Estimating Concurrence via Entanglement Witnesses

The interest on quantum entanglement has dramatically increased during the last two decades due to the emerging field of quantum information theory [1, 2]. It turns out that quantum entanglement may be used as basic resources in quantum information processing and communication. The prominent examples are quantum cryptography, quantum teleportation, quantum error correction codes and quantum computation. Hence it is of basic importance from both experimental and theoretical point of view to provide methods of detecting and quantifying entanglement [3–5]. There are no universal criteria to detect quantum entanglement and there are few measures of entanglement (based on the notion of entropy [6], entanglement of formation [7], concurrence [8], robustness [9, 10], geometrical measures [11, 12] and others). They can be calculated for pure quantum states or for the very limited class of mixed states possessing some symmetry properties [13–16]. Therefore, the great effort is directed to obtain methods of estimation of particular entanglement measures and to find relations between them [17–22]. On the other hand, what is measured in an experiment it is an expectation value of some observables, hence the estimations based on such quantities are most welcome [23–25]. For example, it was recently shown [25] that the concurrence for pure bipartite state |ψ〉 living in a Hilbert space H can be obtained as follows