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makarov k mathematics open quantum systems 2022

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Textbook in PDF format This book presents new developments in the open quantum systems theory with emphasis on applications to the (frequent) measurement theory.In the first part of the book, the uniqueness theorems for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions are discussed. The major theme involves an intrinsic characterization of the simplest symmetric operator solutions to the Heisenberg uncertainty relations, the problem posed by Jorgensen and Muhly, followed by the proof of the uniqueness theorems for the simplest solutions to the restricted Weyl commutation relations. The detailed study of unitary invariants of the corresponding dissipative and symmetric operators opens up a look at the classical Stone-von Neumann uniqueness theorem from a new angle and provides an extended version of the uniqueness result relating various realizations of a differentiation operator on the corresponding metric graphs. The second part of the book is devoted to mathematical problems of the quantum measurements under continuous monitoring. Among the topics discussed are the complementarity of the Quantum Zeno effect and Exponential Decay scenario in frequent quantum measurements, and a rigorous treatment, within continuous monitoring paradigm, of the celebrated 'double-slit experiment' where the renowned exclusive and interference measurement alternatives approach in quantum theory is presented in a way that is accessible for mathematicians. One of the striking applications of the generalized (1-stable) central limit theorem is the mathematical evidence of exponential decay of unstable states of the quantum pendulum under continuous monitoring. The main goal of this monograph is to develop a mathematical framework that accommodates an adequate description of the results of continuous observation of quantum systems. It is well known that a quantum observation/measurement always affects the system subject to it, and therefore the system can no longer be considered completely isolated. Instead, it should be treated as part of a more general system in which the presence of an observer/measuring device is taken into account. As such, the initial system should be regarded as an open quantum system that interacts with a part of the larger system. In the abstract setting, dealing with open systems assumes the presence of communication channels through which the interaction is carried out, both between parts of the system in question and with the outside world. A special case of open systems is the class of dissipative systems, the dynamics of which are governed by a strongly continuous semigroup of contractions. By applying the canonical dilation procedure, such open systems can always be viewed as a part of a larger closed system: the resulting Hamiltonian of the dilated system is chosen to be the self-adjoint dilatation of the generator of the semigroup, while the space of (pure) states of the larger system can be identified with the extended Hilbert space where the dilated operator has been realized as a self-adjoint operator. Despite its mathematical attractiveness, this dilation method is incompatible with the physical requirement that the total energy of the obtained large isolated system must be bounded from below. The transition in the opposite direction is usually associated with the reduction of the unitary evolution onto a subspace, in most cases lacks the semigroup property and needs a special consideration. The situation changes dramatically if the reduced description of the evolution is accompanied by continuous monitoring of the system. Under certain circumstances, the exponential decay of the states under continuous monitoring can be justified even if the energy distribution of the state is semi-bounded from below and the spectrum of the system is discrete. Therefore, within the continuous monitoring paradigm, one can bypass applying the Weisskopf-Wigner method that can only give an approximate description of the decay processes only. In the same time, the exponential decay under the continuous monitoring scenario fills in the gap between the quantum Zeno effect (Turing’s paradox) and the anti-Zeno phenomenon, and what is also important is, it gives a fresh look at the descent of dissipative operators and opens up new perspectives for their applications in quantum theory

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makarov k mathematics open quantum systems 2022