Concurrence-based entanglement measure for Werner States

Quantum entanglement is playing very significant roles in quantum information processing such as quantum cryptography, quantum teleportation and quantum computation [1]. This motivates an increasing interest in the study of operational detection and quantification of entanglement for various quantum systems. Despite of a great deal of efforts in recent years, for the moment only partial solutions are known to detect and quantify entanglement for generic mixed state. The crucial entanglement measure concurrence, firstly proposed by Hill and Wootters [2, 3], has recently been shown to play an essential role in describing quantum phase transitions in various interacting quantum manybody systems [4], affecting macroscopic properties of solids significantly [5] and revealing distinct scaling behavior for different types of multipartite entanglement [6]. The concurrence was then generalized by Uhlmann, Rungta et al, and by Albeverio and Fei [7] to arbitrary bipartite quantum system. Multi-variable concurrence vectors are also introduced in [8, 9] and possible multipartite generalizations are given in [10]. However, even the problem of obtaining only lower bound of concurrence has required considerable efforts [8, 11]. This problem has been advanced significantly in [12], providing an algebraic lower bound which can be optimized further by numerical approaches, and in [13] through an entirely analytical derivation of a complementary tightly lower bound. In addition, nice analytical results are also given for isotropic states [14] and rotationally symmetric states [15]. An important class of quantum states are the Werner states [16, 17], which appear in realistic quantum computing devices and quantum communication environments, e.g. transmitting perfect entangled states through a noisy depolarizing channel. An effective experimental generation of these states has been recently demonstrated in [18]. An analytical expression has been derived in [17] for entanglement of formation (EOF), which quantifies the minimally required physical resources to prepare a Werner state. The greatest cross norm is also obtained for the Werner states [19]. It is believed [20] that there is a novel connection between the concurrence and their EOF, through a parameter that depicts the Werner state completely. One expects that the situation would be similar to the case of two qubits where EOF is an analytic monotone function of concurrence [3]. However, for Werner states why such a parameter plays the role of concurrence is not yet well understood. There is also no rigorous and clear proof of this fact in the literatures, for high dimensions. In this letter we first find an analytic expression of concurrence for Werner states in arbitrary dimensions, which complements many of the existing analytic results. We then show how EOF is exactly related to the concurrence. We demonstrate that, surprisingly, an optimal ensemble decomposition will achieve concurrence, tangle and EOF simultaneously for the Werner states. Thus the expected connection is rigorously proved and shown to be natural.