Motion Caused by Magnetic Field in Lobachevsky Space

We study motion of a relativistic particle in the 3-dimensional Lobachevsky space in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motion are found and equations of motion are solved exactly in the special cylindrical coordinates. Motion on surface of the cylinder of constant radius is considered in detail.Comment: 4 page

Classical Particle in Presence of Magnetic Field, Hyperbolic Lobachevsky and Spherical Riemann Models

Motion of a classical particle in 3-dimensional Lobachevsky and Riemann spaces is studied in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in Euclidean space. In both cases three integrals of motions are constructed and equations of motion are solved exactly in the special cylindrical coordinates on the base of the method of separation of variables. In Lobachevsky space there exist trajectories of two types, finite and infinite in radial variable, in Riemann space all motions are finite and periodical. The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding 4-potential description is demonstrated explicitly. The role of the symmetry is clarified in classification of all possible solutions, based on the geometric symmetry group, SO(3,1) and SO(4) respectively

Synthesis of the Einstein-Podolsky-Rosen entanglement in a sequence of two single-mode squeezers

Synthesis of the Einstein-Podolsky-Rosen entangled state --- the primary entangled resource in continuous-variable quantum-optical information processing --- is a technological challenge of great importance. Here we propose and implement a new scheme of generating this state. Two nonlinear optical crystals, positioned back-to-back in the waist of a pump beam, function as single-pass degenerate optical parametric amplifiers and produce single-mode squeezed vacuum states in orthogonal polarization modes, but in the same spatiotemporal mode. A subsequent pair of waveplates acts as a beam splitter, entangling the two polarization modes to generate the Einstein-Podolsky-Rosen state. This technique takes advantage of the strong nonlinearity associated with type-I phase-matching configuration while at the same time eliminating the need for actively stabilizing the optical phase between the two squeezers, which typically arises if these squeezers are spatially separated. We demonstrate our method in an experiment, preparing a 1.4 dB two-mode squeezed state and characterizing it via two-mode homodyne tomography.Comment: 4 pages, 3 figure

Post-processing procedure for industrial quantum key distribution systems

We present algorithmic solutions aimed on post-processing for industrial quantum key distribution systems with hardware sifting. The main steps of the procedure are error correction, parameter estimation, and privacy amplification. Authentication of a classical public communication channel is also considered.Comment: 5 pages; presented at the 3rd International School and Conference "Saint-Petersburg OPEN 2016" (Saint-Petersburg, March 28-30, 2016

Undoing the effect of loss on quantum entanglement

Entanglement distillation is a process via which the strength and purity of quantum entanglement can be increased probabilistically. It is a key step in many quantum communication and computation protocols. In particular, entanglement distillation is a necessary component of the quantum repeater, a device which counters the degradation of entanglement that inevitably occurs due to losses in a communication line. Here we report an experiment on distilling the Einstein-Podolsky-Rosen (EPR) state of light, the workhorse of continuous-variable entanglement, using the technique of noiseless amplification. In contrast to previous implementations, the entanglement enhancement factor achievable by our technique is not fundamentally limited and permits recovering an EPR state with a macroscopic level of entanglement no matter how low the initial entanglement or how high the loss may be. In particular, we recover the original level of entanglement after one of the EPR modes has passed through a channel with a loss factor of 20. The level of entanglement in our distilled state is higher than that achievable by direct transmission of any state through a similar loss channel. This is a key bench-marking step towards the realization of a practical continuous-variable quantum repeater and other CV quantum protocols.Comment: 8 pages, 5 figure