The Variational Definition Of Negativity And Its Application

Quantum entanglement is a joint consequence of the superposition principle which is one of the essence supposed in quantum mechanics. It is an important property that distinguishes the quantum from classical world. Quantum entanglement plays key role and has been widely applied in quantum information processing, Such as quantum communication, quantum computing and so on. More and more studies have been focused on the quantization of entanglement. Even though researchers worked out many results about the separable criterion and entanglement measure, each result has its limitations. In the lower dimensions, the quantization of bipartite entanglement has been figured out. However, good understanding of entanglement (especially of mixed states) is still an open question.In this paper, we present a variational definition of the pure-state negativity, by which we derive an analytic lower bound for bipartite mixed-state entanglement in terms of the method of the convex roof construction. It is shown that this lower bound can be directly measured by one single projection operator or a few local observables in experiment without the requirement of simultaneous multiple copies of states. In particular, we show that the lower bound can serve as exact entanglement measure for isotropic states.Entanglement of superpositions was first addressed and by Linden et al who found an upper bound on the entanglement of the superposition in terms of the entanglement of the two individual states being superposed by employing the von Neumann entropy of the reduced matrix as the measure of entanglement. Later although a lot of authors considered the same question based on different entanglement measure, the bound on the entanglement of superposition usually had different tightness. In this paper, we considered the entanglement of superpositions based on a variational definition of negativity. we find a tight upper bound of negativity of superpositions in terms of the negativities of the two quantum states superposed. It is shown that the upper bound has not only very simple formulation, but also a better tightness. In addition, our upper bound can be extended to the case with more than two quantum states superposed.