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arXiv · 1311.4649

Describing orbit space of global unitary actions for mixed qudit states

Vladimir Gerdt·Arsen Khvedelidze·Yuri Palii
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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 $\mathfrak{P}_+$.

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Vladimir Gerdt, Arsen Khvedelidze, Yuri Palii. 2013-11-19. Describing orbit space of global unitary actions for mixed qudit states. https://arxiv.org/abs/1311.4649

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