Tensor Fields on Orbits of Quantum States and Applications

摘要: On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes tensor field constructions. First, as pull-back fields order from modified Hermitian fields, constructed the property having vertical distributions C_0-principal bundle H_0 over projective space P(H) in kernel. And second, directly group, as left-invariant representation-dependent operator-valued (LIROVTs) arbitrary being evaluated quantum state. Within NP-hard problem of deciding whether given state n-level bi-partite system is entangled or separable (Gurvits, 2003), show that both constructions admit geometric approach to this problem, evades traditional ambiguity defining metrical structures convex set mixed states. In particular considering manifolds associated orbits passing through selected state when acted upon local group U(n)xU(n) Schmidt coefficient decomposition inducing transformations, find following results: the case pure states Schmidt-equivalence are Lagrangian submanifolds define maximal This implies stronger statement as one proposed Bengtsson (2007). Moreover, Riemannian split and provide quantitative characterization of entanglement recover measure Schlienz Mahler (1995). case highlight relation between LIROVTs class computable separability criteria based Bloch-representation (de Vicente, 2007).