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Spin nonclassicality via variance

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Abstract

Although variance, as one of the most fundamental and ubiquitous quantities in quantifying uncertainty, has been widely used in both classical and quantum physics, there are still new applications awaiting exploration. In this work, by interchanging the roles of the state variable and the observable variable, i.e., by formally regarding any state as an observable (which is rational because any state is a priori a Hermitian operator ) and considering the average variance of this state (now in the position of an observable ) in all spin coherent states, we introduce a quantifier of spin nonclassicality with respect to a resolution of identity induced by spin coherent states. This quantifier is easy to compute and it admits various operational interpretations, such as the purity deficit, the Tsallis 2-entropy deficit, and the squared norm deficit between the Wigner function and the Husimi function. We reveal several intuitive properties of this quantifier, connect it to the phase-space distribution uncertainty, and illustrate it with some prototypical examples. Various extensions are further indicated.

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References

  1. L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett., 4, 205–207 (1979).

    Article  ADS  Google Scholar 

  2. L. Mandel, “Non-classical states of the electromagnetic field,” Phys. Scr., 1986, 34–42 (1986).

    Article  Google Scholar 

  3. M. Hillery, “Nonclassical distance in quantum optics,” Phys. Rev. A, 35, 725–732 (1987).

    Article  ADS  Google Scholar 

  4. C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A, 44, R2775–R2778 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  5. N. Lütkenhaus and S. M. Barnett, “Nonclassical effects in phase space,” Phys. Rev. A, 51, 3340–3342 (1995).

    Article  ADS  Google Scholar 

  6. V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and A. Wünsche, “Hilbert–Schmidt distance and non-classicality of states in quantum optics,” J. Modern Opt., 47, 633–654 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett., 88, 153601, 4 pp. (2002).

    Article  ADS  Google Scholar 

  8. V. V Dodonov and V. I. Man’ko (eds.), Theory of Nonclassical States of Light, Taylor & Francis, London (2003).

    Book  Google Scholar 

  9. J. M. C. Malbouisson and B. Baseia, “On the measure of nonclassicality of field states,” Phys. Scr., 67, 93–98 (2003).

    Article  ADS  MATH  Google Scholar 

  10. A. Kenfack and K. Życzkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B: Quantum Semiclass. Opt., 6, 396–404 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  11. J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett., 94, 173602, 4 pp. (2005).

    Article  ADS  Google Scholar 

  12. M. Boca, I. Ghiu, P. Marian, and T. A. Marian, “Quantum Chernoff bound as a measure of nonclassicality for one-mode Gaussian states,” Phys. Rev. A, 79, 014302, 4 pp. (2009).

    Article  ADS  Google Scholar 

  13. A. Mari, K. Kieling, B. M. Nielsen, E. S. Polzik, and J. Eisert, “Directly estimating nonclassicality,” Phys. Rev. Lett., 106, 010403, 4 pp. (2011).

    Article  ADS  Google Scholar 

  14. C. Gehrke, J. Sperling, and W. Vogel, “Quantification of nonclassicality,” Phys. Rev. A, 86, 052118, 8 pp. (2012).

    Article  ADS  Google Scholar 

  15. W. Vogel and J. Sperling, “Unified quantification of nonclassicality and entanglement,” Phys. Rev. A, 89, 052302, 6 pp. (2014).

    Article  ADS  Google Scholar 

  16. H. C. F. Lemos, A. C. L. Almeida, B. Amaral, and A. C. Oliveira, “Roughness as classicality indicator of a quantum state,” Phys. Lett. A, 382, 823–836 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. B. Yadin, F. C. Binder, J. Thompson, V. Narasimhachar, M. Gu, and M. S. Kim, “Operational resource theory of continuous-variable nonclassicality,” Phys. Rev. X, 8, 041038, 18 pp. (2018).

    Google Scholar 

  18. S. Bose, “Wehrl-entropy-based quantification of nonclassicality for single-mode quantum optical states,” J. Phys. A: Math. Theor., 52, 025303, 17 pp. (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. S. Luo and Y. Zhang, “Quantifying nonclassicality via Wigner–Yanase skew information,” Phys. Rev. A, 100, 032116, 8 pp. (2019).

    Article  ADS  MathSciNet  Google Scholar 

  20. S. Luo and Y. Zhang, “Detecting nonclassicality of light via Lieb’s concavity,” Phys. Lett. A, 383, 125836, 5 pp. (2019).

    Article  MathSciNet  Google Scholar 

  21. S. Luo and Y. Zhang, “Quantumness of bosonic field states,” Internat. J. Theor. Phys., 59, 206–215 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  22. Y. Zhang and S. Luo, “Quantum states as observables: their variance and nonclassicality,” Phys. Rev. A, 102, 062211, 6 pp. (2020).

    Article  ADS  MathSciNet  Google Scholar 

  23. O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A, 78, 042112, 9 pp. (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. O. Giraud, P. Braun, and D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys., 12, 063005, 23 pp. (2010).

    Article  ADS  MATH  Google Scholar 

  25. T. Kiesel, W. Vogel, S. L. Christensen, J.-B. Béguin, J. Appel, and E. S. Polzik, “Atomic nonclassicality quasiprobabilities,” Phys. Rev. A, 86, 042108, 5 pp. (2012).

    Article  ADS  Google Scholar 

  26. M. Oszmaniec and M. Kuś, “On detection of quasiclassical states,” J. Phys. A: Math. Theor., 45, 244034, 13 pp. (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. F. Bohnet-Waldraff, D. Braun, and O. Giraud, “Quantumness of spin-\(1\) states,” Phys. Rev. A, 93, 012104, 10 pp. (2016).

    Article  ADS  Google Scholar 

  28. H. Dai and S. Luo, “Information-theoretic approach to atomic spin nonclassicality,” Phys. Rev. A, 100, 062114, 10 pp. (2019).

    Article  ADS  Google Scholar 

  29. P. W. Atkins and J. C. Dobson, “Angular momentum coherent states,” Proc. Roy. Soc. Lond. Ser. A, 321, 321–340 (1971).

    Article  ADS  MathSciNet  Google Scholar 

  30. J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A: Gen. Phys., 4, 313–332 (1971).

    Article  ADS  Google Scholar 

  31. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A, 6, 2211–2237 (1972).

    Article  ADS  Google Scholar 

  32. R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. (N. Y.), 74, 391–463 (1972).

    Article  ADS  MathSciNet  Google Scholar 

  33. A. M. Perelomov, “Coherent states for arbitrary Lie group,” Commun. Math. Phys., 26, 222–236 (1972).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. A. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin (1986).

    Book  MATH  Google Scholar 

  35. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Japan (3), 22, 264–314 (1940).

    MATH  Google Scholar 

  36. K. Nemoto and B. C. Sanders, “Superpositions of \(SU(3)\) coherent states via a nonlinear evolution,” J. Phys. A: Math. Gen., 34, 2051–2062 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. C. Tsallis, “Possible generalization of Boltzmann–Gibbs statistics,” J. Stat. Phys., 52, 479–487 (1988).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. Nemoto, “Wigner functions for arbitrary quantum systems,” Phys. Rev. Lett., 117, 180401, 5 pp. (2016).

    Article  ADS  Google Scholar 

  39. E. P. Wigner and M. M. Yanase, “Information contents of distributions,” Proc. Nat. Acad. Sci. USA, 49, 910–918 (1963).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. S. L. Luo, “Quantum versus classical uncertainty,” Theoret. and Math. Phys., 143, 681–688 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. S. Luo and Y. Sun, “Quantum coherence versus quantum uncertainty,” Phys. Rev. A, 96, 022130, 5 pp. (2017).

    Article  ADS  MathSciNet  Google Scholar 

  42. S. Luo and Y. Sun, “Coherence and complementarity in state-channel interaction,” Phys. Rev. A, 98, 012113, 8 pp. (2018).

    Article  ADS  Google Scholar 

  43. V. V. Dodonov, I. A. Malkin, and V. I. Manko, “Even and odd coherent states and excitations of a singular oscillator,” Physica, 72, 597–615 (1974).

    Article  ADS  MathSciNet  Google Scholar 

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Funding

This work was supported by the National Key R&D Program of China (Grant No. 2020YFA0712700), and the National Natural Science Foundation of China (Grant Nos. 11875317 and 61833010).

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Correspondence to Shunlong Luo.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 208, pp. 74-84 https://doi.org/10.4213/tmf10039.

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Zhang, Y., Luo, S. Spin nonclassicality via variance. Theor Math Phys 208, 916–925 (2021). https://doi.org/10.1134/S0040577921070060

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