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arXiv:1412.3239 · Reviews in Mathematical Physics

Structure of uniformly continuous quantum Markov semigroups

Julien Deschamps, Franco Fagnola, Emanuela Sasso, Veronica Umanità

Published Feb 1, 2016• 43 citations
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Abstract

The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterization of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.

DOI
10.1142/s0129055x16500033

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The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterization of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.
We establish the structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra and apply the result to derive a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. We deduce the structure of invariant states and a method for finding decoherence-free subsystems and subspaces. The relationship between environment induced decoherence and the decoherence-free subalgebra is also discussed.
In this thesis I analyze the structure of decoherence-free subalgebra N(T) of a uniformly continuous covariant Quantum Markov Semigroup (QMS) with respect to a representation ? of a compact group G on a Hilbert space h. In particular, I obtain that, when ? is irreducible, N(T) is isomorphic to (?(k) ? 1m)^d for suitable Hilbert spaces k and m, and an integer d related to the connected components of G. I extend this result when ? is reducible and N(T) is atomic by the decomposition of h due to the Peter?Weyl theorem. The thesis is then concluded by exploring the possibility of extending these results beyond the direct sum by using the concept of direct integral.
Abstract We demonstrate a method for finding the decoherence-free subalgebra N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra B(Γ(Cd)){\mathcal {B}}(\Gamma (\mathbb {C}^d))B(Γ(Cd)) B ( Γ ( C d ) ) of all bounded operator on the Fock space Γ(Cd)\Gamma (\mathbb {C}^d)Γ(Cd) Γ ( C d ) on Cd\mathbb {C}^dCd C d . We show that N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) is a type I von Neumann algebra L∞(Rdc;C)⊗‾B(Γ(Cdf))L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))L∞(Rdc​;C)⊗​B(Γ(Cdf​)) L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers dc,df≤dd_c,d_f\le ddc​,df​≤d d c , d f ≤ d . This result is illustrated by some applications and examples.