Maximally entangled mixed states of two qubits

In this paper we investigate how much entanglement in a mixed two-qubit system can be created by global unitary transformations. The class of states for which no more entanglement can be created by global unitary operations is clearly a generalization of the class of Bell states to mixed states, and gives strict bounds on how the degree of mixing of a state limits its entanglement. This question is of considerable interest as entanglement is the magic ingredient of quantum information theory and experiments always deal with mixed states. Recently, Ishizaka and Hiroshima @1# independently considered the same question. They proposed a class of states and conjectured that the entanglement of formation @2# and the negativity @3# of these states could not be increased by any global unitary operation. Here we rigorously prove their conjecture and furthermore prove that the states they proposed are the only ones having the property of maximal entanglement. Closely related to the issue of generalized Bell states is the question of characterizing the set of separable density matrices @5#, as the entangled states closest to the maximally mixed state necessarily have to belong to the proposed class of maximal entangled mixed states. We can thus give a complete characterization of all nearly entangled states lying on the boundary of the sphere of separable states surrounding the maximally mixed state. As a by-product this gives an alternative derivation of the well-known result of Zyczkowski et al. @3# that all states for which the inequality Tr(r 2 )< 1