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arXiv:1911.03956

Iterates of quantum operations

József Zsolt Bernád

Published Nov 10, 2019• 0 citations
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Abstract

Iterates of quantum operations and their convergence are investigated in the context of mean ergodic theory. We discuss in detail the convergence of the iterates and show that the uniform ergodic theorem plays an essential role. Our results will follow from some general theorems concerning completely positive maps, mean ergodic operators, and operator algebras on Hilbert spaces. A few examples of both finite and infinite dimensional Hilbert spaces are presented as well.

DOI
10.48550/arxiv.1911.03956

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Iterates of quantum operations and their convergence are investigated in the context of mean ergodic theory. We discuss in detail the convergence of the iterates and show that the uniform ergodic theorem plays an essential role. Our results will follow from some general theorems concerning completely positive maps, mean ergodic operators, and operator algebras on Hilbert spaces. A few examples of both finite and infinite dimensional Hilbert spaces are presented as well.
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Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.
We develop a Perron-Frobenius type theory for products of random quantum channels acting on finite-dimensional matrix algebras sampled from a stationary and ergodic stochastic process, which, in keeping with the literature, we call ergodic quantum processes. This serves as a unifying framework for many models, including i.i.d., Markovian, periodic, and quasiperiodic models. We establish various characterizations of irreducibility, from which we recover a number of general ergodic theorems. We then analyze some specific examples, and, in particular, give a refinement of our theory in the i.i.d. case.