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arXiv:2112.13781 · Milan Journal of Mathematics

The Decoherence-Free Subalgebra of Gaussian Quantum Markov Semigroups

Julián Agredo, Franco Fagnola, Damiano Poletti

Published Jun 1, 2022• 13 citations

Abstract

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.

DOI: 10.1007/s00032-022-00355-0

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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.

Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

We construct a quantum Markov semigroup on the von Neumann algebra B ( L 2 (ℝ + ;ℂ)) of all bounded operators on a Hilbert space L 2 (ℝ + ;ℂ) with the following property: the biggest *-subalgebra of B ( L 2 (ℝ + ;ℂ)) contained in the domain of the infinitesimal generator is not σ-weakly dense. Our semigroup is an extension of a classical Markov semigroup on L ∞ (ℝ + ;ℂ) with the same property. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.