Separability of rank two quantum states on multiple quantum spaces with different dimensions

We consider the separability of rank two quantum states on multiple quantum spaces with different dimensions. The sufficient and necessary conditions for separability of these multiparty quantum states are explicitly presented. A nonseparability inequality is also given, for the case where one of the eigenvectors corresponding to nonzero eigenvalues of the density matrix is maximally entangled. Quantum entanglement is one of the most striking features of quantum phenomena [1]. It was first recognized by Schrödinger [2] and Einstein, Podolsky and Rosen [3], where a description of the world called local realism was suggested. Bell proved that the local realism implies constraints on the predictions of spin correlations in the form of inequalities (Bell’s inequalities) [4]. The feature of quantum mechanics called nonlocality is one of the most apparent manifestations of quantum entanglement. Nonlocality has been given a lot of attention in foundational considerations, in the discussion of Bell type inequalities and hidden variable models, see e.g. [5]. Nonlocal correlations in quantum systems imply a kind of entanglement among the quantum subsystems. The recent development of quantum information theory showed that quantum entanglement can have important practical applications (see e.g. [6]). It is playing very important roles in quantum information processing such as quantum computation [7], quantum teleportation [8, 9, 10, 11] (for experimental realization see [12]), dense coding [13] and quantum cryptographic schemes [14, 15, 16]. Due to interaction with environment, in real conditions one encounters mixed states rather than pure ones. They can still possess some residual entanglement. More specially, a mixed state is considered to be entangled if it is not a mixture of product states [17]. In mixed states the quantum correlations are weakened, hence the manifestations of mixed-state entanglement can be very subtle [17, 18, 19]. To investigate the structure of mixed-state entanglement some beautiful works have been done in quantifying entanglement [20, 21, 22, 23, 24] for bipartite systems and multipartite systems (see e.g. [25, 26]). However most proposed measures of entanglement for bipartite systems involve extremizations which are difficult to handle analytically. For multipartite systems, one even does not know how to define the measures. Till now there is no general criterion that allows one to distinguish whether a mixed state is separable or not. The separability of pure states for bipartite systems is quite well understood (cf. [27]). For mixed states, some progress has been achieved in understanding the separability and entanglement problem for bipartite systems (cf. [28]), e.g., the proper definition of separable and