We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding the invariants of the local unitary group. The conditions, as given by the Luna–Richardson theorem, for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed. As an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non-trivial. On the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations. This canonical form and the invariants are directly determined by the symmetry group. Further, we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties.
The unitary $ \mathrm{U}(d)$-equivalence relation between elements of the space
P+ of mixed states of
d-dimensional quantum system defines the orbit space $ \mathfrak{P}_+/ \mathrm{U}(d)\,$ and provides its description in terms the ring
R[P+]U(d) of
U(d)-invariant polynomials. We prove that the semi-algebraic structure of $ \mathfrak{P}_+/ \mathrm{U}(d)\, $ is determined completely by two basic properties of density matrices, their semi-positivity and Hermicity. Particularly, it is shown that the Processi-Schwarz inequalities in elements of integrity basis for
R[P+]U(d) defining the orbit space, are identically satisfied for all elements of
P+.
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.