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Classification of multipartite entanglement of all finite dimensionality.
We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions
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
Transformations among Pure Multipartite Entangled States via Local Operations Are Almost Never Possible
Local operations assisted by classical communication (LOCC) constitute the
free operations in entanglement theory. Hence, the determination of LOCC
transformations is crucial for the understanding of entanglement. We
characterize here almost all LOCC transformations among pure multipartite
multilevel states. Combined with the analogous results for qubit states shown
by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a
characterization of almost all local transformations among multipartite pure
states. We show that nontrivial LOCC transformations among generic, fully
entangled, pure states are almost never possible. Thus, almost all multipartite
states are isolated. They can neither be deterministically obtained from
local-unitary-inequivalent (LU-inequivalent) states via local operations, nor
can they be deterministically transformed to pure, fully entangled
LU-inequivalent states. In order to derive this result, we prove a more general
statement, namely, that, generically, a state possesses no nontrivial local
symmetry. We discuss further consequences of this result for the
characterization of optimal, probabilistic single copy and probabilistic
multi-copy LOCC transformations and the characterization of LU-equivalence
classes of multipartite pure states.Comment: 13 pages main text + 10 pages appendix, 1 figure; close to published
versio
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
Closed formula for the relative entropy of entanglement in all dimensions
The relative entropy of entanglement is defined in terms of the relative
entropy between an entangled state and its closest separable state (CSS). Given
a multipartite-state on the boundary of the set of separable states, we find a
closed formula for all the entangled state for which this state is a CSS. Quite
amazing, our formula holds for multipartite states in all dimensions. In
addition we show that if an entangled state is full rank, then its CSS is
unique. For the bipartite case of two qubits our formula reduce to the one
given in Phys. Rev. A 78, 032310 (2008).Comment: 8 pages, 1 figure, significantly revised; theorem 1 is now providing
necessary and sufficient conditions to determine if a state is CS
Three-particle templates for boosted Higgs
We explore the ability of three-particle templates to distinguish color
neutral objects from QCD background. This method is particularly useful to
identify the standard model Higgs, as well as other massive neutral particles.
Simple cut-based analysis in the overlap distributions of the signal and
background is shown to provide a significant rejection power. By combining with
other discriminating variables, such as planar flow, and several variables that
depend on the partonic template, three-particle templates are used to
characterize the influence of gluon emission and color flow in collider events.
The performance of the method is discussed for the case of a highly boosted
Higgs in association with a leptonically-decaying W boson.Comment: 32 pages, 13 figures. v2: Acknowledgments added, typos fixed. v3:
added comparison to filtering method, minor correction and acknowledgment
added. The version to appear in Phys. Rev.
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