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)$O(A_KU)$ storage at each node in the network where $U$ is the expansion number and $A_K$ is the number of nodes at the $K$-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 $E$ be an optimal elliptic curve defined over $\mathbb{Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $E$. 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 $E$ 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 $2$." 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 $2$. We also present a partial classification of level-$2$ root systems, namely the Coxeter $d$-polytopes of level-$2$ with $d+2$ 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

Schmidt number of pure states in bipartite quantum systems as an algebraic-geometric invariant

Our previous work about algebraic-geometric invariants of the mixed states are extended and a stronger separability criterion is given. We also show that the Schmidt number of pure states in bipartite quantum systems, a classical concept, is actually an algebraic-geometric invariant.Comment: 8 pages, no figure, minor changes in v

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