Characterization of separability and entanglement in \(2times{D}\)- and \(3times{D}\)-dimensional systems by single-qubit and single-qutrit unitary transformations

We investigate the geometric characterization of pure state bipartite entanglement of \(2times{D}\)- and \(3times{D}\)-dimensional composite quantum systems. To this aim, we analyze the relationship between states and their images under the action of particular classes of local unitary operations. We find that invariance of states under the action of single-qubit and single-qutrit transformations is a necessary and sufficient condition for separability. We demonstrate that in the \(2times{D}\)-dimensional case the von Neumann entropy of entanglement is a monotonic function of the minimum squared Euclidean distance between states and their images over the set of single qubit unitary transformations. Moreover, both in the \(2times{D}\)- and in the \(3times{D}\)-dimensional cases the minimum squared Euclidean distance exactly coincides with the linear entropy and thus as well with the tangle measure of entanglement in the $(2times{D}$-dimensional case). These results provide a geometric characterization of entanglement measures originally established in informational frameworks. Consequences and applications of the formalism to quantum critical phenomena in spin systems are discussed.