The Decoherence-Free Subalgebra of Gaussian Quantum Markov Semigroups

Abstract

AbstractWe 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.