Generalization of Sato equation and systems of multidimensional nonlinear Partial Differential Equations

This paper develops one of the methods for study of nonlinear Partial Differential equations. We generalize Sato equation and represent the algorithm for construction of some classes of nonlinear Partial Differential Equations (PDE) together with solutions parameterized by the set of arbitrary functions.Comment: 10 page

On integration of multidimensional generalizations of classical CC- and SS-integrable nonlinear partial differential equations

We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of (1+1)-dimensional $C$-integrable and (2+1)-dimensional $S$-integrable $N$-wave equations are derived among examples. Examples of multidimensional second order PDEs are represented as well.Comment: 31 page

Partial structural restoring of two-qubit transferred state

We consider the communication line with two-qubit sender and receiver, the later is embedded into the four-qubit extended receiver. Using the optimizing unitary transformation on the extended receiver we restore the structure of the non-diagonal part of an arbitrary initial sender's state at the remote receiver at certain time instant. Obstacles for restoring the diagonal part are discussed. We represent examples of such structural restoring in a communication line of 42 spin-1/2 particles.Comment: 17 pages, 2 figure

Informational correlation between two parties of a quantum system: short spin-1/2 chains with XY Hamiltonian

We introduce the informational correlation $E^{AB}$ between two interacting quantum subsystems $A$ and $B$ of a quantum system as the number of arbitrary parameters $\varphi_i$ of a unitary transformation $U^A$ (locally performed on the subsystem $A$) which may be detected in the subsystem $B$ by the local measurements. This quantity indicates whether the state of the subsystem $B$ may be effected by means of the unitary transformation applied to the subsystem $A$. Emphasize that $E^{AB}\neq E^{BA}$ in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem $B$. However, there is some non-reducible part of $E^{AB}(t)$ which may not be decreased by any unitary transformation of the subsystem $B$ at a fixed time instant $t$. Two examples of the informational correlations between two parties of the four node spin-1/2 chain are studied.Comment: 45 pages, 2 figure

A variant of the Dressing Method applied to nonintegrable multidimensional nonlinear Partial Differential Equations

We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single linear Partial Differential Equation (PDE) with potentials expressed through the field of the nonlinear PDE. The absence of linear overdetermined system associated with nonlinear PDE creates an obstacle to obtain evolution of the spectral data (or dressing functions): evolution is defined by nonlinear IDE (or PDE in particular case). As an example, we consider generalization of the dressing method applicable to integrable (2+1)-dimensional $N$-wave and Davey-Stewartson equations. Although represented algorithm does not supply an analytic particular solutions, this approach may have a perspective development.Comment: 14 page

Construction of particular solutions to nonlinear equations of Mathematical Physics by using matrix algebraic equation

The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between algebraic matrix equations and PDE. Example of (2+2)-dimensional generalization of Burgers equation is given.Comment: 17 page

The minimal entanglement of bipartite decompositions as a witness of strong entanglement in a quantum system

We {characterize the multipartite entanglement in a quantum system by the quantity} which vanishes if only the quantum system may be decomposed into two weakly entangled subsystems, unlike measures of multipartite entanglement introduced before. We refer to this {quantity} as the minimal entanglement of bipartite decompositions (MEBD). Big MEBD means that the system may not be decomposed into two weakly entangled subsystems. MEBD allows one to define, for instance, whether the given quantum system may be a candidate for a quantum register, where the above decomposition is undesirable. A method of lower estimation of MEBD is represented. Examples of big MEBD in spin-1/2 chains governed by the $H_{dz}$ Hamiltonian in the strong external magnetic field are given.Comment: 11 pages, 1 figur

Partially integrable generalizations of classical integrable models by combination of characteristics method and Hopf-Cole transformation

We represent an integration algorithm combining the characteristics method and Hopf-Cole transformation. This algorithm allows one to partially integrate a large class of multidimensional systems of nonlinear Partial Differential Equations (PDEs). A specific generalization of the equation describing the dynamics of two-dimensional viscous fluid and a generalization of the Korteweg-de Vries equation are examples of such systems. The richness of available solution space for derived nonlinear PDEs is discussed.Comment: 7 page

Remote creation of a one-qubit mixed state through a short homogeneous spin-1/2 chain

We consider a method of remote mixed state creation of a one-qubit subsystem (receiver) in a spin-1/2 chain governed by the nearest-neighbor $XY$-Hamiltonian. Owing to the evolution of the chain along with the variable local unitary transformation of the one- or two-qubit sender, a large variety of receiver states can be created during some time interval starting with a fixed initial state of the whole quantum system. These states form the creatable region of the receiver's state-space. It is remarkable that, having the two-qubit sender, a large creatable region may be covered at a properly fixed time instant $t_0$ using just the variable local unitary transformation of the sender. In this case we have completely local control of the remote state creation. In general, for a given initial state, there are such receiver's states that may not be created using the above tool. These states form the unavailable region. In turn, this unavailable region might be the creatable region of another sender. Thus, in future, we have a way to share the whole receiver's state-space among the creatable regions of several senders. The effectiveness of remote state creation is characterized by the density function of the creatable region.Comment: 30 pages, 5 figure

Integration of nonlinear Partial Differential Equations by using matrix algebraic systems

The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE. Admittable solutions involve arbitrary functions of either single or several variables.Comment: 13 page