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.
References
L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett., 4, 205–207 (1979).
L. Mandel, “Non-classical states of the electromagnetic field,” Phys. Scr., 1986, 34–42 (1986).
M. Hillery, “Nonclassical distance in quantum optics,” Phys. Rev. A, 35, 725–732 (1987).
C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A, 44, R2775–R2778 (1991).
N. Lütkenhaus and S. M. Barnett, “Nonclassical effects in phase space,” Phys. Rev. A, 51, 3340–3342 (1995).
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).
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).
V. V Dodonov and V. I. Man’ko (eds.), Theory of Nonclassical States of Light, Taylor & Francis, London (2003).
J. M. C. Malbouisson and B. Baseia, “On the measure of nonclassicality of field states,” Phys. Scr., 67, 93–98 (2003).
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).
J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett., 94, 173602, 4 pp. (2005).
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).
A. Mari, K. Kieling, B. M. Nielsen, E. S. Polzik, and J. Eisert, “Directly estimating nonclassicality,” Phys. Rev. Lett., 106, 010403, 4 pp. (2011).
C. Gehrke, J. Sperling, and W. Vogel, “Quantification of nonclassicality,” Phys. Rev. A, 86, 052118, 8 pp. (2012).
W. Vogel and J. Sperling, “Unified quantification of nonclassicality and entanglement,” Phys. Rev. A, 89, 052302, 6 pp. (2014).
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).
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).
S. Bose, “Wehrl-entropy-based quantification of nonclassicality for single-mode quantum optical states,” J. Phys. A: Math. Theor., 52, 025303, 17 pp. (2019).
S. Luo and Y. Zhang, “Quantifying nonclassicality via Wigner–Yanase skew information,” Phys. Rev. A, 100, 032116, 8 pp. (2019).
S. Luo and Y. Zhang, “Detecting nonclassicality of light via Lieb’s concavity,” Phys. Lett. A, 383, 125836, 5 pp. (2019).
S. Luo and Y. Zhang, “Quantumness of bosonic field states,” Internat. J. Theor. Phys., 59, 206–215 (2020).
Y. Zhang and S. Luo, “Quantum states as observables: their variance and nonclassicality,” Phys. Rev. A, 102, 062211, 6 pp. (2020).
O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A, 78, 042112, 9 pp. (2008).
O. Giraud, P. Braun, and D. Braun, “Quantifying quantumness and the quest for Queens of Quantum,” New J. Phys., 12, 063005, 23 pp. (2010).
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).
M. Oszmaniec and M. Kuś, “On detection of quasiclassical states,” J. Phys. A: Math. Theor., 45, 244034, 13 pp. (2012).
F. Bohnet-Waldraff, D. Braun, and O. Giraud, “Quantumness of spin-\(1\) states,” Phys. Rev. A, 93, 012104, 10 pp. (2016).
H. Dai and S. Luo, “Information-theoretic approach to atomic spin nonclassicality,” Phys. Rev. A, 100, 062114, 10 pp. (2019).
P. W. Atkins and J. C. Dobson, “Angular momentum coherent states,” Proc. Roy. Soc. Lond. Ser. A, 321, 321–340 (1971).
J. M. Radcliffe, “Some properties of spin coherent states,” J. Phys. A: Gen. Phys., 4, 313–332 (1971).
F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A, 6, 2211–2237 (1972).
R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. (N. Y.), 74, 391–463 (1972).
A. M. Perelomov, “Coherent states for arbitrary Lie group,” Commun. Math. Phys., 26, 222–236 (1972).
A. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin (1986).
K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Japan (3), 22, 264–314 (1940).
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).
C. Tsallis, “Possible generalization of Boltzmann–Gibbs statistics,” J. Stat. Phys., 52, 479–487 (1988).
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).
E. P. Wigner and M. M. Yanase, “Information contents of distributions,” Proc. Nat. Acad. Sci. USA, 49, 910–918 (1963).
S. L. Luo, “Quantum versus classical uncertainty,” Theoret. and Math. Phys., 143, 681–688 (2005).
S. Luo and Y. Sun, “Quantum coherence versus quantum uncertainty,” Phys. Rev. A, 96, 022130, 5 pp. (2017).
S. Luo and Y. Sun, “Coherence and complementarity in state-channel interaction,” Phys. Rev. A, 98, 012113, 8 pp. (2018).
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).
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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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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DOI: https://doi.org/10.1134/S0040577921070060