Lower Bound of Concurrence and Distillation for Arbitrary Dimensional Bipartite Quantum States

We present an analytical lower bound of concurrence for arbitrary dimensional bipartite quantum states. This lower bound may be used to improve all the known lower bounds of concurrence. Moreover, the lower bound gives rise to an operational sufficient criterion of distillability of quantum entanglement. The significance of our result is illustrated by quantitative evaluation of entanglement for entangled states that fail to be identified by the usual concurrence estimation method, and by showing the distillability of mixed states that can not be recognized by other distillability criteria. Introduction. — Quantum entanglement is a striking feature of quantum systems and plays essential roles in some physical processes such as quantum phase transitions in various interacting quantum many-body systems. Quantum entangled states are the key physical resources in many quantum information processing. An important issue in the theory of quantum entanglement is to recognize and quantify the entanglement for a given quantum state. Concurrence is one of the most important measures of quantum entanglement [1–6]. For mixed two-qubit states, an analytical formula of concurrence has been derived [1]. For general high dimensional case, due to the extremizations involved in the computation, only a few analytic formulas of concurrence have been found for some special symmetric states [7]. To estimate the concurrence for general mixed states, efforts have been made toward the analytical lower bounds of concurrence. In Ref. [8] a lower bound of con-currence that can be tightened by numerical optimization over some parameters has been derived. In Ref. [9] analytic lower bounds of concurrence for any dimensional mixed bipartite quantum states have been presented by using the positive partial transposition (PPT) and realignment separability criteria. These bounds are exact for some special classes of states and can be used to detect many bound entangled states. In Ref. [10] another lower bound of concurrence for even dimensional bipar-tite states has been presented from a new separability criterion Ref. [11]. A lower bound of concurrence based on local uncertainty relations criterion is derived in Ref. [12]. This bound is further optimized in Ref. [13]. In Refs. [14, 15] the authors presented lower bounds of concurrence for bipartite systems in terms of a different approach, which has a close relationship with the distillability of bipartite quantum states. In Ref. [16] an explicit analytical lower bound of concurrence is obtained by using positive maps, which is better than the ones in Refs. [9, 10] …