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The study of open quantum systems relies on the notion of unital completely positive semigroups on $C^*$-algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on operator systems. We show that any continuous unital completely positive semigroup on matricial system can be extended to a semigroup on a finite-dimensional $C^*$-algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique., Comment: 21 pages, 1 figure, typos corrected, journal reference added

The motion of a vertically positioned bicycle is considered when a horizontal control force, which maybe both internal and external in relation to the bicycle, is applied to its pedal. Tangential forces of dryfriction obeying the Euler { Coulomb law act at points of contact of the wheels with the horizontal supportplane. The constraint at the points of contact of the wheels with the support is assumed to be unilateral.The problem of determining the acceleration of the centre of mass of the bicycle and the realized motionsof its wheels (with or without slip, and with or without detachment) with dierent values of the design parameters and control force is solved. This approach is usefull for control of vehicle and optimization of its dynamical parameters. Cases of non-uniqueness of motion the Painleve paradox are found.

We study dissipative translationally invariant free-fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach, we solve the Lindblad equation and find the density matrix at all times for arbitrary time dependence of the Liouvillian. We then investigate the Liouvillian spectral properties and derive a generic criterion for the closure of the dissipative gap, which is believed to be linked with nonequilibrium dissipative phase transitions. We illustrate our findings with a few exotic examples. Particularly, we show the presence of gapless modes with a linear spectrum for fermions with long-range hopping, which might be related to nonunitary conformal field theories. The predicted effects can be probed in experiments with ultracold atomic and quantum-optical systems using currently available experimental facilities., Comment: 13 pages, 2 figures

In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs). We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism. Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator. We apply the MIC-POVM-based probability representation to the digital quantum computing model. In particular, for the case of spin-$1/2$ evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics. One of the most important findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach. Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum (NISQ) devices., Comment: 19 pages, 8 figures, 2 tables

We perform a quantum information analysis for multi-mode Gaussian approximate position measurements, underlying noisy homodyning in quantum optics. The "Gaussian maximizer" property is established for the entropy reduction of these measurements which provides explicit formulas for computations including their entanglement-assisted capacity. The case of one mode is discussed in detail., Comment: 18 pages, one figure

It was shown in arXiv:0906.2527, that in finite-dimensional Hilbert spaces each operator system corresponds to some channel, for which this operator system will be an operator graph. This work is devoted to finding necessary and sufficient conditions for this property to hold in infinite-dimensional case., Comment: 9 pages; author name corrected, typos corrected, journal reference added