67,926 research outputs found
Some Good Error-Correcting Codes from Algebraic-Geometric Codes
Algebraic-geometric codes on Garcia-Stichtenoth family of curves are used to
construct the asymptotically good quantum codes.Comment: 7 page
Quantum entanglement without eigenvalue spectra
From the consideration of measuring bipartite mixed states by separable pure
states, we introduce algebraic sets in complex projective spaces for bipartite
mixed states as the degenerating locus of the measurement. These algebraic sets
are independent of the eigenvalues and only measure the "position" of
eigenvectors of bipartite mixed states. They are nonlocal invariants ie.,
remaining invariant after local unitary transformations. The algebraic sets
have to be the sum of the linear subspaces if the mixed states are separable,
and thus we give a "eigenvalue-free" criterion of separability.Based on our
criterion, examples are given to illustrate that entangled mixed states which
are invariant under partial transposition or fufill entropy and disorder
criterion of separability can be constructed systematically.We reveal that a
large part of quantum entanglement is independent of eigenvalue spectra and
develop a method to measure this part of quantum enatnglement.The results are
extended to multipartite case.Comment: 23 pages, no figure, multipartite case was included in v
Schmidt numbers of low rank bipartite mixed states
We prove a lower bound for Schmidt numbers of bipartite mixed states. This
lower bound can be applied easily to low rank bipartite mixed states. From this
lower bound it is known that generic low rank bipartite mixed states have
relatively high Schmidt numbers and thus entangled. We can compute Schmidt
numbers exactly for some mixed states by this lower bound as shown in Examples.
This lower bound can also be used effectively to determine that some mixed
states cannot be convertible to other mixed states by local operations and
classical communication.
The results in this paper are proved by ONLY using linear algebra, thus the
results and proof are easy to understand.Comment: 10 pages, no figur
Strongly Resilient Non-Interactive Key Predistribution For Hierarchical Networks
Key establishment is the basic necessary tool in the network security, by
which pairs in the network can establish shared keys for protecting their
pairwise communications. There have been some key agreement or predistribution
schemes with the property that the key can be established without the
interaction (\cite{Blom84,BSHKY92,S97}). Recently the hierarchical cryptography
and the key management for hierarchical networks have been active topics(see
\cite{BBG05,GHKRRW08,GS02,HNZI02,HL02,Matt04}. ). Key agreement schemes for
hierarchical networks were presented in \cite{Matt04,GHKRRW08} which is based
on the Blom key predistribution scheme(Blom KPS, [1]) and pairing. In this
paper we introduce generalized Blom-Blundo et al key predistribution schemes.
These generalized Blom-Blundo et al key predistribution schemes have the same
security functionality as the Blom-Blundo et al KPS. However different and
random these KPSs can be used for various parts of the networks for enhancing
the resilience. We also presentkey predistribution schemes from a family
hyperelliptic curves. These key predistribution schemes from different random
curves can be used for various parts of hierarchical networks. Then the
non-interactive, identity-based and dynamic key predistributon scheme based on
this generalized Blom-Blundo et al KPSs and hyperelliptic curve KPSs for
hierarchical networks with the following properties are constructed.
1) storage at each node in the network where is the expansion
number and is the number of nodes at the -th level of the hierarchical
network; 2)Strongly resilience to the compromising of arbitrary many leaf and
internal nodes; 3)Information theoretical security without random oracle.Comment: 6 page
Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves
Let be an optimal elliptic curve defined over . The critical
subgroup of is defined by Mazur and Swinnerton-Dyer as the subgroup of
generated by traces of branch points under a modular
parametrization of . We prove that for all rank two elliptic curves with
conductor smaller than 1000, the critical subgroup is torsion. First, we define
a family of critical polynomials attached to and describe two algorithms to
compute such polynomials. We then give a sufficient condition for the critical
subgroup to be torsion in terms of the factorization of critical polynomials.
Finally, a table of critical polynomials is obtained for all elliptic curves of
rank two and conductor smaller than 1000, from which we deduce our result.Comment: fixed typos; added definition of degree of a rational function in
section 2; deleted first remark after lemma 2.
Explicit RIP Matrices in Compressed Sensing from Algebraic Geometry
Compressed sensing was proposed by E. J. Cand\'es, J. Romberg, T. Tao, and D.
Donoho for efficient sampling of sparse signals in 2006 and has vast
applications in signal processing. The expicit restricted isometry property
(RIP) measurement matrices are needed in practice. Since 2007 R. DeVore, J.
Bourgain et al and R. Calderbank et al have given several deterministic
cosntrcutions of RIP matrices from various mathematical objects. On the other
hand the strong coherence property of a measurement matrix was introduced by
Bajwa and Calderbank et al for the recovery of signals under the noisy
measuremnt. In this paper we propose new explicit construction of real valued
RIP measurement matrices in compressed sensing from algebraic geometry. Our
construction indicates that using more general algebraic-geometric objects
rather than curves (AG codes), RIP measurement matrices in compressed sensing
can be constructed with much smaller coherence and much bigger sparsity orders.
The RIP matrices from algebraic geometry also have a nice asymptotic bound
matching the bound from the previous constructions of Bourgain et al and the
small-bias sets. On the negative side, we prove that the RIP matrices from
DeVore's construction, its direct algebraic geometric generalization and one of
our new construction do not satisfy the strong coherence property. However we
give a modified version of AG-RIP matrices which satisfies the strong coherence
property. Therefore the new RIP matrices in compressed sensing from our
modified algebraic geometric construction can be used for the recovery of
signals from the noisy measurement.Comment: 21 pages, submitte
Even More Infinite Ball Packings from Lorentzian Root Systems
Boyd (1974) proposed a class of infinite ball packings that are generated by
inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of
root systems in Lorentz space. In particular, he showed that the space-like
weight vectors correspond to a ball packing if and only if the associated
Coxeter graph is of "level ." In Maxwell's work, the simple roots form a
basis of the representations space of the Coxeter group. In several recent
studies, the more general based root system is considered, where the simple
roots are only required to be positively independent. In this paper, we propose
a geometric version of "level" for the root system to replace Maxwell's graph
theoretical "level." Then we show that Maxwell's results naturally extend to
the more general root systems with positively independent simple roots. In
particular, the space-like extreme rays of the Tits cone correspond to a ball
packing if and only if the root system is of level . We also present a
partial classification of level- root systems, namely the Coxeter
-polytopes of level- with facets.Comment: 26 pages, 8 figures, 4 tables. Draf
Constraints on the mixing of states on bipartite quantum systems
We give necessary conditions for the mixing problem in bipartite case, which
are independent of eigenvalues and based on algebraic-geometric invariants of
the bipartite states. One implication of our results is that for some special
bipartite mixed states, only special mixed states in a measure zero set can be
used to mix to get them. The results indicate for many physical problems on
composite quantum systems the description based on majorization of eigenvalues
is not sufficientComment: 11 pages, no figur
New Invariants and Separability criterion of the Mixed States: Bipartite Case
We introduce algebraic sets in the complex projective spaces for the mixed
states in bipartite quantum systems as their invariants under local unitary
operations. The algebraic sets of the mixed state have to be the union of the
linear subspaces if the mixed state is separable. Some examples are given and
studied based on our criterionComment: 12 pages, no figur
Reducing the Computation of Linear Complexities of Periodic Sequences over
The linear complexity of a periodic sequence over plays an
important role in cryptography and communication [12]. In this correspondence,
we prove a result which reduces the computation of the linear complexity and
minimal connection polynomial of a period sequence over to the
computation of the linear complexities and minimal connection polynomials of
period sequences. The conditions and
are required for the result to hold. Some applications of
this reduction in fast algorithms to determine the linear complexities and
minimal connection polynomials of sequences over are presented.Comment: 10 pages. To appear in IEEE Transactions on Innformation Theor
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