Quantum discord for qubit-qudit systems

We present two formulae to calculate quantum discord, a kind of quantum correlation, between a qubit and a second party of arbitrary dimension d. The first formula is the original entropic definition and the second is a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X-state, the density matrix having non-zero elements only along the diagonal and anti-diagonal, the entropic calculation can also be carried out analytically. Such states of the full bipartite qubit-qudit system may be named 'extended X-states', whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. In the case of N = 2, extended X-states encompass the entire 15-dimensional parameter space, that is, they represent the full qubit-qubit system.