Decoherence of Quantum Markov Systems

Zhang, Tian Li

Published Aug 1, 2013• 0 citations

Abstract

The main purpose of this thesis is to discuss quantum decoherence in the frame- work of operator algebras. We will define several types of decoherence of quantum Markov systems based on the original notion introduced by Ph. Blanchard and R. Olkiewicz; then we will give a number of suffiecient conditions for a quantum Markov system to display decoherence, and explain the relation between them.

DOI: 10.82465/4920

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Decoherence of Quantum Markov Systems

Zhang, Tian Li

Published Aug 1, 2013• 0 citations

Publisher

Abstract

DOI: 10.82465/4920

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Abstract We demonstrate a method for finding the decoherence-free subalgebra

{\mathcal {N}}({\mathcal {T}})

N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra

{\mathcal {B}}(\Gamma (\mathbb {C}^d))

B ( Γ ( C d ) ) of all bounded operator on the Fock space

\Gamma (\mathbb {C}^d)

Γ ( C d ) on

\mathbb {C}^d

C d . We show that

{\mathcal {N}}({\mathcal {T}})

N ( T ) is a type I von Neumann algebra

L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))

L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers

d_c,d_f\le d

d c , d f ≤ d . This result is illustrated by some applications and examples.

This paper consists of

3

parts. The first part only considers classical processes and introduces two different extensions of the notion of hidden Markov process. In the second part, the notion of quantum hidden process is introduced. In the third part it is proven that, by restricting various types of quantum Markov chains to appropriate commutative sub-algebras (diagonal sub-algebras) one recovers all the classical hidden process and, in addition, one obtains families of processes which are not usual hidden Markov process, but are included in the above mentioned extensions of these processes. In this paper we only deal with processes with an at most countable state space.