eu ON EXPONENTIAL CONVERGENCE OF GENERIC QUANTUM MARKOV SEMIGROUPS IN A WASSERSTEIN-TYPE DISTANCE

We investigate about exponential convergence for generic quantum Markov semigroups using an generalization of the Lipschitz seminorm and a noncommutative analogue of Wasserstein distance. We show turns out to be closely related with classical convergence rate of reductions to diagonal subalgebras of the given generic quantum Markov semigroups.In particular we compute the convergence rates of generic quantum Markov semigroups. AMS Subject Classification: 81S22, 60J27


Introduction
We consider the von Neumann algebra B(h) of all linear bounded operators on a given complex separable Hilbert space h and a Quantum Markov semigroup (QMS) T = (T t ) t≥0 which acts on B(h), i.e., T is a weakly * -continuous semigroup of completely positive, preserving, normal maps on B(h).Quantum Markov semigroups (QMS) are a non-commutative extension of Markov semigroups defined in classical probability, they represent an evolution without memory of a microscopic system in accordance with the laws of quantum physics and fit into the framework of open quantum systems.The semigroup T corresponds to the Heisenberg picture in the sense that given any observable x, T t (x) describes its evolution at time t.In this way, given a density matrix ρ, its dynamics (Schrodinger picture) is given by the semigroup T * t (ρ), where tr(ρT t (x)) = tr(T * t (ρ)x).
Several aspects of temporal evolutions described by QMSs have been investigated.By example, in [4], [5], and [6], the exponential speed of convergence of the quantum Markov semigroup is studied using the quantum L 2 -spectral gap (gap(L)).In [3] a Wasserstein-type distance, denoted by W d , has been defined and applied to measure deviations from equilibrium, in other words, to define an entropy production index (see [16,17]).W d is a non commutative analogue of the classical Wasserstein distance w d used in optimal transport (see [12], [22], [23]).
In this paper we use a generalization of the Lipschitz seminorm and a noncommutative analogue of Wasserstein distance to study exponential convergence of generic QMSs.This research is motivated by the exploration of relation between exponential convergence of QMSs and his classical reductions given by classical Markov semigroups.The exponential convergence in the classical case is represented by a Wasserstein curvature (or Chen exponent) σ d linked with the classical Wasserstein distance (see [8], [9], [19] , [22]).Moreover, we show that in the generic QMSs case the exponent convergence is related with σ d and the parameters of QMS.
The paper is organized as follows.In Section 2 we recall the basic aspects about classical Wasserstein distance.We recall generalization of the Lipschitz seminorm and a noncommutative analogue of Wasserstein distance introduced in [3].After, some useful estimates on norms of commutators are showed in the Section 4. Finally, we apply these estimates.Specifically, we see in Section 5 that if T is a generic quantum Markov semigroup and Where µ n , λ n are coefficients generator of semigroup and σ d is a rate convergence of classical reduction of semigroup.

Exponential Convergence: Classical Case
We start this section by reviewing the Wasserstein distance and Wasserstein curvature for classical Markov processes.
Let (Ω, (F t ) t≥0 , F, P) be a filtered probability space, E a Polish space endowed with metric d, and A = L ∞ (E).Consider a E-valued cadlag Markov process {(X t ) t≥0 , (P x ) x∈E }, with (T t ) t≥0 associated Markov semigroup acting on A as follows The predual semigroup of (T t ) t≥0 acts on probability measures µ as We denote by P d (E) the space of probability measures ν on E such that E d(x, y)ν(dy) < +∞ for some (or equivalently for all) x ∈ E.
Moreover, we consider Lip d (E) the space of Lipschitz functions on E with a Lipschitz seminorm defined by Remark 1.Under the previous assumptions, if a Markov kernel P t (x, •) belongs to P d (E) for all t > 0 and for all x ∈ E then T t (f ) is well defined for all f ∈ Lip d (E).
We can therefore define Remark 2. Is easy to see that i.e, the supremum does not change if we restrict to self-adjoint elements.
Note that σ d (0) = 0.By the semigroup property of T t , it follows that the function σ d (t) is super-additive so that the following limit is well defined: Moreover, the number σ d is the best (maximal) constant ∆ in the contraction inequality Definition 1.The number σ d given by ( 1) is called Wasserstein curvature of the process (X t ) t≥0 with respect to metric d.This notion of curvature was introduced by Joulin [19], [20] and Ollivier [21] and is connected to the notion of Ricci curvature on Riemannian manifolds [24].In this remainder of this section, we will assume implicitly that the Markov kernel P t (x, •) belongs to the space P d (E) for all t > 0, x ∈ E.
The coefficient σ d is linked with the classical Wasserstein distance.
Remark 3. The classical Wasserstein distance is defined by where (M, d) is a metric space and Ξ(µ, ν) is the set of all Borel probability measures ϑ on M × M such that for all measurable subsets A, B ⊆ M When M is a separable space and µ, ν ∈ P d (M ) the Kantorovich-Rubinstein theorem provides another representation for the Wasserstein metric: (for a proof of the Kantorovich-Rubinstein theorem see for example [12], Theorem 11.8.2 p.421).
By remark 3, the Wasserstein curvature σ d is the best (maximal) constant ∆ in the inequality where µ 1 and µ 2 are σ-finite measures.Then σ d is the best (maximal) constant ∆ holding simultaneously the inequalities ( 2) and (3).

An Non Commutative Extension of the Lipschitz Seminorm and a Wasserstein-Type Distance
We start our discussion about a non commutative extension of the Lipschitz seminorm and a Wasserstein-type distance.In the quantum case, we consider h complex separable Hilbert space with orthonormal basis fixed (e k ) k∈V (V is a finite or countable set).
In [3], we proposed a quantum version of w d , we recall the definition.
Definition 2. The quantum Wasserstein distance between two states is defined by: and d a distance defined on the set V .
Note that the usual deriviation δ ml (a) = [(|e m e l | + |e l e m |), a] satisfies δ ml = d(m, l)δ d ml .We collect here some preliminary results on the quantum Wasserstein distance that we need in the sequel.
Let θ be the phase of the complex number tr ((σ 1 − σ 2 )a) so that The operator y = (e −iθ a+e iθ a * )/2 is clearly self-adjoint and has Lipschitz norm smaller than 1, indeed sup m,l Moreover, by (4), This completes the proof.
We call diagonal algebra, and denote it by D the Abelian algebra generated by one-dimensional projections |e k e k |.Let E : B(h) → D be the conditional expectation with range D defined by and let E * be the predual map on trace class operators with range l 1 (V ) Since the norm of an anti self-adjoint matrix is the largest eigenvalue, computing the norm of the above 2 × 2 matrix (thought of as an operator on the linear span of e n , e m ) we find so, by item (a), we see that δ d mn (E(x)) ≤ p nm δ d mn (x)p nm ,then Let σ 1 , σ 2 be states on B(h) then E * (σ 1 ), E * (σ 2 ) are diagonal states with respect to (e j ) j∈V (i.e.measures probabilities on V ), then for all pairs (σ 1 , σ 2 ) of states.Then the restriction of W d to the diagonal subalgebra of B(h) coincides with the classical Wasserstein distance w d .

Estimates of Lipschitz Seminorm
In this section we prove some useful estimates on the norms of commutators δ mn (x).These estimates turn out to be useful for computing the exponential convergence rate of a generic quantum Markov semigroup.We begin by some simple lemma.
Lemma 5. Let e, f be two unit vectors in h and a, b ∈ h.Then In particular x is a rank-two self-adjoint operator that can be represented by the 2 × 2 matrix Recalling that x 2 = x * x and computing the biggest eigenvalue we obtain the squared norm of x.The last inequalities immediately follow from the Schwarz inequality | a, b | ≤ a • b .
The previous Lemma will be used to deduce bounds of δ nm (x) .
Proposition 6.For all n = m and all x ∈ B(h) we have Proof.Note that the above series converge because x is a bounded operator.Computing where x n • and x m • (resp.x • n and x • m ) denote the n and m row (resp.column) vector of x.Keeping into account cancellations for i, j = n, m we find then where The proof is completed writing explicitly the norms of the four vectors φ n , ψ m , ξ n , η m .
Proposition 7.For all n = m and all x ∈ B(h) we have Let p mn be the orthogonal projection onto the subspace generated by e n and e m Clearly, for all unit vector u ∈ h we have Note that vectors ξ n , η m in ( 8) are orthogonal to e n , e m and so the right-hand side is equal to Maximizing the right-hand side on the unit sphere in h we find and the claimed inequality follows computing the norms of φ n and ψ m For a self-adjoint x we can also find an upper bound for the norm of δ nm (x) as a multiple of the right hand side of (9).Theorem 8.For all self-adjoint ∈ B(h) and all n, m we have where It suffices to apply Propositions 6 and 7 noting that, for a self-adjoint operator x j =m,n

Lipschitz Seminorm and Generic QMSs
Generic QMS arise in the stochastic limit of a open discrete quantum system with generic Hamiltonian, interacting with Gaussian fields through a dipole type interaction (see Refs. [1], [2] and [6]).The generator is given by and operators G, L kj given by We denote by D, and call it the diagonal subalgebra, the Abelian subalgebra of B(h) of operators x such that e j , xe k = 0 for all k = j ∈ V and D of f the operator space of off-diagonal operators namely the closed (in the norm, strong and weak* topologies) subspace of x ∈ B(h) such that e k , xe k = 0 for all k ∈ V .Finally, we also denote by (P t ) t≥0 the strongly continuous contraction semigroup on B(h) generated by G (see (10) and Theorem 3.1 of [6]).The diagonal algebra D is clearly isometrically isomorphic to the Banach space l ∞ (V ).Identifying D with l ∞ (V ) and taking the restrictions of T t to D we find a weakly-* continuous classical sub-Markov semigroup T = (T t ) t≥0 on l ∞ (V ).Its generator L is characterized (see [13] Lemma 2.19) by A straightforward computation shows that the operator L satisfies L jk = γ − jk , for all j, k with κ k < κ j , The following properties are important in this section.
Theorem 9. Let (T t ) t≥0 be a generic QMS then: (a) The Abelian subalgebra D and the operator space D of f are T t -invariant for all t ≥ 0. Moreover T t (x) = P * t xP t for all x ∈ D of f .
(b) The spectral gap of a generic generator L is always equal to the spectral gap of the corresponding diagonal restriction L.
For a proof of the previous theorem, see Theorems 3 and 15 of [6].
Lemma 10.For all selfadjoint x ∈ D of f and all t ≥ 0 we have and Proof.Clearly In other words, the action of P * t • P t on matrix elements x jk of x corresponds to multiplication by a scalar.As a consequence, by Proposition 6, for all n = m, δ mn (P * t xP t ) 2 is smaller than

Unfortunately
x nm e it(κn−κm) − x mn e −it(κn−κm) 2 is not dominated by any multiple of |x nm − x mn | 2 (this happens, for instance, when x nm = x mn ∈ R), therefore we bring into action another derivation δ nm ′ where, for instance m Since x is self-adjoint, by Proposition 7, we have x nm e it(κn−κm) − x mn e −it(κn−κm) 2 = 2 ℑ(x nm e it(κn−κm) ) and the max{•, •} term is dominated by δ nm (x) 2 .The estimate of the norm δ nm (P * t xP t ) now follows from the elementary inequality (r + s) 1/2 ≤ r 1/2 + s 1/2 for all r, s ≥ 0.
Lemma 11.For all selfadjoint x ∈ D of f and all t ≥ 0 we have where c is given by (11) and Proof.Note that, for all n = m, we have The conclusion is now immediate.

Proposition 4 .
For all x ∈ B(h) it follows that (a) E(x) LIP d = sup m,l∈V,m =l 1 d(m,l) |x(l) − x(m)|.(b) E(x) LIP d ≤ x LIP d Proof.(a) If x ∈ B(h) then E(x) = s∈V x(s)|e s e s | where x(s) ∈ C and the convergence of the sum is in the weak* topology, then d(n, m)δ d mn (E(x)) = (x(l) − x(m))|e l e m | − (x(l) − x(m))|e m e l |.Since the norm of an anti self-adjoint matrix is the largest eigenvalue, computing the norm of the above 2 × 2 matrix (thought as an operator on the linear span of e l , e m ) we find E(x) LIP d = sup m,l∈V,m =l δ d ml (E(x)) = sup m,l∈V,m =l 1 d(m, l) (x(l) − x(m))|e l e m | − (x(l) − x(m))|e m e l | = sup m,l∈V,m =l 1 d(m, l) |x(l) − x(m)| 2 |e m e m | + |x(l) − x(m)| 2 |e l e l | = sup m,l∈V,m =l 1 d(m, l) |x(l) − x(m)| = x(•) LIP d .(b) First notice that if n = m and x = ij x ij |e i e j | then d(n, m)δ d mn (x) = (x nm − x mn )|e m e m | + (x mn − x nm )|e n e n | + (x nn − x mm )|e m e n | + j =n j =m x nj |e m e j | + (x mm − x nn )|e n e m | + j =n j =m x mj |e n e j | − i =n i =m x im |e i e n | − i =m i =n x in |e i e m |.Let p nm be a projection defined by p nm = |e n e n | + |e m e m | so d(n, m)p nm δ d mn (x)p nm = (x nm − x mn )|e m e m | + (x mn − x nm )|e n e n | + (x nn − x mm )|e m e n | + (x mm − x nn )|e n e m | .

Remark 4 .
A straightforward application of Theorem 8 shows that our Wasserstein norm is equivalent to the Hilbert-Schmidt norm for a finite V (with card(V ) ≥ 2) and, more generally, for a set V with a distance d such that inf m,l∈V,m =l d(m, l) > 0 and sup m,l∈V d(m, l) < ∞.

Remark 5 .
It is worth noticing here that if the set V = N and the distance is d(n, m) = |n − m|, then r = 1.Given the structure of generic QMS T is clear that T restricted to D defines a classical semigroup T t satisfying an inequality with classical Wasserstein curvature σ d Moreover: Proposition 12. Let T be a generic QMS and let k := min n =m µ n + λ n + µ m + λ m 2 ∧ σ d .e −tk x LIP d for all t > 0 and for all x ∈ B(h), x selfadjoint.Proof.Let E : B(h) → D, where D is the diagonal subalgebra and E ⊥ := I − E : B(h) → D of f then, by Lemma 11 and propostion 4, we obtain that T t (x) LIP d ≤ T t (E(x)) LIP d + T t (E ⊥ (x)) LIP d ≤ e −σ d t E(x) LIP d + P * t E ⊥ (x)P t LIP d ≤ e −σ d t x LIP d + 2 √ 2(2 + r)e −ct E ⊥ x LIP d ≤ e −σ d t x LIP d + 2 √ 2(2 + r)e −ct ( E(x) LIP d + x LIP d ) −(σ d ∧c)t x LIP d with c = 1 2 min{λ n + µ n + λ m + µ m |n = m}.−tk we obtaint that W d (T * t (ρ 1 ), T * t (ρ 2 )) n =m µ n + λ n + µ m + λ m 2 ∧ σ d .x LIP d ≤1 tr(ρ 1 − ρ 2 )T t (x) = (4 √ 2 + 1 + 2 √ 2 r)e −tk W d (ρ 1 , ρ 2 ),