Volkert, Georg F. (2010): Tensor Fields on Orbits of Quantum States and Applications. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes of tensor field constructions. First, as pullback tensor fields of order two from modified Hermitian tensor fields, constructed on Hilbert spaces by means of the property of having the vertical distributions of the C_0principal bundle H_0 over the projective Hilbert space P(H) in the kernel. And second, directly constructed on the Lie group, as leftinvariant representationdependent operatorvalued tensor fields (LIROVTs) of arbitrary order being evaluated on a quantum state. Within the NPhard problem of deciding whether a given state in a nlevel bipartite quantum system is entangled or separable (Gurvits, 2003), we show that both tensor field constructions admit a geometric approach to this problem, which evades the traditional ambiguity on defining metrical structures on the convex set of mixed states. In particular by considering manifolds associated to orbits passing through a selected state when acted upon by the local unitary group U(n)xU(n) of Schmidt coefficient decomposition inducing transformations, we find the following results: In the case of pure states we show that Schmidtequivalence classes which are Lagrangian submanifolds define maximal entangled states. This implies a stronger statement as the one proposed by Bengtsson (2007). Moreover, Riemannian pullback tensor fields split on orbits of separable states and provide a quantitative characterization of entanglement which recover the entanglement measure proposed by Schlienz and Mahler (1995). In the case of mixed states we highlight a relation between LIROVTs of order two and a class of computable separability criteria based on the Blochrepresentation (de Vicente, 2007).
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  Geometric quantum mechanics, Quantum entanglement 
Subjects:  600 Natural sciences and mathematics > 510 Mathematics 600 Natural sciences and mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  19. July 2010 
1. Referee:  Dürr, Detlef 
Persistent Identifier (URN):  urn:nbn:de:bvb:19118130 
MD5 Checksum of the PDFfile:  d7c80ac3a46a32562b07ecfca266df5e 
Signature of the printed copy:  0001/UMC 18737 
ID Code:  11813 
Deposited On:  04. Aug 2010 06:26 
Last Modified:  16. Oct 2012 08:40 