The number of irreducible polynomials of degree n over Fq with given trace and constant terms

AbstractWe study the number Nγ(n,c,q) of irreducible polynomials of degree n over Fq where the trace γ and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of Nγ(n,c,q) varying c and γ. We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any Nγ(n,c,q) when n=a(q−1) for a nonzero constant term c and a nonzero trace γ. As a byproduct, we give a simple and explicit formula for the number N(n,c,q) of irreducible polynomials over Fq of degree n=q−1 with a prescribed primitive constant term c

Implementing Quantum Gates using the Ferromagnetic Spin-J XXZ Chain with Kink Boundary Conditions

We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent DMRG method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin-3/2 XXZ chains, such as Not, Hadamard, Pi-8, Phase, and C-Not, with fidelity levels exceeding 99%.Comment: Updated Acknowledgement

Numerical evidence of chiral magnetic effect in lattice gauge theory

The chiral magnetic effect is the generation of electric current of quarks along external magnetic field in the background of topologically nontrivial gluon fields. There is a recent evidence that this effect is observed by the STAR Collaboration in heavy ion collisions at RHIC. In our paper we study qualitative signatures of the chiral magnetic effect using quenched lattice simulations. We find indications that the electric current is indeed enhanced in the direction of the magnetic field both in equilibrium configurations of the quantum gluon fields and in a smooth gluon background with nonzero topological charge. In the confinement phase the magnetic field enhances the local fluctuations of both the electric charge and chiral charge densities. In the deconfinement phase the effects of the magnetic field become smaller, possibly due to thermal screening. Using a simple model of a fireball we obtain a good agreement between our data and experimental results of the STAR Collaboration.Comment: 14 pages, 14 figures, uses RevTeX 4.0; revision: references and comments added, figures corrected, published versio

From Ground States to Local Hamiltonians

Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest neighbour interactions on some type of lattices. A necessary condition for a space $V$ to be the ground-state space of some local Hamiltonian with a given interaction pattern, is that the maximally mixed state supported on $V$ is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have $V$ satisfying the necessary condition mentioned above as its ground-state space, by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.Comment: 11 pages, 2 figures, to be published in PR

Decay of Superconducting and Magnetic Correlations in One- and Two-Dimensional Hubbard Models

In a general class of one and two dimensional Hubbard models, we prove upper bounds for the two-point correlation functions at finite temperatures for electrons, for electron pairs, and for spins. The upper bounds decay exponentially in one dimension, and with power laws in two dimensions. The bounds rule out the possibility of the corresponding condensation of superconducting electron pairs, and of the corresponding magnetic ordering. Our method is general enough to cover other models such as the t-J model.Comment: LaTeX, 8 pages, no figures. A reference appeared after the publication is adde

Mutual Exclusion Statistics in Exactly Solvable Models in One and Higher Dimensions at Low Temperatures

We study statistical characterization of the many-body states in exactly solvable models with internal degrees of freedom. The models under consideration include the isotropic and anisotropic Heisenberg spin chain, the Hubbard chain, and a model in higher dimensions which exhibits the Mott metal-insulator transition. It is shown that the ground state of these systems is all described by that of a generalized ideal gas of particles (called exclusons) which have mutual exclusion statistics, either between different rapidities or between different species. For the Bethe ansatz solvable models, the low temperature properties are well described by the excluson description if the degeneracies due to string solutions with complex rapidities are taken into account correctly. {For} the Hubbard chain with strong but finite coupling, charge-spin separation is shown for thermodynamics at low temperatures. Moreover, we present an exactly solvable model in arbitrary dimensions which, in addition to giving a perspective view of spin-charge separation, constitutes an explicit example of mutual exclusion statistics in more than two dimensions

Cluster Property and Robustness of Ground States of Interacting Many Bosons

We study spatial correlation functions of local operators of interacting many bosons confined in a box of a large, but volume V, for various `ground states' whose energy densities are almost degenerate. The ground states include the coherent state of interacting bosons (CSIB), the number state of interacting bosons (NSIB), and the number-phase squeezed state of interacting bosons, which interpolates between the CSIB and NSIB. It was shown previously that only the CSIB is robust (i.e., does not decohere for a macroscopically long time) against the leakage of bosons into an environment. We show that for the CSIB the spatial correlation of any local operators A(r) and B(r') (which are localized around r and r', respectively) vanishes as |r - r' | \sim V^{1/3} \to \infty, i.e., the CSIB has the `cluster property.' In contrast, the other ground states do not possess the cluster property. Therefore, we have successfully shown that the robust state has the cluster property. This ensures the consistency of the field theory of bosons with macroscopic theories.Comment: We have replaced the manuscript in order to update the reference list and to fix typos. (5 pages, no figures) In the final manuscript, a few sentences have added for more detailed explanation. Journal PDF at http://jpsj.jps.or.jp/journal/JPSJ-71-1.htm

Appearance and Stability of Anomalously Fluctuating States in Shor's Factoring Algorithm

We analyze quantum computers which perform Shor's factoring algorithm, paying attention to asymptotic properties as the number L of qubits is increased. Using numerical simulations and a general theory of the stabilities of many-body quantum states, we show the following: Anomalously fluctuating states (AFSs), which have anomalously large fluctuations of additive operators, appear in various stages of the computation. For large L, they decohere at anomalously great rates by weak noises that simulate noises in real systems. Decoherence of some of the AFSs is fatal to the results of the computation, whereas decoherence of some of the other AFSs does not have strong influence on the results of the computation. When such a crucial AFS decoheres, the probability of getting the correct computational result is reduced approximately proportional to L^2. The reduction thus becomes anomalously large with increasing L, even when the coupling constant to the noise is rather small. Therefore, quantum computations should be improved in such a way that all AFSs appearing in the algorithms do not decohere at such great rates in the existing noises.Comment: 11 figures. A few discussions were added in verion 2. Version 3 is the SAME as version 2; only errors during the Web-upload were fixed. Version 4 is the publised version, in which several typos are fixed and the reference list is update

Does the 2D Hubbard Model Really Show d-Wave Superconductivity?

Some issues concerning the question if the two-dimensional Hubbard model really show d-wave superconductivity are briefly discussed.Comment: Revtex, no figure