On the Distance Between Unitary Propagators of Quantum Systems of Differing Dimensions

A distance measure is presented between two unitary propagators of quantum systems of differing dimensions along with a corresponding method of computation. A typical application is to compare the propagator of the actual (real) process with the propagator of the desired (ideal) process; the former being of a higher dimension then the latter. The proposed measure has the advantage of dealing with possibly correlated inputs, but at the expense of working on the whole space and not just the information bearing part as is usually the case, i.e., no partial trace operation is explicitly involved. It is also shown that the distance measure and an average measure of channel fidelity both depend on the size of the same matrix: as the matrix size increases, distance decreases and fidelity increases.