In the standard geometric approach, the entanglement of a pure state is $sin^2theta$, where $theta$ is the angle between the entangled state and the closest separable state of products of normalised qubit states. We consider here a generalisation of
this notion by considering separable states that consist of products of unnormalised states of different dimension. The distance between the target entangled state and the closest unnormalised product state can be interpreted as a measure of the entanglement of the target state. The components of the closest product state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector. For several cases where the target state has a large degree of symmetry, we solve the system of equations analytically, and look specifically at the limit where the number of qubits is large.
We show via an explicit example that quantum anomalies can lead to decoherence of a single quantum qubit through phase relaxation. The anomaly causes the Hamiltonian to develop a non-self-adjoint piece due to the non-invariance of the domain of the H
amiltonian under symmetry transformation. The resulting decoherence originates completely from the dynamics of the system itself and not, as usually considered, from interactions with the environment.
In the standard geometric approach to a measure of entanglement of a pure state, $sin^2theta$ is used, where $theta$ is the angle between the state to the closest separable state of products of normalized qubit states. We consider here a generalizati
on of this notion to separable states consisting of products of unnormalized states of different dimension. In so doing, the entanglement measure $sin^2theta$ is found to have an interpretation as the distance between the state to the closest separable state. We also find the components of the closest separable state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector.
