  We propose novel mixed states in two qubits, ``maximally entangled mixed
states'', which have a property that the amount of entanglement of these states
cannot be increased further by applying any unitary operations. The property is
proven when the rank of the states is less than 4, and confirmed numerically in
the other general cases. The corresponding entanglement of formation specified
by its eigenvalues gives an upper bound of that for density matrices with same
eigenvalues.
