  Quantum discord, a kind of quantum correlation, is defined as the difference
between quantum mutual information and classical correlation in a bipartite
system. It has been discussed so far for small systems with only a few
independent parameters. We extend here to a much broader class of states when
the second party is of arbitrary dimension d, so long as the first, measured,
party is a qubit. We present two formulae to calculate quantum discord, the
first relating to the original entropic definition and the second to 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 so as to appear visually like the letter X, the entropic
calculation can 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. Our results
also apply to states of a N-qubit system, where "extended X states" consist of
(2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N
qubits. While these are still smaller than the total number (2^(2N) - 1) of
states of N qubits, the number of parameters involved is nevertheless large. In
the case of N = 2, they encompass the entire 15-dimensional parameter space,
that is, the extended X states for N = 2 represent the full qubit-qubit system.
