On the representation of linear evolution operators

Yingyu Zhang, Yixing Wang

Abstract Based on the Kraus operator-sum representation of the analytical solution of the diffusion equation, we obtain the evolution of a general linear state in the diffusion channel. Also, we study the quantum statistical properties of the initial general linear state and its von-Neumann entropy evolution in the diffusion channel, especially find that the entropy evolution is influenced by the diffusion noise and the thermal parameter but without the displacement.

T. Yu. Mikhaîlova, Vladimir I. Pupyshev

Mihail Vladimirovich Karasev

Sergey B. Bravyi

Notions of a Gaussian state and a Gaussian linear map are generalized to the case of anticommuting (Grassmann) variables. Conditions under which a Gaussian map is trace preserving and (or) completely positive are formulated. For any Gaussian map an explicit formula relating correlation matrices of input and output states is presented. This formalism allows to develop the Lagrangian representation for fermionic linear optics (FLO). It covers both unitary operations and the single-mode projectors associated with FLO measurements. Using the Lagrangian representation we reduce a classical simulation of FLO to a computation of Gaussian integrals over Grassmann variables. Explicit formulas describing evolution of a quantum state under FLO operations are put forward.

Il'dar Abdullovich Ibragimov, Nataliya V. Smorodina, Михаил Михайлович Фаддеев

Octavio Castaños, Ram n L pez-Pe a, Margarita A Man ko +1 more

The probability representation of angular momentum states and the connection of the representation with the formalism of the star-product quantization procedure are reviewed. The Schrödinger equation of a general time-dependent Hamiltonian, linear in angular momentum operators, and the evolution equation for the density operator in the probability representation are solved analytically by means of the formalism of linear time-dependent constants of motion. These analytical solutions define wavefunctions and tomograms of states (generic Dicke states), which contain the atomic coherent states as a particular case. The statistical properties of these new states are also evaluated. General forms of analytically solvable Hamiltonians are established in terms of the Euler angle parametrization of the three-dimensional rotations.