Magnetic quantum correlation in the 1D transverse-field XXZ model

One-dimensional spin-1/2 systems are well-known candidates to study the quantum correlations between particles. In the condensed matter physics, studies often are restricted to the 1st neighbor particles. In this work, we consider the 1D XXZ model in a transverse magnetic field (TF) which is not integrable except at specific points. Analytical expressions for quantum correlations (entanglement and quantum discord) between spin pairs at any distance are obtained for both zero and finite temperature, using an analytical approach proposed by Caux et al. [PRB 68, 134431 (2003)]. We compare the efficiency of the QD with respect to the entanglement in the detection of critical points (CPs) as the neighboring spin pairs go farther than the next nearest neighbors. In the absence of the TF and at zero temperature, we show that the QD for spin pairs farther than the 2nd neighbors is able to capture the critical points while the pairwise entanglement is absent. In contrast to the pairwise entanglement, two-site quantum discord is effectively long-range in the critical regimes where it decays algebraically with the distance between pairs. We also show that the thermal quantum discord between neighbor spins possesses strong distinctive behavior at the critical point that can be seen at finite temperature and, therefore, spotlights the critical point while the entanglement fails in this task.Comment: 8 pages, 6 figures, Accepted to Phys. Rev.

Algebraic List-decoding of Subspace Codes

Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain vector space over a finite field. The Koetter-Kschischang construction of subspace codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang subspace codes. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer $L$, our list-$L$ decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at most $L - \frac{L(L+1)}{2}R$ where $R$ is the normalized packet rate. Just as in the case of Sudan's list-decoding algorithm, this exceeds the previously best known error-correction radius $1-R$, demonstrated by Koetter and Kschischang, for low rates $R$

Effects of a space modulation on the behavior of a 1D alternating Heisenberg spin-1/2 model

The effects of a magnetic field ($h$) and a space modulation ($\delta$) on the magnetic properties of a one dimensional antiferromagnetic-ferromagnetic Heisenberg spin-1/2 model have been studied by means of numerical exact diagonalization of finite size systems, nonlinear sigma model and bosonization approach. The space modulation is considered on the antiferromagnetic couplings. At $\delta=0$, the model is mapped to a gapless L\"{u}ttinger liquid phase by increasing the magnetic field. However, the space modulation induces a new gap in the spectrum of the system and the system experiences different quantum phases which are separated by four critical fields. By opening the new gap a magnetization plateau appears at ${1/2}M_{sat}$. The effects of the space modulation are reflected in the emergence of a plateau in other physical functions such as the F-dimer and the bond dimer order parameters, and the pair-wise entanglement.Comment: 19 pages, 9 figures, Accepted by Journal of Phys.: Condens. Matte

Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

Suppose Alice wishes to send messages to Bob through a communication channel C_1, but her transmissions also reach an eavesdropper Eve through another channel C_2. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of Eve's equivocation ratio. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all $\epsilon > 0$, there exist coding schemes of rate $R \ge C_s - \epsilon$ that asymptotically achieve both the reliability and the security objectives. However, his proof of this result is based upon a nonconstructive random-coding argument. To date, despite a considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C_2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Herein, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C_1 and C_2 are symmetric and binary-input, and C_2 is degraded with respect to C_1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will be satisfied unless the main channel C_1 is noiseless, although we believe it can be always satisfied in practice.Comment: 15 pages, to appear in the IEEE Transactions on Information Theor