Various problems concerning the geometry of the space u ∗ (H) of Hermitian operators on a Hilbert space H 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 D(H) of density states on an n-dimensional Hilbert space H is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space D k (H) 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 D(H), viewed as a subset of the dual u ∗ (H) to the Lie algebra of the unitary group U(H), at every point must be tangent to the strata D k (H) it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition H = H 1 ⊗ H 2, an abstract criterion of entanglement is proved.
Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space 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 of density states on an n -dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank- k states, k = 1, ..., n , is a smooth manifold of (real) dimension 2 nk − k 2 − 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.
We review the geometry of the space of quantum states
S(H) of a finite-level quantum system with Hilbert space
H from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups: the special unitary group
SU(H) and its complexification
SL(H), the complex special linear group. A stratum of the stratification generated by
SU(H) is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by
SL(H) is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of
SL(H) which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this
SL(H)-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on
S(H) that preserve the von Neumann entropy and the convex structure of
S(H), while the latter give rise to invertible maps on
S(H) that preserve the von Neumann entropy but not the convex structure of
S(H). A similar decomposition is given for the
SL(H)-action generating the stratification of
S(H) into manifolds of quantum states with the same rank, where gradient vector fields preserve the rank but do not preserve entropy. Some comments on multipartite quantum systems are made. It is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the action of the complex special linear group associated with the partition.
Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of
C∗ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given
C∗ C∗ -algebra
A A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states
S S of a possibly infinite-dimensional, unital
C∗ C∗ -algebra
A A is partitioned into the disjoint union of the orbits of an action of the group
G G of invertible elements of
A A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space
H H are smooth, homogeneous Banach manifolds of
G=GL(H) G=GL(H) , and, when
A A admits a faithful tracial state
τ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through
τ τ is a smooth, homogeneous Banach manifold for
G G .
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).