Quantifying Entanglement

We have witnessed great advances in quantum information theory in recent years. There are two distinct directions in which progress is currently being made: quantum computation and error correction on the one hand (for a short survey see [1,2]) and non–locality, Bell’s inequalities and purification on the other hand [3,4]. There has also been a number of papers relating the two (e.g. [5,6]). Our present work belongs to this second group. Recently it was realised that the CHSH (Clauser-HorneShimony-Holt) form of Bell’s inequalities are not a sufficiently good measure of quantum correlations in the sense that there are states which do not violate the CHSH inequality, but on the other hand can be purified by local interactions and classical communications to yield a state that does violate the CHSH inequality [3]. Subsequently, it was shown that the only states of two two–level systems which cannot be purified are those that can be written as the sum over density operators which are direct products states of the two subsystems [7]. Therefore, although it is possible to say whether a quantum state is entangled or not, the amount of entanglement cannot easily be determined for general mixed states. Bennett et al [5] have recently proposed a measure of entanglement for a general mixed state of two quantum subsystems. However, this measure has the disadvantage that it is hard to compute for a general state, even numerically. In this letter we specify conditions which any measure of entanglement has to satisfy and construct a whole class of ‘good’ entanglement measures. Our measures are geometrically intuitive. Unless stated otherwise the following considerations apply to a system composed of two quantum subsystem of arbitrary dimensions. First we define the term purification procedure more precisely. There are three distinct ingredients in any protocol that aims at increasing correlations between two quantum subsystems locally: Local general measurements (LGM): these are performed by the two parties (A and B) separately and are described by two sets of operators satisfying the completeness relations ∑ iA † iAi = I and ∑ j B † jBj = I. The joint action of the two is described by ∑ ij Ai⊗Bj , which again describes a local general measurement. Classical communication (CC): this means that the actions of A and B can be classically correlated. This can be described by a complete measurement on the whole space A + B which, as opposed to local general measurements, is not necessarily decomposable into a direct product of two operators as above, each acting on only one subsystem. If ρAB is the joint state of subsystems A and B then the transformation involving ‘LGM+CC’ would look like