Multiple-quantum operator algebra spaces and description for unitary time evolution of multilevel spin systems

A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.     

Entanglement capacity of two-qubit unitary operator for rank two mixed states

The entanglement capacity of two-qubit unitary operator acting on rank two mixed states in concurrence is discussed. The condition of perfect entangler is the same as that acting on pure states and the entanglement capacity is the mixing parameter v1. For non-perfect entangler, the upper and lower bound of the entanglement capacity are given.     

A three-parameter deformation of the Weyl-Heisenberg algebra: differential calculus and invariance

We define a three-parameter deformation of the Weyl-Heisenberg algebra that generalizes the q-oscillator algebra. By a purely algebraical procedure, we set up on this quantum space two differential calculi that are shown to be invariant on the same quantum group, extended to a ten-generator Hopf-star-algebra. We prove that when the values of the parameters are related, the two differential calculi reduce to one that is invariant under two quantum groups.     

Extended Bell inequality and maximum violation

The original formula of Bell inequality(BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the statedensity operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by PB is always less than or at most equal to one for the local realistic model(P_B~(lc)≤ 1)regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as P_~(max )B =2, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.     

Extended number state basis and number-phase intelligent states of light field I. Mapping and operator ordering approach to quantum phase problem

The quantum phase problem for single mode optical fields is solved and original definitions of all possible phase operators are given. Measures of phase dispersion are reconsidered and the uncertainty relations are modified. Properties of phase representation, phase distribution and those of the quasidistribution for number and phase are studied.     

Coherent states,phase states,and condensed states

We study phase properties of generalized coherent states obtained from usual Fock coherent states by adapting classical methods of statistical mechanics, in particular, the well-known procedure of thermodynamical limit. Moreover, we show that there exists a close connection between these states and the states describing boson systems with condensation properties.     

Kane model,dynamical algebra and coherent states

Dynamical invariance algebra is developed for the Kane model. On the basis of the Heisenberg-Weyl algebra of invariants explicit expressions for interband coherent states and Green functions are derived.     

Multiparty quantum secret sharing with Bell states and Bell measurements

We present a multiparty quantum secret sharing scheme and analyze its security. In this scheme, the sender Alice takes EPR pairs in Bell states as quantum resources. In order to obtain the shared key, all participants only need to perform Bell measurements, not to perform any local unitary operation. The total efficiency in this scheme approaches 100% as the classical information exchanged is not necessary except for the eavesdropping checks.     

The q-boson operator algebra and q-Hermite polynomials

The q-boson algebra is defined as an associative algebra with generators and relations. Some examples are given, and then the q-boson algebra is extended such that the roots of the diagonal generators are also defined. It is shown that a family of transformations exist mapping one set of standard generators of the q-boson algebra to another set of standard generators. Using such a transformation, one obtains expressions for q-bosons for which the kth q-boson state is expressed in terms of a q-Hermite polynomial p _k (x; q) which reduces to the ordinary Hermite polynomial of degree k when q=1.     

BRST operator and an open gauge algebra

We show that in gauge theories of a general type with an open generator gauge algebra a nilpotent BRST does not exist.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 99–102, February, 1987.     

On generalized electromagnetism and Dirac algebra

Using a framework of Dirac algebra, the Clifford algebra appropriate for Minkowski space-time, the formulation of classical electromagnetism including both electric and magnetic charge is explored. Employing the two-potential approach of Cabibbo and Ferrari, a Lagrangian is obtained that is dyality invariant and from which it is possible to derive by Hamilton's principle both the symmetrized Maxwell's equations and the equations of motion for both electrically and magnetically charged particles. This latter result is achieved by defining the variation of the action associated with the cross terms of the interaction Lagrangian in terms of a surface integral. The surface integral has an equivalent path-integral form, showing that the contribution of the cross terms is local in nature. The form of these cross terms derives in a natural way from a Dirac algebraic formulation, and, in fact, the use of the geometric product of Dirac algebra is an essential aspect of this derivation. No kinematic restrictions are associated with the derivation, and no relationship between magnetic and electric charge evolves from the (classical) formulation. However, it is indicated that in bound states quantum mechanical considerations will lead to a version of Dirac's quantization condition. A discussion of parity violation of the generalized electromagnetic theory is given, and a new approach to the incorporation of this violation into the formalism is suggested. Possibilities for extensions are mentioned.Work supported by the Department of Energy, contract DE-AC03-76SF00515.     

Extended supersymmetries and the Dirac operator

We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges that exist and restrictions on the geometry of the underlying spaces as well as the admissible gauge field configurations. From the superalgebra with two or more real supercharges we infer the existence of integrability conditions and obtain a corresponding superpotential. This potential can be used to deform the supercharges and to determine zero modes of the Dirac operator. The general results are applied to the Kähler spaces CPⁿ.     

Extended conformal algebra and the Yang-Baxter equation

Working in the context of classical Toda field theory, we derive their extended conformal symmetry algebra from their integrability properties.     

Cryptanalysis of multiparty quantum secret sharing with Bell states and Bell measurements

The security of multiparty quantum secret sharing with Bell states and Bell measurements [R.H. Shi et al., Opt. Commun. 283 (2010) 2476] is analyzed. It is shown that the first agent and the last one can gain access to the dealer's secret if they collaborate in this protocol. Therefore, this protocol does not satisfy the security requirement of quantum secret sharing.     

Quantum secret sharing between multiparty and multiparty with Bell states and Bell measurements

We present a quantum secret sharing scheme between multiparty (m members in Group 1) and multiparty (n members in Group 2), and analyze its security. This scheme takes EPR pairs in Bell states as quantum resources. In order to obtain the shared key, all members only need to perform Bell measurements, rather than perform any local unitary operation. The total efficiency in this scheme approaches 100% as the classical information exchanged is not necessary except for the eavesdropping checks.     

Microscopic description and excitation of unitary alalog states

A microscopic investigation in a self-consistent particle-hole model reveals approximate unitary analog states in spite of large symmetry breaking. The K-nucleus elastic scattering and (K^?, π^?) excitation of these states are studied, showing strong surface effects.     

Gauge invariance and the generalized Bondi-Metzner algebra

We examine the algebraic meaning of the Electromagnetic gauge invariance and show that it leads to the new concepts of gauged operators, gauged representations and hence to infinite dimensional extensions of Lie algebras. In particular we prove that the generalized Bondi-Metzner algebra can be interpreted as a gauged Lorentz algebra related to the Electromagnetic gauge.     

Qubit portrait of qudit states and Bell inequalities

A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. In view of the qubit-portrait method, the Bell inequalities for two qubits and two qutrits are discussed within the framework of the probability-representation of quantum mechanics. A semigroup of stochastic matrices is associated with tomographic-probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as an ansatz to provide a necessary condition for the separability of quantum states.