The Decoherence-Free Subalgebra of Gaussian Quantum Markov Semigroups

We demonstrate a method for finding the decoherence-free subalgebra \({\mathcal {N}}({\mathcal {T}})\) of a Gaussian quantum Markov semigroup on the von Neumann algebra \({\mathcal {B}}(\Gamma (\mathbb {C}^d))\) of all bounded operator on the Fock space \(\Gamma (\mathbb {C}^d)\) on \(\mathbb {C}^d\). We show that \({\mathcal {N}}({\mathcal {T}})\) is a type I von Neumann algebra \(L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))\) determined, up to unitary equivalence, by two natural numbers \(d_c,d_f\le d\). This result is illustrated by some applications and examples.