On the quantum Lorentz group

The quantum analog of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowski metric. In this framework we show explicitly the correspondence between the SL(2,C) and Lorentz quantum groups. Five matrices of the quantum Lorentz group are constructed in terms of the R matrix of SL(2,C) group. These matrices satisfy Yang–Baxter equations and two of which have adequate properties tied to the quantum Minkowski space structure as the reality conditions of the coordinates and the symmetrization of the metric. It is also shown that the Minkowski metric leads to invariant and central lengths of four-vectors.     

Quantum principal bundles over quantum real projective spaces

Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U(1)$U\left(1\right)$ as a structure group, the other has the quantum group S_Uq(2)$S{U}_q\left(2\right)$ as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf–Galois extension.     

Faithful compact quantum group actions on connected compact metrizable spaces

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.     

Field theory on the q-deformed fuzzy sphere I

We study the q-deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity. We construct for both cases a differential calculus which is compatible with the star structure, study the integral, and find a canonical frame of one-forms. We then consider actions for scalar field theory, as well as for Yang–Mills and Chern–Simons-type gauge theories. The zero curvature condition is solved.     

Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.     

Poincaré sphere representation for three state systems

We point out that the Poincaré sphere can be used to represent the rays of a three state quantum system. Those interested in geometric phase phenomena may find this representation a useful aid to visualize the global structure of ray space.     

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

On the tomographic picture of quantum mechanics

We formulate necessary and sufficient conditions for a symplectic tomogram of a quantum state to determine the density state. We establish a connection between the (re)construction by means of symplectic tomograms with the construction by means of Naimark positive definite functions on the Weyl-Heisenberg group. This connection is used to formulate properties which guarantee that tomographic probabilities describe quantum states in the probability representation of quantum mechanics.     

Dynamics of the collective excitations of the quantum Hall system

Using the non linear optical technique of 3-pulse 4-wave mixing, we study the dynamics of the collective excitations of the quantum Hall system. We excite the system with 100 fs pulses propagating in directions k₁ and k₃ and then probe its time evolution with a delayed pulse k₂. We measure the non-linear optical response from the lowest Landau level along the direction k₁+k₂−k₃. As function of the time delay of pulse k₂, this signal shows striking beats for short time delays (500 fs), followed by a rise (20 ps) and then a decay (100 ps). We identify the microscopic origin of this dynamics by extending the standard theory of ultra fast nonlinear optics to include the effects of the correlations.     

MMSE detection method in uplink massive MIMO systems based on quantum computing

The minimum mean square error (MMSE) detection method involved matrix inversion operation with excessive computational burden. In this paper, we develop an improved quantum linear system algorithm to solve matrix inversion problem of the MMSE detection method in uplink massive multiple-input and multiple-output (MIMO) systems. In order to apply reasonably the robust computational power of quantum computing, we optimize the preparation of the input state and the extraction of the solution from the final entangled quantum state. We prove that this algorithm can reduce computational complexity to $\mathrm{O}(N\log ?N)$.     

A Quantum Homogeneous Space of Nilpotent Matrices

A quantum deformation of the adjoint action of the special linear group on the variety of nilpotent matrices is introduced. New non-embedded quantum quasi- homogeneous spaces are obtained related to certain maximal coadjoint orbits, and known quantum homogeneous spaces are revisited.     

On the quantum link for transport of logic states

A novel approach for representing logic states in the quantum nodes and transferring the states from one node to another is proposed. Both transmit and receive nodes consist of a rubidium atom (⁸⁷Rb) placed at the center of a two-mode cavity. Representation of logic states by two subspaces of the space of ⁸⁷Rb atom hyperfine states eliminates the need for the transmitting node to change logic state during logic transfer through Raman process. The atom is excited by simultaneous application of two laser beams - one for each subspace. Based on the logic state, the atom emits a photon of appropriate frequency and polarization through Raman process within the corresponding subspace. The emitted photon leaks out of the cavity, reaches the receiving node, and initiates logic dependent transitions there. A simulation platform is developed through the system Hamiltonians for transmit and receive nodes followed by the formulation of the time evolution of the density matrices for the nodes. The efficacy of the simulation approach is emphasized.     

Localization length exponent,critical conductance distribution and multifractality in hierarchical network models for the quantum Hall effect

We study hierarchical network models which have recently been introduced to approximate the Chalker-Coddington model for the integer quantum Hall effect (A.G. Galstyan and M.E. Raikh, PRB 5 (1997) 1422; Arovas et al., PRB 56 (1997) 4751). The hierarchical structure is due to a recursive method starting from a finite elementary cell. The localization-delocalization transition occurring in these models is displayed in the flow of the conductance distribution under increasing system size. We numerically determine this flow, calculate the critical conductance distribution, the critical exponent of the localization length, and the multifractal exponents of critical eigenstates.     

Exact classical and quantum solutions for a covariant oscillator near the black hole horizon in Stueckelberg-Horwitz-Piron theory

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covariance theory. These solutions can be obtained by solving the generalized geodesic equation and Schrödinger-Stueckelberg-Horwitz-Piron (SHP) wave equation for a simple harmonic oscillator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonal and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum p according to a quantum number n. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.     

Electrostatic theory for imaging experiments on local charges in quantum Hall systems

We use a simple electrostatic treatment to model recent experiments on quantum Hall systems, in which charging of localised states by addition of integer or fractionally charged quasiparticles is observed. Treating the localised state as a compressible quantum dot or antidot embedded in an incompressible background, we calculate the electrostatic potential in its vicinity as a function of its charge, and the chemical potential values at which its charge changes. The results offer a quantitative framework for analysis of the observations.     

Entanglement distribution over 150 km in wavelength division multiplexed channels for quantum cryptography

Granting information privacy is of crucial importance in our society, notably in fiber communication networks. Quantum cryptography provides a unique means to establish, at remote locations, identical strings of genuine random bits, with a level of secrecy unattainable using classical resources. However, several constraints, such as non‐optimized photon number statistics and resources, detectors' noise, and optical losses, currently limit the performances in terms of both achievable secret key rates and distances. Here, these issues are addressed using an approach that combines both fundamental and off‐the‐shelves technological resources. High‐quality bipartite photonic entanglement is distributed over a 150 km fiber link, exploiting a wavelength demultiplexing strategy implemented at the end‐user locations. It is shown how coincidence rates scale linearly with the number of employed telecommunication channels, with values outperforming previous realizations by almost one order of magnitude. Thanks to its potential of scalability and compliance with device‐independent strategies, this system is ready for real quantum applications, notably entanglement‐based quantum cryptography.