Ent: A Multipartite entanglement measure, and parameterization of entangled states

A multipartite entanglement measure called the ent is presented and shown to be an entanglement monotone, with the special property of automatic normalization. Necessary and sufficient conditions are developed for constructing maximally entangled states in every multipartite system such that they are true-generalized X states (TGX) states, a generalization of the Bell states, and are extended to general nonTGX states as well. These results are then used to prove the existence of maximally entangled basis (MEB) sets in all systems. A parameterization of general pure states of all ent values is given, and proposed as a multipartite Schmidt decomposition. Finally, we develop an ent vector and ent array to handle more general definitions of multipartite entanglement, and the ent is extended to general mixed states, providing a general multipartite entanglement measure.

Stabilizer codes and equientangled bases from phase states

We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.

A note on locally unextendible non-maximally entangled basis

We study the locally unextendible non-maximally entangled basis (LUNMEB) in $H^{d}\bigotimes H^{d}$. We point out that there exists an error in the proof of the main result of LUNMEB [Quant. Inf. Comput. 12, 0271(2012)], which claims that there are at most d orthogonal vectors in a LUNMEB, constructed from a given non-maximally entangled state. We show that both the proof and the main result are not correct in general. We present a counter example for d=4, in which five orthogonal vectors from a specific non-maximally entangled state are constructed. Besides, we completely solve the problem of LUNMEB for the case of d=2.

Locally unextendible non-maximally entangled basis

We introduce the concept of the locally unextendible non-maximally entangled basis (LUNMEB) in H^d \bigotimes H^d. It is shown that such a basis consists of d orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of (d-1)^2 orthogonal vectors for non-maximally entangled state if it is maximally entangled in (d-1) dimensional subspace. Such a basis plays an important role in determining the number of classical bits that one can send in a superdense coding protocol using a non-maximally entangled state as a resource. By constructing appropriate POVM operators, we find that the number of classical bits one can transmit using a non-maximally entangled state as a resource is (1+p_0\frac{d}{d-1})\log d, where p_0 is the smallest Schmidt coefficient. However, when the state is maximally entangled in its subspace then one can send up to 2\log (d-1) bits. We also find that for d= 3, former may be more suitable for the superdense coding.