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Zapiski Nauchnykh Seminarov POMI, 2014, Volume 421, Pages 68–80
(Mi znsl5750)
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This article is cited in 3 scientific papers (total in 3 papers)
Describing orbit space of global unitary actions for mixed qudit states
V. P. Gerdta, A. M. Khvedelidzebac, Yu. G. Paliida a Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia
b Tbilisi State University, A. Razmadze Mathematical Institute, Tbilisi, Georgia
c School of Natural Sciences, University of Georgia, Tbilisi, Georgia
d Institute of Applied Physics, Moldova Academy of Sciences, Chisinau, Republic of Moldova
Abstract:
The unitary $\mathrm U(d)$-equivalence relation between elements of the space $\mathfrak P_+$ of mixed states of $d$-dimensional quantum system defines the orbit space $\mathfrak P_+/\mathrm U(d)$ and provides its description in terms the ring $\mathbb R[\mathfrak P_+]^{\mathrm U(d)}$ of $\mathrm U(d)$-invariant polynomials. We prove that the semi-algebraic structure of $\mathfrak P_+/\mathrm U(d)$ is determined completely by two basic properties of density matrices, their semi-positivity and Hermicity. Particularly, it is shown that the Processi–Schwarz inequalities in elements of integrity basis for $\mathbb R[\mathfrak P_+]^{\mathrm U(d)}$ defining the orbit space, are identically satisfied for all elements of $\mathfrak P_+$.
Key words and phrases:
density matrix, qudit, unitary group, orbit space, polynomial invariants, syzygy ideal, semialgebraic structure.
Received: 12.11.2013
Citation:
V. P. Gerdt, A. M. Khvedelidze, Yu. G. Palii, “Describing orbit space of global unitary actions for mixed qudit states”, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Zap. Nauchn. Sem. POMI, 421, POMI, St. Petersburg, 2014, 68–80; J. Math. Sci. (N. Y.), 200:6 (2014), 682–689
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https://www.mathnet.ru/eng/znsl5750 https://www.mathnet.ru/eng/znsl/v421/p68
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Abstract page: | 166 | Full-text PDF : | 44 | References: | 28 |
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