., 29, 505 (2008).
M. A. Man’ko and V. I. Man’ko, Theor. Math. Phys., 160, 995 (2009) [tr. Teor. Mat. Fiz., 160, 143 (2009)].
M. A. Man’ko, J. Russ. Laser Res., 30, 514 (2009).
M. A. Man’ko, Phys. Scr., 82, 038109 (2010).
M. A. Man’ko, Phys. Scr., T143, 014017 (2011).
M. A. Man’ko, Theor. Math. Phys., 168, 985 (2011) [tr. Teor. Mat. Fiz., 168, 151 (2011)].
M. A. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 32, 439 (2011).
M. A. Man’ko and V. I. Man’ko, Found. Phys., 41, 330 (2011), [DOI 10.1007/s10701-009-9403-9].
M. A. Man’ko and V. I. Man’ko, Phys. Scr., T147, 014020 (2012).
V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “Entanglement structure of adjoint representation of unitary group and tomography of quantum states,” J. Russ. Laser Res., 24, 507 (2003).
V. N. Chernega and V. I. Man’ko, J. Russ. Laser Res., 28, 103 (2007).
C. Lupo, V. I. Man’ko, and G Marmo, J. Phys. A: Math. Theor., 40, 13091 (2007).
S. N. Filippov and V. I. Man’ko, J. Russ. Laser Res., 30, 55 (2009).
O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 25, 11 (2004).
M. A. Man’ko, V. I. Man’ko, and R. Vilela Mendes, J. Russ. Laser Res., 27, 507 (2006), [arXiv: quant-ph/0602129].
E. H. Lieb and M. B. Ruskai, J. Math. Phys., 14, 1938 (1973).
M. B. Ruskai, “Lieb’s simple proof of concavity (A,B) → TRA p K † B 1−p K and remarks on related inequalities,” arXiv: quant-ph/0404126 v4.
M.A. Man’ko and V.I. Man’ko,“Tomographic entropic inequalities in the probability representation of quantum mechanics,” in: R. Bijker (Ed.), Beauty in Physics: Theory and Experiment, Conference Proceedings, New York (2012), Vol. 1488, p. 110 [arXiv:1208.5695 quant-ph].
M. A. Man’ko, Phys. Scr., T153, 014045 (2013).
M. A. Man’ko and V. I. Man’ko,“Entropy of conditional tomographic probability distributions for classical and quantum systems,” in: Proceedings of the Sixth International Workshop “Spacetime –Matter – Quantum Mechanics from the Planck Scale to Emergent Phenomena (DICE 2012), Castello Pasquini/Castiglioncello, Tuscany, Italy (September 17–21, 2012), J. Phys. Conf. Ser. (2013, in press).
A. E. Rastegin, “Tests for quantum contextuality in terms of q-entropies,” arXiv:1210.6742 quant-ph.
Ya. A. Korennoy and V. I. Man’ko, J. Russ. Laser Res., 32, 74 (2011).
G. G. Amosov, Ya. A. Korennoy, and V. I. Man’ko, Phys. Rev. A, 85, 052119 (2012).
M. Bellini, A. S. Coelho, S. N. Filippov, et al., Phys. Rev. A, 85, 052129 (2012).
V. I. Man’ko, G. Marmo, A. Porzio, et al., Phys. Scr., 83, 045001 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Manuscript submitted by the authors in English on May 5, 2013.
Rights and permissions
About this article
Cite this article
Man’ko, M.A., Man’ko, V.I. Inequalities for nonnegative numbers and information properties of qudit tomograms. J Russ Laser Res 34, 203–218 (2013). https://doi.org/10.1007/s10946-013-9343-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-013-9343-x