On Linear Operator Channels over Finite Fields

Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. There are NO constraints on the distribution of the transformation matrix and the field size. Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant. The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated. Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix. The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics.Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information Theor

Capacity Analysis of Linear Operator Channels over Finite Fields

Motivated by communication through a network employing linear network coding, capacities of linear operator channels (LOCs) with arbitrarily distributed transfer matrices over finite fields are studied. Both the Shannon capacity $C$ and the subspace coding capacity $C_{\text{SS}}$ are analyzed. By establishing and comparing lower bounds on $C$ and upper bounds on $C_{\text{SS}}$, various necessary conditions and sufficient conditions such that $C=C_{\text{SS}}$ are obtained. A new class of LOCs such that $C=C_{\text{SS}}$ is identified, which includes LOCs with uniform-given-rank transfer matrices as special cases. It is also demonstrated that $C_{\text{SS}}$ is strictly less than $C$ for a broad class of LOCs. In general, an optimal subspace coding scheme is difficult to find because it requires to solve the maximization of a non-concave function. However, for a LOC with a unique subspace degradation, $C_{\text{SS}}$ can be obtained by solving a convex optimization problem over rank distribution. Classes of LOCs with a unique subspace degradation are characterized. Since LOCs with uniform-given-rank transfer matrices have unique subspace degradations, some existing results on LOCs with uniform-given-rank transfer matrices are explained from a more general way.Comment: To appear in IEEE Transactions on Information Theor

The spectrum of the three-dimensional adjoint Higgs model and hot SU(2) gauge theory

We compute the mass spectrum of the SU(2) adjoint Higgs model in 2+1 dimensions at several points located in the (metastable) confinement region of its phase diagram. We find a dense spectrum consisting of an almost unaltered repetition of the glueball spectrum of the pure gauge theory, and additional bound states of adjoint scalars. For the parameters chosen, the model represents the effective finite temperature theory for pure SU(2) gauge theory in four dimensions, obtained after perturbative dimensional reduction. Comparing with the spectrum of screening masses obtained in recent simulations of four-dimensional pure gauge theory at finite temperature, for the low lying states we find quantitative agreement between the full and the effective theory for temperatures as low as T = 2 Tc. This establishes the model under study as the correct effective theory, and dimensional reduction as a viable tool for the description of thermodynamic properties. We furthermore compare the perturbative contribution O(g.T) with the non-perturbative contributions O(g^2.T) and O(g^3.T) to the Debye mass. The latter turns out to be dominated by the scale g^2.T, whereas higher order contributions are small corrections.Comment: LaTeX. Typos corrected and references adde

Impurity scattering and transport of fractional Quantum Hall edge state

We study the effects of impurity scattering on the low energy edge state dynamic s for a broad class of quantum Hall fluids at filling factor $\nu =n/(np+1)$, for integer $n$ and even integer $p$. When $p$ is positive all $n$ of the edge modes are expected to move in the same direction, whereas for negative $p$ one mode moves in a direction opposite to the other $n-1$ modes. Using a chiral-Luttinger model to describe the edge channels, we show that for an ideal edge when $p$ is negative, a non-quantized and non-universal Hall conductance is predicted. The non-quantized conductance is associated with an absence of equilibration between the $n$ edge channels. To explain the robust experimental Hall quantization, it is thus necessary to incorporate impurity scattering into the model, to allow for edge equilibration. A perturbative analysis reveals that edge impurity scattering is relevant and will modify the low energy edge dynamics. We describe a non-perturbative solution for the random $n-$channel edge, which reveals the existence of a new disorder-dominated phase, characterized by a stable zero temperature renormalization group fixed point. The phase consists of a single propagating charge mode, which gives a quantized Hall conductance, and $n-1$ neutral modes. The neutral modes all propagate at the same speed, and manifest an exact SU(n) symmetry. At finite temperatures the SU(n) symmetry is broken and the neutral modes decay with a finite rate which varies as $T^2$ at low temperatures. Various experimental predictions and implications which follow from the exact solution are described in detail, focusing on tunneling experiments through point contacts.Comment: 19 pages (two column), 5 post script figures appended, 3.0 REVTE

Scalar-gauge dynamics in (2+1) dimensions at small and large scalar couplings

We present the results of a detailed calculation of the excitation spectrum of states with quantum numbers J^{PC}=0++, 1-- and 2++ in the three-dimensional SU(2) Higgs model at two values of the scalar self-coupling and for fixed gauge coupling. In the context of studies of the electroweak phase transition at finite temperature these couplings correpond to tree-level, zero temperature Higgs masses of 35 GeV and 120 GeV, respectively. We also study the properties of Polyakov loop operators, which serve to test the confining properties of the model in the symmetric phase. At both values of the scalar coupling we obtain masses of bound states consisting entirely of gauge degrees of freedom ("W-balls"), which are very close to those obtained in the pure gauge theory. We conclude that the previously observed, approximate decoupling of the scalar and gauge sectors of the theory persists at large scalar couplings. We study the crossover region at large scalar coupling and present a scenario how the confining properties of the model in the symmetric phase are lost inside the crossover by means of flux tube decay. We conclude that the underlying dynamics responsible for the observed dense spectrum of states in the Higgs region at large couplings must be different from that in the symmetric phase.Comment: 36 pages, LaTeX, 13 postscript files, to be included with epsf; improved presentation, updated references, conclusions unchanged; version to appear in Nucl. Phys.

Coarse-grained distinguishability of field interactions

Information-theoretical quantities such as statistical distinguishability typically result from optimisations over all conceivable observables. Physical theories, however, are not generally considered valid for all mathematically allowed measurements. For instance, quantum field theories are not meant to be correct or even consistent at arbitrarily small lengthscales. A general way of limiting such an optimisation to certain observables is to first coarse-grain the states by a quantum channel. We show how to calculate contractive quantum information metrics on coarse-grained equilibrium states of free bosonic systems (Gaussian states), in directions generated by arbitrary perturbations of the Hamiltonian. As an example, we study the Klein-Gordon field. If the phase-space resolution is coarse compared to h-bar, the various metrics become equal and the calculations simplify. In that context, we compute the scale dependence of the distinguishability of the quartic interaction

Subfactors and quantum information theory

We consider quantum information tasks in an operator algebraic setting, where we consider normal states on von Neumann algebras. In particular, we consider subfactors $\mathfrak{N} \subset \mathfrak{M}$, that is, unital inclusions of von Neumann algebras with trivial center. One can ask the following question: given a normal state $\omega$ on $\mathfrak{M}$, how much can one learn by only doing measurements from $\mathfrak{N}$? We argue how the Jones index $[\mathfrak{M}:\mathfrak{N}]$ can be used to give a quantitative answer to this, showing how the rich theory of subfactors can be used in a quantum information context. As an example we discuss how the Jones index can be used in the context of wiretap channels. Subfactors also occur naturally in physics. Here we discuss two examples: rational conformal field theories and Kitaev's toric code on the plane, a prototypical example of a topologically ordered model. There we can directly relate aspects of the general setting to physical properties such as the quantum dimension of the excitations. In the example of the toric code we also show how we can calculate the index via an approximation with finite dimensional systems. This explicit construction sheds more light on the connection between topological order and the Jones index.Comment: v2: added more background material, some corrections and clarifications. 23 pages, submitted to QMath 13 (Atlanta, GA) proceeding

Contacts and Edge State Equilibration in the Fractional Quantum Hall Effect

We develop a simple kinetic equation description of edge state dynamics in the fractional quantum Hall effect (FQHE), which allows us to examine in detail equilibration processes between multiple edge modes. As in the integer quantum Hall effect (IQHE), inter-mode equilibration is a prerequisite for quantization of the Hall conductance. Two sources for such equilibration are considered: Edge impurity scattering and equilibration by the electrical contacts. Several specific models for electrical contacts are introduced and analyzed. For FQHE states in which edge channels move in both directions, such as $\nu=2/3$, these models for the electrical contacts {\it do not} equilibrate the edge modes, resulting in a non-quantized Hall conductance, even in a four-terminal measurement. Inclusion of edge-impurity scattering, which {\it directly} transfers charge between channels, is shown to restore the four-terminal quantized conductance. For specific filling factors, notably $\nu =4/5$ and $\nu=4/3$, the equilibration length due to impurity scattering diverges in the zero temperature limit, which should lead to a breakdown of quantization for small samples at low temperatures. Experimental implications are discussed.Comment: 14 pages REVTeX, 6 postscript figures (uuencoded and compressed

Classical Signal Model for Quantum Channels

Recently it was shown that the main distinguishing features of quantum mechanics (QM) can be reproduced by a model based on classical random fields, so called prequantum classical statistical field theory (PCSFT). This model provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system (e.g., entangled systems), as averages with respect to fluctuations of classical (Gaussian) random fields. In this note we consider some consequences of PCSFT for quantum information theory. They are based on the observation \cite{W} of two authors of this paper that classical Gaussian channels (important in classical signal theory) can be represented as quantum channels. Now we show that quantum channels can be represented as classical linear transformations of classical Gaussian signa