  The common use in literature of orthogonal measurements in obtaining quantum
discord for two-qubit states is discussed and compared with more general
measurements. We prove the optimality of orthogonal measurements for rank 2
states. While for rank 3 and 4 mixed states they are not optimal, we present
strong numerical evidence showing that they give the correct quantum discord up
to minimal corrections. Based on the connection, through purification with an
ancilla, between discord and entanglement of formation (EoF), we give a tight
upper bound for the EoF of a $2\otimes N$ mixed state of rank 2, given by an
optimal decomposition of 2 elements. We also provide an alternative way to
compute the quantum discord for two qubits based on the Bloch vectors of the
state.

Quantum Physics