Rank Of Coefficient Matrix Research Articles

Zero-Error Coding via Classical and Quantum Channels in Sensor Networks.

Today’s sensor networks need robustness, security and efficiency with a high level of assurance. Error correction is an effective communicational technique that plays a critical role in maintaining robustness in informational transmission. The general way to tackle this problem is by using forward error correction (FEC) between two communication parties. However, by applying zero-error coding one can assure information fidelity while signals are transmitted in sensor networks. In this study, we investigate zero-error coding via both classical and quantum channels, which consist of n obfuscated symbols such as Shannon’s zero-error communication. As a contrast to the standard classical zero-error coding, which has a computational complexity of , a general approach is proposed herein to find zero-error codewords in the case of quantum channel. This method is based on a n-symbol obfuscation model and the matrix’s linear transformation, whose complexity dramatically decreases to . According to a comparison with classical zero-error coding, the quantum zero-error capacity of the proposed method has obvious advantages over its classical counterpart, as the zero-error capacity equals the rank of the quantum coefficient matrix. In particular, the channel capacity can reach n when the rank of coefficient matrix is full in the n-symbol multilateral obfuscation quantum channel, which cannot be reached in the classical case. Considering previous methods such as low density parity check code (LDPC), our work can provide a means of error-free communication through some typical channels. Especially in the quantum case, zero-error coding can reach both a high coding efficiency and large channel capacity, which can improve the robustness of communication in sensor networks.

Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix

We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite (n-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the 2⊗2⊗2⊗4 system is discussed in terms of the generalized method, and 22 different SLOCC families are found.

A note on AMP algoritm and its extension

The ABS methods are of direct finite algorithms which solve a linear systems of equations at most in number of equations. Amini, Mahdavi-Amiri and Peyghami in [3] present an ABS type algorithm so that two new equations are satisfied in every iteration when coefficient matrix is a full row rank matrix. Their article has some problems. In this paper we first point to them and then we extend their algorithm so that it is applicable to any arbitrary rank of coefficient matrix.