Geodesic
The global indicator of classicality of an arbitrary $N$-level quantum system
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 5-23
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

It is commonly accepted that a deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce the global indicator $\mathcal{Q}_N$ for quantification of “classicality-quantumness” correspondence in the form of the functional on the orbit space $\mathcal{O}[\mathfrak{P}_N]$ of the $SU(N)$ group adjoint action on the state space $\mathfrak{P}_N$ of an $N$-dimensional quantum system. The indicator $\mathcal{Q}_{N}$ is defined as a relative volume of a subspace $\mathcal{O}[\mathfrak{P}^{(+)}_N] \subset \mathcal{O}[\mathfrak{P}_N],$ where the Wigner quasiprobability distribution is positive. An algebraic structure of $\mathcal{O}[\mathfrak{P}^{(+)}_N]$ is revealed and exemplified by a single qubit $(N=2)$ and single qutrit $(N=3)$. For the Hilbert-Schmidt ensemble of qutrits the dependence of the global indicator on the moduli parameter of the Wigner quasiprobability distribution has been found. 
@article{ZNSL_2019_485_a0,
     author = {V. Abgaryan and A. Khvedelidze and A. Torosyan},
     title = {The global indicator of classicality of an arbitrary $N$-level quantum system},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--23},
     year = {2019},
     volume = {485},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a0/}
}
TY  - JOUR
AU  - V. Abgaryan
AU  - A. Khvedelidze
AU  - A. Torosyan
TI  - The global indicator of classicality of an arbitrary $N$-level quantum system
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2019
SP  - 5
EP  - 23
VL  - 485
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a0/
LA  - en
ID  - ZNSL_2019_485_a0
ER  - 
%0 Journal Article
%A V. Abgaryan
%A A. Khvedelidze
%A A. Torosyan
%T The global indicator of classicality of an arbitrary $N$-level quantum system
%J Zapiski Nauchnykh Seminarov POMI
%D 2019
%P 5-23
%V 485
%U http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a0/
%G en
%F ZNSL_2019_485_a0
V. Abgaryan; A. Khvedelidze; A. Torosyan. The global indicator of classicality of an arbitrary $N$-level quantum system. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 5-23. http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a0/

[1] H. C. F. Lemos, A. C. L. Almeida, B. Amaral, A. C. Oliveira, “Roughness as classicality indicator of a quantum state”, Phys. Lett., A 382 (2018), 823–836 | DOI | MR | Zbl

[2] A. Kenfack, K. $\dot{\text{Z}}$yczkowski, “Negativity of the Wigner function as an indicator of non-classicality”, J. Opt. B: Quantum Semiclass. Opt., 6 (2004), 396–404 | DOI | MR

[3] L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence”, Opt. Lett., 4 (1979), 205–207 | DOI

[4] V. Veitch, Ch. Ferrie, D. Gross, J. Emerson, “Negative quasi-probability as a resource for quantum computation”, New J. Phys., 14 (2012) | DOI

[5] F. Albarelli, M. G. Genoni, G. A. Matteo, A. Ferraro, “Resource theory of quantum non-Gaussianity and Wigner negativity”, Phys. Rev., A 98 (2018), 052350 | DOI

[6] E. P. Wigner, Quantum-mechanical distribution functions revisited. Part I: Physical Chemistry. Part II: Solid State Physics, 1997, 251–262 | DOI

[7] Ch. Ferrie, R. Morris, J. Emerson, “Necessity of negativity in quantum theory”, Phys. Rev., A 82 (2010), 044103 | DOI | MR

[8] R. F. O'Connell, E. P. Wigner, “Quantum-Mechanical Distribution Functions: Conditions for Uniqueness”, Phys. Lett., A 83 (1981), 145–148 | DOI | MR

[9] R. F. O'Connell, E. P. Wigner, “Manifestations of Bose and Fermi statistics on the quantum distribution function for systems of spin-$0$ and spin-$1/2$ particles”, Phys. Rev., A 30 (1984), 2613 | DOI | MR

[10] V. Abgaryan, A. Khvedelidze, On families of Wigner functions for $N$-level quantum systems, 2018, arXiv: 1708.05981 | Zbl

[11] V. Abgaryan, A. Khvedelidze, A. Torosyan, “On moduli space of the Wigner quasiprobability distributions for N-dimensional quantum systems”, Zap. Nauchn. Semin. POMI, 468, 2018, 177–201 | MR

[12] M. Hillery, “Nonclassical distance in quantum optics”, Phys. Rev., A 35 (1987), 725 | DOI

[13] V. V. Dodonov, O. V. Man'ko, V. I. Man'ko, A. Wünsche, “Hilbert–Schmidt distance and non-classicality of states in quantum optics”, J. Mod. Opt., 47:4 (2000), 633–654 | DOI | MR | Zbl

[14] P. Marian, T. A. Marian, H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field”, Phys. Rev. Lett., 88:15 (2002), 153601 | DOI

[15] K. $\dot{\text{Z}}$yczkowski, H.-J. Sommers, “Hilbert–Schmidt volume of the set of mixed quantum states”, J. Phys. A: Math. Gen., 36 (1011), 10115 | MR

[16] I. Daubechies, “Continuity statements and counterintuitive examples in connection with Weyl quantization”, J. Math. Phys., 24 (1982), 1453 | DOI | MR

[17] I. Bengtsson, A. Ericsson, M. Kus, W. Tadej, K. $\dot{\text{Z}}$yczkowski, “Birkhoff's Polytope and Unistochastic Matrices, $N=3$ and $N=4$”, Commun. Math. Phys., 259 (2005), 307–324 | DOI | MR | Zbl

[18] R. Bhatia, “Majorisation and Doubly Stochastic Matrices”, Matrix Analysis, Graduate Texts in Mathematics, 169, Springer, 1997 | DOI | MR