Abstract:
In (Phys Lett A, 2002, 297: 4–8) an entanglement criterion for finite-dimensional bipartite systems is proposed: If ρ AB is a separable state, then Tr(ρ A 2 ) Tr(ρ 2) and Tr(ρ B 2 ) Tr(ρ 2). In the present paper this criterion is extended to infinite-dimensional bipartite and multipartite systems. The reduction criterion presented in (Phys Rev A, 1999, 59: 4206–4216) is also generalized to infinite-dimensional case. Then it is shown that the former criterion is weaker than the later one.

Abstract:
It is shown that the order property of pure bipartite states under SLOCC (stochastic local operations and classical communications) changes radically when dimensionality shifts from finite to infinite. In contrast to finite dimensional systems where there is no pure incomparable state, the existence of infinitely many mutually SLOCC incomparable states is shown for infinite dimensional systems even under the bounded energy and finite information exchange condition. These results show that the effect of the infinite dimensionality of Hilbert space, the ``infinite workspace'' property, remains even in physically relevant infinite dimensional systems.

Abstract:
We propose a entanglement measure for pure $M \otimes N$ bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a $2 \otimes 2$ system, via a $2 \otimes 3$ system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure, and for experimentally measuring the entanglement. The form of the measure is similar to generalized concurrence. In the case of $2 \otimes 3$ systems, we prove that our measure, that is directly measurable, equals the concurrence. It is also shown that in order to measure the entanglement, it is sufficient to measure the projections of the state onto a maximum of $M(M-1)N(N-1)/2$ Bell states.

Abstract:
There has been spectacular progress in the field of quantum information in recent decades. The development of this field highlights the importance of the role of entanglement in quantum computing, quantum teleportation and quantum cryptography. These notes serve to provide a gentle introduction to the entanglement of bipartite states. In these notes, we introduce Bell's theorem in the form derived by Clauser, Horne, Shimony and Holt. We discuss the Schmidt decomposition and the Peres-Horodecki criterion in the entanglement of pure and mixed bipartite states. Finally, we describe a teleportation protocol as an illustration of the use of entangled states.

Abstract:
The PHC criterion and the realignment criterion for pure states in infinite-dimensional bipartite quantum systems are given. Furthermore, several equivalent conditions for pure states to be separable are generalized to infinite-dimensional systems.

Abstract:
Based on set theoretic ordering properties, a general formulation for constructing a pair of convertibility monotones, which are generalizations of distillable entanglement and entanglement cost, is presented. The new pair of monotones do not always coincide for pure bipartite infinite dimensional states under SLOCC (stochastic local operations and classical communications), demonstrating the existence of SLOCC incomparable pure bipartite states, a new property of entanglement in infinite dimensional systems, with no counterpart in the corresponding finite dimensional systems.

Abstract:
Entanglement in a class of bipartite generalized coherent states is discussed. It is shown that a positive parameter can be associated with the bipartite generalized coherent states so that the states with equal value for the parameter are of equal entanglement. It is shown that the maximum possible entanglement of 1 bit is attained if the positive parameter equals $\sqrt{2}$. The result that the entanglement is one bit when the relative phase between the composing states is $\pi$ in bipartite coherent states is shown to be true for the class of bipartite generalized coherent states considered.

Abstract:
We construct a family of bipartite states of arbitrary dimension whose eigenvalues of the partially transposed matrix can be inferred directly from the block structure of the global density matrix. We identify from this several subfamilies in which the PPT criterion is both necessary and sufficient. A sufficient criterion of separability is obtained, which is fundamental for the discussion. We show how several examples of states known to be classifiable by the PPT criterion indeed belong to this general set. Possible uses of these states in numerical analysis of entanglement and in the search of PPT bound entangled states are briefly discussed.

Abstract:
We present a study of the entanglement properties of Gaussian cluster states, proposed as a universal resource for continuous-variable quantum computing. A central aim is to compare mathematically-idealized cluster states defined using quadrature eigenstates, which have infinite squeezing and cannot exist in nature, with Gaussian approximations which are experimentally accessible. Adopting widely-used definitions, we first review the key concepts, by analysing a process of teleportation along a continuous-variable quantum wire in the language of matrix product states. Next we consider the bipartite entanglement properties of the wire, providing analytic results. We proceed to grid cluster states, which are universal for the qubit case. To extend our analysis of the bipartite entanglement, we adopt the entropic-entanglement width, a specialized entanglement measure introduced recently by Van den Nest M et al., Phys. Rev. Lett. 97 150504 (2006), adapting their definition to the continuous-variable context. Finally we add the effects of photonic loss, extending our arguments to mixed states. Cumulatively our results point to key differences in the properties of idealized and Gaussian cluster states. Even modest loss rates are found to strongly limit the amount of entanglement. We discuss the implications for the potential of continuous-variable analogues of measurement-based quantum computation.

Abstract:
We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitatively the degree of entanglement for any bipartite quantum state.