Quantum teleportation

In the framework of an algebraic approach, we consider a quantum teleportation procedure. It turns out that using the quantum measurement nonlocality hypothesis is unnecessary for describing this procedure. We study the question of what material objects are information carriers for quantum teleportation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 79–98, October, 2008.     

Spectra of radiation for spinless and spin relativistic particles in strong magnetic fields

Synchro-curvature radiation put forward by Zhang and Cheng is a new and universal radiation mechanism, it describes in detail the radiation properties of a relativistic charged particle moving in a curved magnetic field. This new radiation generalizes all the classical results of ordinary synchrotron and curvature radiations and reveals inherent linkage and unification between them. Additionly, a general, simple and unitary formula is provided for discussing the radiation problem in research of pulsars. However, the magnitude of the magnetic field of a pulsar is so strong (10⁷-10⁹T) that the quantum effects cannot be neglected. The GFWW method developed recently by Lieu and Axford is applied to generalize the results of Zhang and Cheng. The quantum limited synchro-curvature radiation spectra for spinless K-G particles and spin-1/2 Dirac particles are presented, respectively. Their radiation properties are also discussed. Project supported by the National Natural Science Foundation of China and the National “Climbing Project”.     

Semi-classical quantum theory for cyclotron radiation

A semi-classical quantum theory of the cyclotron radiation of the nonrelativistic thermal electrons in a very strong magnetic field is presented. The basic formulae of the absorption coefficient of cyclotron resonancek _vand the absorption (scattering) cross-section of cyclotron resonance σ _v have been derived under the quadrupole approximation. σ _v is an important quantity in the study of the “magnetic inverse-Compton scattering”. It is shown that σ _v is greatly larger than the Thomson cross-sectron σ_T, which is important in discussing the magnetic inverse-Compton scattering of the relativistic electrons in a very strong magnetic field. Project supported by the National Natural Science Foundation of China and the Climbing Plan.     

Quantum trajectories of an oscillator interacting with an electromagnetic field, a classical force, and a heat bath

We consider a solvable problem describing the dynamics of a quantum oscillator interacting with an electromagnetic field, a classical force, and a heat bath. We propose a general method for solving Markovian master equations, the method of quantum trajectories. We construct the stochastic evolution operator involving the stochastic analogue of the Baker-Hausdorff formula and calculate the system density matrix for an arbitrary initial state. As a physical application, we evaluate the influence of the environment at a finite temperature on the accuracy of measuring a weak classical force by the interference method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 444–459, March, 2009.     

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s _λ (M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 14–46, April, 2006.     

Bernstein’s analyticity theorem for quantum differences

We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f’s quantum differences are nonnegative then f has a power series representation on I. Further, if the quantum differences have fixed sign on I then f is analytic on I.     

Twisted Homology of Quantum SL(2)

We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the “twisted” Hochschild dimension coincides with the classical dimension of , thus avoiding the “dimension drop” in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such automorphism is the canonical modular automorphism arising from the Haar functional. In addition, we identify the twisted cyclic cohomology classes corresponding to the three covariant differential calculi over quantum SU(2) discovered by Woronowicz.     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.     

Yang-Baxter algebra and generation of quantum integrable models

We discover an operator-deformed quantum algebra using the quantum Yang-Baxter equation with the trigonometric R-matrix. This novel Hopf algebra together with its q→1 limit seems the most general Yang-Baxter algebra underlying quantum integrable systems. We identify three different directions for applying this algebra in integrable systems depending on different sets of values of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 470–485, June, 2007.     

Regularity criterion for a mathematical model of the laser

There exists a deep relationship between the nonexplosion conditions for Markov evolution in classical and quantum probability theories. Both of these conditions are equivalent to the preservation of the unit operator (total probability) by a minimal Markov semigroup. In this work, we study the Heisenberg evolution describing an interaction between the chain ofN two-level atoms andn-mode external Bose field, which was considered recently by J. Alli and J. Sewell. The unbounded generator of the Markov evolution of observables acts in the von Neumann algebra. For the generator of a Markov semigroup, we prove a nonexplosion condition, which is the operator analog of a similar condition suggested by R. Z. Khas’minski and later by T. Taniguchi for classical stochastic processes. For the operator algebra situation, this condition ensures the uniqueness and conservativity of the quantum dynamical semigroup describing the Markov evolution of a quantum system. In the regular case, the nonexplosion condition establishes a one-to-one relation between the formal generator and the infinitesimal operator of the Markov semigroup. Translated fromMatematicheskie Zemetki, Vol. 67, No. 5, pp. 788–796, May, 2000.     

A pure quantum mechanical theory of parity effect in tunneling and evolution of spins

In recent years, the spin parity effect in magnetic macroscopic quantum tunneling has attracted extensive attention. Using the spin coherent-state path-integral method it is shown that if the HamiltonianH of a single-spin system hasM - fold rotational symmetry around z-axis, the tunneling amplitude 〈−S|e ^Ht |S〉 vanishes when S, the quantum number of spin, is not an integer multiple ofM/2, where |m〉 (m=-S, -S +1, ⋯, S) are the eigenstates of S^z. Not only is a pure quantum mechanical approach adopted to the above result, but also is extended to more general cases where the quantum system consists ofN spins, the quantum numbers of which can take any values, including the single-spin system, ferromagnetic particle and antiferromagnetic particle as particular instances, and where the states involved are not limited to the extreme ones. The extended spin parity effect is that if the Hamiltonian ℋ of the system ofN spins also has the above symmetry, then 〈m′_N⋯m′₂ m′₁|e^{−H t} |m ₁ m ₂⋯m _N vanishes when ∑ _i=1 ^N (m _i−m′₁) not an integer multiple ofM, where |m ₁ m ₂⋯m _N〉=∏ _α=1 ^N |m _a 〉 are the eigenstates of S _a ^z . In addition, it is argued that for large spin the above result, the so-called spin parity effect, does not mean the quenching of spin tunneling from the direction of ⊕-z to that of ±z. Project supported by the National Natural Science Foundation of China (Grant Nos. 19674002, 19677101).     

Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data

The aim of this paper is to answer an important issue in quantum mechanics, namely to estimate the purity of a quantum state of a light beam. Estimation of the purity is based on the results of quantum homodyne measurements performed on independent identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, which can take negative values and must satisfy certain constraints of positivity imposed by quantum physics. We estimate the integrated squared Wigner function by a kernel-based second order U — statistic. This quadratic functional is a physical measure of the purity of the state. We also give an adaptive estimator, which does not depend on the smoothness parameters. We establish upper bounds of the minimax risk over a class of infinitely differentiable functions.      

Quantum commutative algebras and their duals

If an algebraA is quantum commutative with respect to the action of a quasitriangular Hopf algebraH, then the monoidal structure on the category_H ^ℳ of modules overH induces a rnonoidal structure on the category_A#H ^ℳ of modules over the associated smash productA # H. The condition under which the braiding structure of_H ^ℳ induces a braiding structure on_A#H ^ℳ is further investigated. Dually, the notion of quantum cocommutativity is introduced, and similar result in this dual situation is obtained.     

Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT

To study the representation category of the triplet W-algebra that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category C _p of finite-dimensional representations of the restricted quantum group Ū _q sℓ(2) at . We fully describe the category C _p by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the -and Ū _q sℓ(2)-representation categories is conjectured for all p = 2 and proved for p = 2. The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal R-matrix with the braiding matrix. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 398–427, September, 2006.     

Geometric torsions and invariants of manifolds with a triangulated boundary

Geometric torsions are torsions of acyclic complexes of vector spaces consisting of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a three-dimensional manifold with a triangulated boundary. These invariants can be naturally combined into a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued at their common boundary, these vectors undergo scalar multiplication, i.e., they satisfy Atiyah’s axioms of a topological quantum field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 98–114, January, 2009.