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تسجيل مستخدم جديدRepeatable procedures and maps in open quantum dynamics
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Examples of repeatable procedures and maps are found in the open quantum dynamics of one qubit that interacts with another qubit. They show that a mathematical map that is repeatable can be made by a physical procedure that is not.
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Two kinds of maps that describe evolution of states of a subsystem coming from dynamics described by a unitary operator for a larger system, maps defined for fixed mean values and maps defined for fixed correlations, are found to be quite different f
or the same unitary dynamics in the same situation in the larger system. An affine form is used for both kinds of maps to find necessary and sufficient conditions for inverse maps. All the different maps with the same homogeneous part in their affine forms have inverses if and only if the homogeneous part does. Some of these maps are completely positive; others are not, but the homogeneous part is always completely positive. The conditions for an inverse are the same for maps that are not completely positive as for maps that are. For maps defined for fixed mean values, the homogeneous part depends only on the unitary operator for the dynamics of the larger system, not on any state or mean values or correlations. Necessary and sufficient conditions for an inverse are stated several different ways: in terms of the maps of matrices, basis matrices, density matrices, or mean values. The inverse maps are generally not tied to the dynamics the way the maps forward are. A trace-preserving completely positive map that is unital can not have an inverse that is obtained from any dynamics described by any unitary operator for any states of a larger system.
Simple examples are used to introduce and examine symmetries of open quantum dynamics that can be described by unitary operators. For the Hamiltonian dynamics of an entire closed system, the symmetry takes the expected form which, when the Hamiltonia
n has a lower bound, says that the unitary symmetry operator commutes with the Hamiltonian operator. There are many more symmetries that are only for the open dynamics of a subsystem. Examples show how these symmetries alone can reveal properties of the dynamics and reduce what needs to be done to work out the dynamics. A symmetry of the open dynamics of a subsystem can even imply properties of the dynamics for the entire system that are not implied by the symmetries of the dynamics of the entire system. The symmetries are generally not related to constants of the motion for the open dynamics of the subsystem. There are many symmetries that cannot be seen in the Schrodinger picture as symmetries of dynamical maps of density matrices for the subsystem. There are symmetries of the open dynamics of a subsystem that depend only on the dynamics. In the simplest examples, these are also symmetries of the dynamics of the entire system. There are many more symmetries, of a new kind, that also depend on correlations, or absence of correlations, between the subsystem and the rest of the entire system, or on the state of the rest of the entire system.
We consider the description of quantum noise within the framework of the standard Copenhagen interpretation of quantum mechanics applied to a composite system environment setting. Averaging over the environmental degrees of freedom leads to a stochas
tic quantum dynamics, described by equations complying with the constraints arising from the statistical structure of quantum mechanics. Simple examples are considered in the framework of open system dynamics described within a master equation approach, pointing in particular to the appearance of the phenomenon of decoherence and to the relevance of quantum correlation functions of the environment in the determination of the action of quantum noise.
We extend the concept of superadiabatic dynamics, or transitionless quantum driving, to quantum open systems whose evolution is governed by a master equation in the Lindblad form. We provide the general framework needed to determine the control strat
egy required to achieve superadiabaticity. We apply our formalism to two examples consisting of a two-level system coupled to environments with time-dependent bath operators.
Dependent symmetries, symmetries that depend on the situation of the subsystem in a larger closed system, are explored by looking at simple examples. This is a new kind of symmetry in the open quantum dynamics of a subsystem Each symmetry implies a
particular form for the results of the open dynamics. The forms exhibit the symmetries very simply. It is shown directly, without assuming anything about the symmetry, that the dynamics produces the form, but knowing the symmetry and the form it implies can reduce what needs to be done to work out the dynamics; pieces can be deduced from the symmetry rather that calculated from the dynamics. Symmetries can be related to constants of the motion in new ways. A quantity might be a dependent constant of the motion, constant only for particular situations of the subsystem in the larger system. In particular, a generator of dependent symmetries could represent a quantity that is a dependent constant of the motion for the same situations as for the symmetries. The examples present a variety of possibilities. Sometimes a generator of dependent symmetries does represent a dependent constant of the motion. Sometimes it does not. Sometimes no quantity is a dependent constant of the motion. Sometimes every quantity is.
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