Geometry of quantum systems: density states and entanglement

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dc.creatorGrabowski, Janusz
dc.creatorKuś, Marek
dc.creatorMarmo, Giuseppe
dc.date2005-07-18
dc.date2005-11-17
dc.date.accessioned2026-07-07T06:42:17Z
dc.date.available2026-07-07T06:42:17Z
dc.descriptionVarious problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space $\cD(\cH)$ of density states on an $n$-dimensional Hilbert space $\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\cD^k(\cH)$ of rank-$k$ states, $k=1,...,n$, is a smooth manifold of (real) dimension $2nk-k^2-1$ and this stratification is maximal in the sense that every smooth curve in $\cD(\cH)$, viewed as a subset of the dual $u^*(\cH)$ to the Lie algebra of the unitary group $U(\cH)$, at every point must be tangent to the strata $\cD^k(\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\cH=\cH^1\ot\cH^2$, an abstract criterion of entanglement is proved.
dc.descriptionLatex, 26 pages, minor corrections, published version
dc.identifierhttps://arxiv.org/abs/math-ph/0507045
dc.identifierhttp://arxiv.org/abs/math-ph/0507045
dc.identifierJ.Phys. A38 (2005) 10217-10244
dc.identifierdoi:10.1088/0305-4470/38/47/011
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/101983
dc.subjectMathematical Physics
dc.subjectQuantum Physics
dc.titleGeometry of quantum systems: density states and entanglement
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