3,598 research outputs found
On independent permutation separability criteria
Recently P. Wocjan and M. Horodecki [quant-ph/0503129] gave a
characterization of combinatorially independent permutation separability
criteria. Combinatorial independence is a necessary condition for permutations
to yield truly independent criteria meaning that that no criterion is strictly
stronger that any other. In this paper we observe that some of these criteria
are still dependent and analyze why these dependencies occur. To remove them we
introduce an improved necessary condition and give a complete classification of
the remaining permutations. We conjecture that the remaining class of criteria
only contains truly independent permutation separability criteria. Our
conjecture is based on the proof that for two, three and four parties all these
criteria are truly independent and on numerical verification of their
independence for up to 8 parties. It was commonly believed that for three
parties there were 9 independent criteria, here we prove that there are exactly
6 independent criteria for three parties and 22 for four parties.Comment: Revtex4, 7 pages, minor correction
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
Absolutely Maximally Entangled (AME) states are those multipartite quantum
states that carry absolute maximum entanglement in all possible partitions. AME
states are known to play a relevant role in multipartite teleportation, in
quantum secret sharing and they provide the basis novel tensor networks related
to holography. We present alternative constructions of AME states and show
their link with combinatorial designs. We also analyze a key property of AME,
namely their relation to tensors that can be understood as unitary
transformations in every of its bi-partitions. We call this property
multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom
Holographic duality from random tensor networks
Tensor networks provide a natural framework for exploring holographic duality
because they obey entanglement area laws. They have been used to construct
explicit toy models realizing many interesting structural features of the
AdS/CFT correspondence, including the non-uniqueness of bulk operator
reconstruction in the boundary theory. In this article, we explore the
holographic properties of networks of random tensors. We find that our models
naturally incorporate many features that are analogous to those of the AdS/CFT
correspondence. When the bond dimension of the tensors is large, we show that
the entanglement entropy of boundary regions, whether connected or not, obey
the Ryu-Takayanagi entropy formula, a fact closely related to known properties
of the multipartite entanglement of assistance. Moreover, we find that each
boundary region faithfully encodes the physics of the entire bulk entanglement
wedge. Our method is to interpret the average over random tensors as the
partition function of a classical ferromagnetic Ising model, so that the
minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the
analog of a bulk field, we find that our model reproduces the expected
corrections to the Ryu-Takayanagi formula: the minimal surface is displaced and
the entropy is augmented by the entanglement of the bulk field. Increasing the
entanglement of the bulk field ultimately changes the minimal surface
topologically in a way similar to creation of a black hole. Extrapolating bulk
correlation functions to the boundary permits the calculation of the scaling
dimensions of boundary operators, which exhibit a large gap between a small
number of low-dimension operators and the rest. While we are primarily
motivated by AdS/CFT duality, our main results define a more general form of
bulk-boundary correspondence which could be useful for extending holography to
other spacetimes.Comment: 57 pages, 13 figure
Entanglement Detection by Local Orthogonal Observables
We propose a family of entanglement witnesses and corresponding positive maps
that are not completely positive based on local orthogonal observables. As
applications the entanglement witness of the bound entangled state
[P. Horodecki, Phys. Lett. A {\bf 232}, 333 (1997)] is explicitly constructed
and a family of -dimensional bound entangled states is designed so that the
entanglement can be detected by permuting local orthogonal observables. Further
the proposed physically not implementable positive maps can be physically
realized by measuring a Hermitian correlation matrix of local orthogonal
observables.Comment: 4 pages, 1 figur
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