Quantum Cryptography,Quantum Communication,and Quantum Computer in a Noisy Environment

First, we study several information theories based on quantum computing in a desirable noiseless situation. (1) We present quantum key distribution based on Deutsch’s algorithm using an entangled state. (2) We discuss the fact that the Bernstein-Vazirani algorithm can be used for quantum communication including an error correction. Finally, we discuss the main result. We study the Bernstein-Vazirani algorithm in a noisy environment. The original algorithm determines a noiseless function. Here we consider the case that the function has an environmental noise. We introduce a noise term into the function f(x). So we have another noisy function g(x). The relation between them is g(x) = f(x) ± O(??). Here O(??) ? 1 is the noise term. The goal is to determine the noisy function g(x) with a success probability. The algorithm overcomes classical counterpart by a factor of N in a noisy environment.     

Efficient Quantum Algorithm for the Parity Problem of a Certain Function

Based on a particular mathematical structure of a certain function f(x) under our attention, we present a novel quantum algorithm. The algorithm allows one to determine the property of a certain function. In our study, it is f(x) = f(?x). Therefore, there would be a question here, “How fast can we succeed in this?” All we need to do is only the evaluation of a single quantum state \(|\overbrace {0,0,\ldots ,0,1}^{N}\rangle \) (N ≥?2). Only using that with a little amount of information, we can derive the global property f(x) = f(?x). Our quantum algorithm overcomes a classical counterpart by a factor of the order of 2^N.     

Quantum Mechanics and Imprecise Probability

An extension of the Born rule, the quantum typicality rule, has recently been proposed [B. Galvan in Found. Phys. 37:1540–1562 (2007)]. Roughly speaking, this rule states that if the wave function of a particle is split into non-overlapping wave packets, the particle stays approximately inside the support of one of the wave packets, without jumping to the others. In this paper a formal definition of this rule is given in terms of imprecise probability. An imprecise probability space is a measurable space endowed with a set of probability measures ℘. The quantum formalism and the quantum typicality rule allow us to define a set of probabilities on (X ^T ,ℱ), where X is the configuration space of a quantum system, T is a time interval and ℱ is the σ-algebra generated by the cylinder sets. Thus, it is proposed that a quantum system can be represented as the imprecise stochastic process , which is a canonical stochastic process in which the single probability measure is replaced by a set of measures. It is argued that this mathematical model, when used to represent macroscopic systems, has sufficient predictive power to explain both the results of the statistical experiments and the quasi-classical structure of the macroscopic evolution.     

Lower Bound of Minimal Time Evolution in Quantum Mechanics

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S ², must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2×2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector P of the evolving quantum state with the vector of the 2×2 hermitian Hamiltonians and it is shown that the Hamiltonian H can be parameterized by two independent parameters and Θ.     

Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Let be the spectrum of in L ²(ℝ), where q is an even almost-periodic complex-valued function with bounded primitive and derivative. It is known that , where is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to the first asymptotic correction is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of Δμ _n .      

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

(1+1)-Dirac Particle with Position-Dependent Mass in Complexified Lorentz Scalar Interactions: Effectively $\mathcal{PT}$ -Symmetric

The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the “quasi-parity” on the Dirac particles’ spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S(z)=S(x−ib) (an inversely linear plus linear, leading to a symmetric oscillator model), and S(x)=S _r (x)+iS _i (x) (a -symmetric Scarf II model) are considered. Moreover, a first-order intertwining differential operator and an η-weak-pseudo-Hermiticity generator are presented and a complexified -symmetric periodic-type model is used as an illustrative example.     

Quantum Groups GLp,q(2)- and $\mathit{SU}_{q_{1}/q_{2}}(2)$ -Invariant Bosonic Gases: A Comparative Study

We discuss the algebras, representations, and thermodynamics of quantum group bosonic gas models with two different symmetries: GL _p,q (2) and . We establish the nature of the basic numbers which follow from these GL _p,q (2)- and -invariant bosonic algebras. The Fock space representations of both of these quantum group invariant bosonic oscillator algebras are analyzed. It is concisely shown that these two quantum group invariant bosonic particle gases have different algebraic and high-temperature thermo-statistical properties.     

Wavelet Transformation and Wigner-Husimi Distribution Function

We find that the optical wavelet transformation can be used to study the Husimi distribution function in phase space theory of quantum optics. We prove that the Husimi distribution function of a quantum state |ψ〉 is just the modulus square of the wavelet transform of with ψ(x) being the mother wavelet up to a Gaussian function. Thus a convenient approach for calculating various Husimi distribution functions of miscellaneous quantum states is presented.     

Semiclassical analysis of edge state energies in the integer quantum Hall effect

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separately, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x = x_c and an infinite wall at x = 0, and appears in numerous physical contexts. The eigenvalues E_n(x_c) for a given quantum number n are solutions of the equation S(E,x_c)=π[n+ γ(E,x_c)] where S is the WKB action and 0 < γ < 1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of γ[E_n(x_c),x_c] ≡γ_n(x_c) on x_c is analyzed between its two extreme values as x_c ↦-∞ far inside the sample and as x_c ↦∞ far outside the sample. The edge-state energiesE_n(x_c) obey an almost exact scaling law of the form and the scaling function f(y) is explicitly elucidated.     

Two-Parameter Deformed SUSY Algebra for Fibonacci Oscillators

We construct a two-parameter deformed SUSY algebra for the system of n ordinary fermions and n(q ₁,q ₂)-deformed bosons called Fibonacci oscillators with -symmetry. We then analyze the Fock space representation of the algebra constructed. We obtain the total deformed Hamiltonian and the energy levels together with their degeneracies for the system. We also consider some physical applications of the Fibonacci oscillators with -symmetry, and discuss the main reasons to consider two distinct deformation parameters.     

The Algebraic Structure of an Approximately Universal System of Quantum Computational Gates

Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the basic algebraic properties of this system by introducing the notion of Shi-Aharonov quantum computational structure. We show that the quotient of this structure is isomorphic to a structure based on a particular set of complex numbers (the closed disc with center and radius ). Dedicated to Pekka Lahti.     

Singular Energy Distributions in Driven and Undriven Granular Media

We study the kinetic theory of driven and undriven granular gases, taking into account both translational and rotational degrees of freedom. We obtain the high-energy tail of the stationary bivariate energy distribution, depending on the total energy E and the ratio of rotational energy E _w to total energy. Extremely energetic particles have a unique and well-defined distribution f(x) which has several remarkable features: x is not uniformly distributed as in molecular gases; f(x) is not smooth but has multiple singularities. The latter behavior is sensitive to material properties such as the collision parameters, the moment of inertia and the collision rate. Interestingly, there are preferred ratios of rotational-to-total energy. In general, f(x) is strongly correlated with energy and the deviations from a uniform distribution grow with energy. We also solve for the energy distribution of freely cooling Maxwell Molecules and find qualitatively similar behavior.     

The Ensemble Quantum State of a Single Particle

The derivation of the statistical nature of the quantum mechanical wave function is presented within the formalism of quantum mechanics and the second quantization. The statistical wave function may be derived for non relativistic bosons, non relativistic fermions, and relativistic bosons by employing the commuting field operator . For relativistic electrons a strictly anticommuting must be employed to derive the statistical wave function (spinor). The discussion at the end of the paper aims to show the physical plausibility of a statistical wave function.     

Physical Propositions and Quantum Languages

The word proposition is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct two languages ℒ ^* (x) and ℒ(x) with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice of propositions of quantum logic (QL) can be obtained by selecting a subset of p-testable propositions within the Boolean lattice of all propositions associated with sentences of ℒ(x). Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different metalinguistic concepts (truth and verifiability according to QM, respectively). Moreover one can construct a quantum language ℒ _TQ (x) whose Lindenbaum–Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples of physical systems, while standard QL does not bear this interpretation.     

Baxterization of Solutions to Reflection Equation with Hecke R-matrix

Let R be a Hecke solution to the Yang–Baxter equation and K be a reflection equation matrix with coefficients in an associative algebra . Let R(x) be the baxterization of R and suppose that K satisfies a polynomial equation with coefficients in the center of . We construct solutions to the reflection equation with spectral parameter relative to R(x), in the form of polynomials in K.     

Dual parametrization of GPDs versus the double distribution Ansatz

We establish a link between the dual parametrization of GPDs and a popular parametrization based on the double distribution Ansatz, which is in prevalent use in phenomenological applications. We compute several first forward-like functions that express the double distribution Ansatz for GPDs in the framework of the dual parametrization and show that these forward-like functions make the dominant contribution into the GPD quintessence function. We also argue that the forward-like functions with 1 contribute to the leading singular small-x_Bj behavior of the imaginary part of DVCS amplitude. This makes the small-x_Bj behavior of independent of the asymptotic behavior of PDFs. Assuming analyticity of Mellin moments of GPDs in the Mellin space we are able to fix the value of the D -form factor in terms of the GPD quintessence function N(x, t) and the forward-like function Q ₀(x, t) .     

Constructing Four-Photon States for Quantum Communication and Information Processing

We provide a method for constructing a set of four-photon states suitable for quantum communication applications. Among these states is a set of concatenated quantum code states that span a decoherence-free subspace that is robust under collective-local as well as global dephasing noise. This method requires only the use of spontaneous parametric down-conversion, quantum state post-selection, and linear optics. In particular, we show how this method can be used to produce all sixteen elements of the second-order Bell gem , which includes these codes states and is an orthonormal basis for the Hilbert space of four qubits composed entirely of states that are fully entangled under the four-tangle measure.     

Letter: Riemannian Metric of the Invariance Quantum Group of the Fermion Algebra

We calculate the bi-invariant metric of FIO(2), the inhomogeneous invariance quantum group of the fermion algebra. We find that this metric is identical to that of the bi-invariant metric of GL(2, R) ⁺ × SU (1, 1). However, the quantum group manifold is restricted to a region of the GL(2, R) manifold.     

Finite- N effects for ideal polymer chains near a flat impenetrable wall

This paper addresses the statistical mechanics of ideal polymer chains next to a hard wall. The principal quantity of interest, from which all monomer densities can be calculated, is the partition function, G _N(z) , for a chain of N discrete monomers with one end fixed a distance z from the wall. It is well accepted that in the limit of infinite N , G _N(z) satisfies the diffusion equation with the Dirichlet boundary condition, G _N(0) = 0 , unless the wall possesses a sufficient attraction, in which case the Robin boundary condition, G _N(0) = - G _N ^′(0) , applies with a positive coefficient, . Here we investigate the leading N ^-1/2 correction, G _N(z) . Prior to the adsorption threshold, G _N(z) is found to involve two distinct parts: a Gaussian correction (for z aN ^1/2 with a model-dependent amplitude, A , and a proximal-layer correction (for z a described by a model-dependent function, B(z) .