ExponentialL 2-convergence of quantum Markov semigroups on $$\mathcal{B}(h)$$

Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

“…Yet, we notice that this fact alone is not sufficient and, further, the involved constants get worse when considering the quantum case. This is quite natural and also happens for uniform exponential convergence and spectral gap (see once again [9]). …”

“…1: L ∞ = B(h) and, for p ∈ [1, ∞), L p (h) as the closure of the algebra B(h) with respect to the norm · p = · p,ρ . We will also have π = ρ and E(x) = − x, L(x) ρ , for any x in the domain of L. Then we know that T is a completely positive semigroup which is contractive with respect to any L p norm (see [9]); moreover, it has spectral gap η := µ 2 −λ 2 2 (see [9,10]). As we previously told, we can consider the restriction of L to the commutative algebra of diagonal operators; this restriction can be seen as an operator G acting on l ∞ (N).…”

“…They admit explicit representation formulae that are natural non-commutative extensions of wellknown classical formulae. Moreover, they enjoy deepest properties, like exponential ergodicity or hypercontractivity (see [4,6,7,9,26]) which are essentially equivalent to the positivity of the spectral gap and to a non-commutative version of logarithmic Sobolev inequalities. They are also relevant from a physical point of view because, if the semigroup is interpreted as a quantum channel, a relationship between the contraction rate and the entropy production can be found [27].…”

“…The operator L is a Lindblad operator and the minimal associated semigroup T (whose existence is guaranteed by Davies' theorem [13]) is conservative and has a unique diagonal invariant state (see [9,17] and references therein) ρ = (1 − ν) n ν n |e n e n |, where ν = λ 2 /µ 2 (we use the Dirac notation: for u and v in h, |u v| denotes the operator on h defined by |u v|(w) = v, w u for any w in h). For x and y in B(h), we will use the scalar product associated with the invariant state ρ, x, y ρ = tr(ρ 1/2 x * ρ 1/2 y); we will also denote by x p = (tr|ρ 1/(2 p) xρ 1/(2 p) | p ) 1/ p , p ≥ 1, the usual L p norms associated with ρ.…”

“…A complete study of the spectral properties of the qOU generator is made in [10] and some other properties, like L p -contractivity and uniform exponential convergence, are studied in [9].…”