Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups

In the celebrated papers [23,24], Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.

Gross discovered the equivalence of log-Sobolev inequalities and hypercontractivity, and also gave a new proof of the hypercontractivity for the classical Ornstein-Uhlenbeck semigroup in the remarkable work [17]. This surprising observation led to the application of hypercontractivity in numerous fields, such as Boolean analysis, concentration inequalities, geometric inequalities, statistical physics, among others. We refer to [18,2] for more information.

Quantum Markov semigroups, which are a generalization of classical Markov semigroups, are a fundamental tool to describe open quantum systems. In [13], Cipriani et al. rigorously proved that a specific unbounded Lindblad-type operator generates a quantum Markov semigroup by virtue of the remarkable quantum semigroup theory established in [11,12]. Such unbounded Lindblad-type operator has been extensively studied in quantum optics models of masers and lasers [14,15]. The quantum Markov semigroup appearing in [13], which is now called quantum Ornstein-Uhlenbeck semigroup, has also been shown to be hypercontractive in the seminal work [8].

The need to construct quantum Markov semigroups on von Neumann algebras, which are symmetric with respect to a nontracial state, is clear for various applications to open quantum systems, quantum statistical mechanics and quantum information (see [19] for more information). In the remarkable paper [19], Ko et al. constructed a family of quantum Ornstein-Uhlenbeck semigroups, which can be seen as a generalization of the quantum Ornstein-Uhlenbeck semigroup appearing in [13,14,15].

In this paper, we aim to study hypercontractivities of the quantum Ornstein-Uhlenbeck semigroups established in [19]. At first, we introduce such quantum Ornstein-Uhlenbeck semigroups. To this end, we present Weyl operators and CCR.

Let H " ℓ 2 pNq, where N " t0, 1, 2, ¨¨¨u is the set of all natural numbers. Let te n u 8 n"0 be the canonical orthonormal basis of H. Then we define the creation and annihilation operators a and a as follows a ˚en " ? n 1 e n1 , ae n " ? n e n´1 , pe ´1 :" 0q. It is clear that they satisfy the canonical commutation relation (CCR) ra, a ˚s " 1, where 1 denotes the identity operator by a slight abuse of notation. The number operator N :" a ˚a acts as N e n " n ¨en . The position and momentum operators are defined as Q " 1 ? 2 pa a˚q , P " 1 ? 2i pa ´a˚q , and so rQ, P s " i ¨1. For any z P C, define the Weyl operators W pzq " e i ? 2 pza ˚zaq " e ipℜz¨Qℑz¨P q . These operators satisfy the Weyl relation W pzq ˚" W p´zq, and @z, w P C, W pzqW pwq " e ´i 2 ℑpzwq W pz wq.

Let A be the C ˚-algebra generated by all Weyl operators. It is well-known that A ⫋ BpHq and A 2 " BpHq. Let ω be the normal faithful state on BpHq, which is defined by

where β ą 0 is a fixed inverse temperature. Note that e ´βN is a trace class operator in BpHq.

ρ " e ´βN Trpe ´βN q " p1 ´e´β qe ´βN .

Indeed, ω is a Gibbs state, and for any x P BpHq, ωpxq " Trpρxq.

We refer the interested reader to [7] for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in [19]. Let the parameters α 1 , α 2 , α 3 P R satisfy the following relations

Let G be the elliptic operator on BpHq given by

3 ˘rQ, rQ, Ass ´iγα 1 pQrP, As rP, AsQq ´iγα 2 α 3 pP rQ, As rQ, AsP q, @ A P BpHq,

˘´1 is the normalized constant. Since P and Q are unbounded operators affiliated to M, the above is a formal expression.

For 0 ă p ď 8, let S p be the Schatten p-class of BpHq with the usual Schatten p-norm denoted by } ¨}p . We use Kosaki’s definition [21] for noncommutative L p spaces. Let L p pρq be the closure of all elements in ρ 1{2 BpHqρ 1{2 with respect to the following norm

Define G p2q on S 2 by G p2q pρ 1{4 xρ 1{4 q " ρ 1{4 Gpxqρ 1{4 , @ x P BpHq.

According to [19], P t " e ´tG p2q (@ t ě 0) is a symmetric semigroup on S 2 . For any x P BpHq, define ρ 1{4 T p8q t pxqρ 1{4 " P t pρ 1{4 xρ 1{4 q.

Then T p8q t is an ergodic quantum Markov semigroup on S 8 , and ω is the unique invariant state associated with T p8q t

. See [19] for more details. For 1 ď p ă 8, denote by T ppq t : L p pρq Ñ L p pρq the usual restriction of T p8q t to L p pρq, by using the same way as in [16]. More explicitly, for p " 2, if x P BpHq, then

Denote by τ the spectral gap of T p2q t , which is defined in (2.1). We present the definition of hypercontractivity. Let pT t q tě0 be an ergodic quantum Markov semigroup with the invariant state induced by ρ. Then pT t q tě0 is said to be hypercontractive if for any 2 ă p ă 8, there exists

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