Tensor Fields on Orbits of Quantum States and Applications

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 pull-back
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_0-principal bundle H_0 over the
projective Hilbert space P(H) in the kernel. And second, directly
constructed on the Lie group, as left-invariant
representation-dependent operator-valued tensor fields (LIROVTs) of
arbitrary order being evaluated on a quantum state. Within the
NP-hard problem of deciding whether a given state in a n-level
bi-partite 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 Schmidt-equivalence classes which
are Lagrangian submanifolds define maximal entangled states. This
implies a stronger statement as the one proposed by Bengtsson
(2007). Moreover, Riemannian pull-back 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
Bloch-representation (de Vicente, 2007).