Quantum Symmetries and Cartan Decompositions in Arbitrary Dimensions

Decompositions of Lie groups are used in systems and control, both to analyse dynamics and to design control algorithms for systems with state varying on a Lie group. In this paper, we investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition, there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd - even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in [ 6 - 8] in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of N qubits to the case where the component subsystems have arbitrary dimensions. We present an example of application of the results to control design for quantum systems.