All Maximally Entangled Four Qubits States

We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bi-partite cuts is maximal. We show that while all the states in the class maximize the average tangle, there are only few states in the class that maximize the average Tsillas or Renyi $\alpha$-entropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average $\alpha$-Tsallis entropy of entanglement for all $\alpha\geq 2$, while the other maximizing it for all $0<\alpha\leq 2$ (including the von-Neumann case of $\alpha=1$). Furthermore, among the maximally entangled four qubits states, there are only 3 maximally entangled states that have the property that for 2, out of the 3 bipartite cuts consisting of 2-qubits verses 2-qubits, the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1ebit. The unique 3 maximally entangled states are the 3 cluster states that are related by a swap operator. We also show that the cluster states are the only states (up to local unitaries) that maximize the average $\alpha$-Renyi entropy of entanglement for all $\alpha\geq 2$.Comment: 15 pages, 2 figures, Revised Version: many references added, an appendix added with a statement of the Kempf-Ness theore

Optimal decomposable witnesses without the spanning property

One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces H_{m,n}=C^m\otimes C^n such that the product vectors obeying =0 do not span H_{m,n}. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of 2\otimes n optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces H_{m,n} with m,n\geq 3.Comment: 11 pages, published version, title modified, some references added, other minor improvement

Entanglement of subspaces in terms of entanglement of superpositions

We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results in [G. Gour, arxiv.org:0704.1521 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures of entanglement other than the entropy of entanglement.Comment: 7 pages, no figure

Classification of unitary highest weight representations for non compact real forms

Using Jakobsen theorems, unitarizability in Hermitian Symmetric Spaces is discussed. The set of all missing highest weights is explicitly calculated and the construction of their corresponding highest weights vectors is studied.Comment: PDF, 35 pages (late submission

A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit-qunit decomposable EW (DEW) W the three statements are equivalent: (i) the set of product vectors obeying \bra{e,f}W\ket{e,f}=0 spans the corresponding Hilbert space, (ii) W is optimal, (iii) W=Q^{\Gamma} with Q denoting a positive operator supported on a completely entangled subspace (CES) and \Gamma standing for the partial transposition. While, implications $(i)\Rightarrow(ii)$ and $(ii)\Rightarrow(iii)$ are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in C^{2}\otimes C^{n} span after partial conjugation the whole space. On the other hand, already in the case of C^{3}\otimes C^{3} Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii), or (ii) does not imply (iii), and the above transparent characterization obeyed by qubit-qunit DEWs, does not hold in general.Comment: 13 pages, proof of lemma 4 corrected, theorem 3 removed, some parts improve

On the dimension of subspaces with bounded Schmidt rank

We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al., which show that in large d x d--dimensional systems there exist random subspaces of dimension almost d^2, all of whose states have entropy of entanglement at least log d - O(1). It is also related to results due to Parthasarathy on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using random subspace techniques. We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to r.Comment: 4 pages, REVTeX4 forma

Unitary derived functor modules with small spectrum

This article does not have an abstract

Maternal transfer of antibodies induced by infection with Eimeria maxima partially protects chickens against challenge with Eimeria tenella

Infection of breeding hens with Eimeria maxima induces production of Eimeria-specific IgG antibodies which are transferred to hatchlings via the egg yolk and confer a high degree of maternal immunity against homologous challenge and partial immunity to infection with another important species, Eimeria tenella. As an example, in an experiment using hatchlings from eggs collected between days 28 and 39 after infection of the hens with 20 000 sporulated E. maxima oocysts, control chicks (challenged with 100 sporulated oocysts) excreted 6·8±1·2 million (mean±s.e., n = 10) or 5·8±1·2 million (n = 8) oocysts of E. maxima or E. tenella, respectively, compared to 0·9±0·4 million (n = 5) E. maxima oocysts or 2·2±0·4 million (n = 9) E. tenella oocysts excreted by hatchlings of infected hens. This represents an 87% reduction in oocyst excretion with regard to E. maxima and a 62% reduction in oocyst excretion with regard to E. tenella in the progeny of the infected hens. In another experiment, eggs were collected from days 28 to 37 and again from days 114 to 123 after infection of the hens with E. maxima and hatchling oocyst excretion rates were 82% and 62%, respectively, reduced for E. maxima and 43% and 41%, respectively, reduced for E. tenella in the progeny of hens infected with E. maxima compared to the progeny of uninfected hens. ELISA and Western blot analyses of maternally-derived IgG revealed a high degree of cross-reactivity to antigens of E. maxima and E. tenella. Thus, maternally-derived, IgG-mediated cross- resistance to different species of Eimeria occurs in the chicken, most likely as a result of cross-recognition of conserved epitopes or protein