1,746 research outputs found
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 -entropy of entanglement. Quite remarkably, we find that up to
local unitaries, there exists two unique states, one maximizing the average
-Tsallis entropy of entanglement for all , while the
other maximizing it for all (including the von-Neumann case of
). 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 -Renyi entropy of entanglement for all .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
The sectional curvature remains positive when taking quotients by certain nonfree actions
We study some cases when the sectional curvature remains positive under the
taking of quotients by certain nonfree isometric actions of Lie groups. We
consider the actions of the groups and such that the quotient space
can be endowed with a smooth structure using the fibrations
and . We prove that the quotient space
carries a metric of positive sectional curvature, provided that the original
metric has positive sectional curvature on all 2-planes orthogonal to the
orbits of the action.Comment: 26 pages, 1 figure. Changed the spelling of the author's nam
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
and 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
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