Complete Gradient Estimates of Quantum Markov Semigroups

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.


Introduction
In the last decades, the theory of optimal transport has made impressive inroads into other disciplines of mathematics, probably most notably with the Lott-Sturm-Villani theory [29,42,43] of synthetic Ricci curvature bounds for metric measure spaces. More recently, optimal transport techniques have also been used to extend this theory to cover also discrete [13,19,31,33] and noncommutative geometries [10,11,35].
The starting point of our investigation are the results from [10,11] and their partial generalizations to the infinite-dimensional case in [22,48]. For a symmetric quantum Markov semigroup (P t ) the authors construct a noncommutative version of the 2-Wasserstein metric, which allows to obtain a quantum analog of the characterization [23,37] of the heat flow as 2-Wasserstein gradient flow of the entropy. On this basis, the geodesic semi-convexity of the entropy in noncommutative 2-Wasserstein space can be understood as a lower Ricci curvature bound for the quantum Markov semigroup, and it can be used to obtain a series of prominent functional inequalities such as a Tala grand inequality, a modified logarithmic Sobolev inequality and the Poincaré inequality [7,11,28,41].
One of the major challenges in the development of this program so far has been to verify semi-convexity in concrete examples, and only few noncommutative examples have been known to date, even less infinite-dimensional ones.
To prove geodesic semi-convexity, the gradient estimate or, equivalently, its integrated form has proven central. They can be seen as noncommutative analogs of the Bakry-Emery gradient estimate and the Γ 2 criterion. Indeed, if the underlying quantum Markov semigroup is the heat semigroup on a complete Riemannian manifold, (GE) reduces to the classical Bakry-Emery estimate As often in noncommutative geometry, tensorization of inequalities is more difficult than that in the commutative case. It is not known whether the gradient estimate in the form (GE) has good tensorization properties. For this reason we introduce (CGE), a complete version of (GE), and establish some of its stability properties. Using these in combination with a variant of the intertwining technique from [11] and a fine analysis of specific generators of Lindblad type, we are able to establish this tensor stable gradient estimate (CGE) for a number of examples for which geodesic convexity was not known before.
Let us briefly outline the content of the individual sections of this article. In Sect. 2 we recall some basics of quantum Markov semigroups, the construction of the noncommutative transport distance W and the connection between the gradient estimate (GE) and the geodesic semi-convexity of the entropy.
In Sect. 3 we extend the intertwining technique from [11,12] to the infinite-dimensional setting. Working with arbitrary operator means, our result does not only cover the gradient estimate implying semi-convexity of the entropy in noncommutative 2-Wasserstein space, but also the noncommutative Bakry-Emery estimate studied in [25]. As examples we show that the Ornstein-Uhlenbeck semigroup on the mixed q-Gaussian algebras satisfies (CGE) with constant K = 1, the heat semigroup on quantum tori satisfies (CGE) with constant K = 0, and that a class of quantum Markov semigroups on discrete group von Neumann algebras and quantum groups O + N , S + N satisfy (CGE) with constant K = 0. Moreover, this intertwining result is also central for the stability properties studied in the next section.
In Sect. 4 we show that the complete gradient estimate is stable under tensor products and free products of quantum Markov semigroups. Besides the applications investigated later in the article, these results also open the door for applications of group transference to get complete gradient estimates for Lindblad generators on matrix algebras.
In Sect. 5 we prove the complete gradient estimate (CGE) with constant K = 1 for quantum Markov semigroups whose generators are of the form where the operators p j are commuting projections. In a number of cases, this result is better than the ones we could obtain by intertwining and yields the optimal constant in the gradient estimate. As examples we show that this result applies to the quantum Markov semigroups associated with the word length function on finite cyclic groups and the nonnormalized Hamming length function on symmetric groups. Using ultraproducts and the stability under free products, we finally extend this result to Poisson-type quantum Markov semigroups on group von Neumann algebras of groups Z * k * Z * l 2 with k, l ≥ 0. In particular, this implies the complete modified logarithmic Sobolev inequality with optimal constant for these groups.
Note added. When preparing this preprint for submission, we were made aware that several of the examples have been obtained independently by Brannan, Gao and Junge (see [7,8]).

The Noncommutative Transport Metric W and Geodesic Convexity of the Entropy
In this section we briefly recall the definition and basic properties of the noncommutative transport distance W associated with a tracially symmetric quantum Markov semigroup. For a more detailed description we refer readers to [48].
Let M be a separable von Neumann algebra equipped with a normal faithful tracial state τ : M → C. Denote by M + the positive cone of M. Given 1 ≤ p < ∞, we define where |x| = (x * x) 1 2 is the modulus of x. One can show that · p is a norm on M. The completion of (M, · p ) is denoted by L p (M, τ ), or simply L p (M). As usual, we put L ∞ (M) = M with the operator norm. In this article, we are only interested in p = 1 and p = 2. In particular, L 2 (M) is a Hilbert space with the inner product x, y 2 = τ (x * y).
A family (P t ) t≥0 of bounded linear operators on M is called a quantum Markov semigroup (QMS) if -P t is a normal unital completely positive map for every t ≥ 0, -P s P t = P s+t for all s, t ≥ 0, -P t x → x in the weak * topology as t 0 for every x ∈ M.
for all x, y ∈ M and t ≥ 0. The generator of (P t ) is the operator L given by Here and in what follows, D(T ) always means the domain of T . For all p ∈ [1, ∞], the τ -symmetric QMS (P t ) extends to a strongly continuous contraction semigroup (P where C • is the opposite algebra of C. We will write aξ and ξ b for L(a)ξ and R(b)ξ , respectively. For A τ -symmetric QMS is called Γ -regular (see [28]) if the representations L and R are normal. Under this assumption, H is a correspondence from M to itself in the sense of Connes [  . By a slight abuse of notation, we write Γ (x, y) for the unique element h ∈ L 1 (M, τ ) such that If (P t ) is Γ -regular, then we can extend L to a map on the operators affiliated with M by defining for any operator x affiliated with M, where u is the partial isometry in the polar decomposition of x and e is the spectral measure of |x|. The same goes for R.
Let Λ be an operator mean in the sense of Kubo and Ando [27], that is, Λ is a map Here and in what follows, by A n A we mean A 1 ≥ A 2 ≥ · · · and A n converges strongly to A. From (b), any operator mean Λ is positively homogeneous: For a positive self-adjoint operator ρ affiliated with M, we definê ρ = Λ(L(ρ), R(ρ)).
Of particular interest for us are the cases when Λ is the logarithmic mean or the arithmetic mean We write · 2 ρ for the quadratic form associated withρ, that is, Given an operator mean Λ, consider the set If Λ is the arithmetic mean Λ ari , then this set coincides with In fact, when Λ = Λ ari , one has ∂a 2 ρ = 1 2 τ ((Γ (a) + Γ (a * ))ρ). If the operator mean Λ is symmetric, then it is dominated by the arithmetic mean and therefore A Γ ⊂ A Λ [27,Theorem 4.5], [48,Lemma 3.24]. The following definition states that this inclusion is dense in a suitable sense.
Of course the arithmetic mean is always regular. In general it seems not easy to check this definition directly, but we will discuss a sufficient condition below.
Let D(M, τ ) be the set of all density operators, that is, 1] such that ξ t ∈ H ρ t for all t ∈ [0, 1], the map t → ∂a, ξ t ρ t is measurable for all a ∈ A Γ and for every a ∈ A Γ one has

Definition 2. Fix an operator mean
For an admissible curve (ρ t ), the vector field (ξ t ) is uniquely determined up to equality a.e. and will be denoted by (Dρ t ). If Λ is a regular mean, the set A Γ can be replaced by A Λ everywhere in Definition 2.

Remark 1. The equation (1) is a weak formulation oḟ
which can be understood as noncommutative version of the continuity equation. Indeed, if (P t ) is the heat semigroup on a compact Riemannian manifold, it reduces to the classical continuity equationρ t + div(ρ t ξ t ) = 0.

Remark 2.
Recently, Li, Junge and LaRacuente [28] introduced a closely related notion of lower Ricci curvature bound for quantum Markov semigroups, the geometric Ricci curvature condition (see also [7,Definition 3.22]). Like CGE, this condition is tensor stable, and it implies CGE for arbitrary operator means [28,Theorem 3.6] (the result is only formulated for the logarithmic mean, but the proof only uses the transformer inequality for operator means).
In the opposite direction, the picture is less clear. For GE, a direct computation on the two-point graph shows that the optimal constant depends on the mean in general. It seems reasonable to expect the same behavior for CGE, which would imply that the optimal constant in CGE for a specific mean is in general bigger than the optimal constant in the geometric Ricci curvature condition. This gradient estimate is closely related to convexity properties of the logarithmic entropy Ent : D(M, τ ) → [0, ∞], Ent(ρ) = τ (ρ log ρ).

As usual let D(Ent
Theorem 1 ([48,Theorem 7.12]). Assume that (P t ) is a Γ -regular QMS. Suppose that Λ = Λ log is the logarithmic mean and is regular for (P t ). If (P t ) satisfies GE(K , ∞), then (a) for every ρ ∈ D(Ent) the curve (P t ρ) satisfies the evolution variational inequality (EVI K ) d dt This gradient flow characterization implies a number of functional inequalities for the QMS, see e.g. [11,Section 8], [48,Section 7], [12,Section 11]. Here we will focus on the modified logarithmic Sobolev inequality and its complete version (see [21,Definition 2.8], [28,Definition 2.12] for the latter).
For ρ, σ ∈ D(M, τ ) the relative entropy of ρ with respect to σ is defined as If N ⊂ M is a von Neumann subalgebra with E : M → N being the conditional expectation, then we define Recall that a conditional expectation E : M → N is a normal contractive positive projection from M onto N which preserves the trace and satisfies For x ∈ D(L 1/2 2 ) ∩ M + the Fisher information is defined as This definition can be extended to x ∈ L 1 + (M, τ ) by setting Recall that the fixed-point algebra of (P t ) is It is a von Neumann subalgebra of M [18, Proposition 3.5].
Definition 5. Let (P t ) be a Γ -regular QMS with the fixed-point subalgebra M fix . For λ > 0, we say that (P t ) satisfies the modified logarithmic Sobolev inequality with constant λ (MLSI(λ)), if 2 ) ∩ M. We say that (P t ) satisfies the complete modified logarithmic Sobolev inequality with constant λ (CLSI(λ)) if (P t ⊗id N ) satisfies the modified logarithmic Sobolev inequality with constant λ for any finite von Neumann algebra N .
For ergodic QMS satisfying GE(K , ∞), the inequality MLSI(2K ) is essentially contained in the proof of [48,Proposition 7.9]. Since (P t ⊗ id N ) is not ergodic (unless N = C), this result cannot imply the complete modified logarithmic Sobolev inequality. However, the modified logarithmic Sobolev inequality for non-ergodic QMS can also still be derived from the gradient flow characterization, as we will see next.
The same is true for the complete gradient estimate and the complete modified logarithmic Sobolev inequality.
for t ≥ 0 (using the continuity of both sides in t).
If K > 0, then MLSI(2K ) follows from a standard argument; see for example [28,Lemma 2.15]. The implication for the complete versions is clear. [7] and plays a central role there in obtaining complete logarithmic Sobolev inequalities.

Gradient Estimates Through Intertwining
Following the ideas from [11,12], we will show in this section how one can obtain gradient estimates for quantum Markov semigroups through intertwining. As examples we discuss the Ornstein-Uhlenbeck semigroup on the mixed q-Gaussian algebras, the heat semigroup on quantum tori, and a family of quantum Markov semigroups on discrete group von Neumann algebras and the quantum groups O + N and S + N . Throughout this section we assume that M is a separable von Neumann algebra with normal faithful tracial state τ and (P t ) is a Γ -regular QMS. We fix the corresponding first order differential calculus (H, L , R, J , ∂). We do not make any assumptions on Λ beyond being an operator mean. In particular, all results from this section apply to the logarithmic mean -thus yielding geodesic convexity by Theorem 1 -as well as the right-trivial mean -thus giving Bakry-Emery estimates.
Proof. Let ρ ∈ M + and a ∈ D(∂). Since Λ is an operator mean, we have [27, Theorem 3.5] As Λ is monotone in both arguments and positively homogeneous, conditions (ii) and (iii) imply Remark 4. The proof shows that assumptions (i)-(iii) still imply if the differential calculus is not the one associated with (P t ). We will use this observation in the proofs of Theorem 3 and Theorem 5.

Remark 5.
A similar technique to obtain geodesic convexity of the entropy has been employed in [11,12]. Our proof using the transformer inequality for operator means is in some sense dual to the monotonicity argument used there (see [38]). Apart from working in the infinite-dimensional setting, let us point out two main differences to the results from these two articles: In contrast to [11], we do not assume that P t is a direct sum of copies of P t (in fact, we do not even assume that H is a direct sum of copies of the trivial bimodule). This enhanced flexibility can lead to better bounds even for finite-dimensional examples (see Example 1). In contrast to [12], our conditions (ii) and (iii) are more restrictive, but they are also linear in ρ, which makes them potentially more feasible to check in concrete examples.
Remark 6. We do not assume that the operators P t form a semigroup or that they are completely positive (if H is realized as a subspace of L 2 (N ) for some von Neumann algebra N ). However, this is the case for most of the concrete examples where we can prove (i)-(iii).

Remark 7.
In particular, the conclusion of the previous theorem holds for all symmetric operator means, and in view of the discussions after Definition 4, it implies that any symmetric operator mean is regular for (P t ).
Under a slightly stronger assumption, conditions (ii) and (iii) can be rewritten in a way that resembles the classical Bakry-Emery criterion. For that purpose define Proof. To see the equivalence of (ii) and (iii), it suffices to notice that J is a bijection and J L(ρ)J = R(ρ) for ρ ∈ M + . The equivalence of (iii) and (2) follows from the identities: for all ρ ∈ M + : As indicated before, our theorem recovers the intertwining result in [11] (in the tracially symmetric case):

Corollary 2.
Assume that H ∼ = j L 2 (M, τ ), L and R act componentwise as left and right multiplication and J acts componentwise as the usual involution. If ∂ j P t = e −K t P t ∂ j , then (P t ) satisfies CGE(K , ∞).
Proof. Let P t = e −K t j P t . Condition (i) from Theorem 2 is satisfied by assumption. Since P t commutes with J , conditions (ii) and (iii) are equivalent. Condition (iii) follows directly from the Kadison-Schwarz inequality: This settles GE(K , ∞). Applying the same argument to (P t ⊗ id N ) then yields the complete gradient estimate. A direct computation shows that P t = e −t I + (1 − e −t )E. Let P t = e −t id H . Since L E = 0, we have ∂ E = 0 and therefore ∂ P t = e −t ∂ = P t ∂, which settles condition (i) from Theorem 2. Conditions (ii) and (iii) with K = 1/2 follow immediately from P t ρ ≥ e −t ρ for ρ ∈ M + . So (P t ) satisfies CGE(1/2, ∞). This result has been independently obtained in [7,Theorem 4.16].
In contrast, if for example p is a projection and [ p, x]]. Clearly, [ p, ·] commutes with L , so that the intertwining criterion from [11] only implies CGE(0, ∞). In fact, in this case we may obtain a better result; see Theorem 5.
Example 2 (Mixed q-Gaussian algebras). Let us recall the mixed q-Gaussian algebras. Our references are [4][5][6]30]. Let H be a real Hilbert space with orthonormal basis (e j ) j∈J . For k ≥ 1, denote by S k the set of permutations of {1, 2, . . . , k}. For k ≥ 2 and 1 ≤ j ≤ k − 1, denote by σ j the adjacent transposition between j and j + 1. For any σ ∈ S k , I (σ ) is the number of inversions of the permutations σ : A k-basis atom is an element of the form e j 1 ⊗ · · · ⊗ e j k . Clearly all the k-basis atoms form a basis of H ⊗k . For any k-basis atom u = e j 1 ⊗ · · · ⊗ e j k , we use the notation that For convenience, in the following we actually assume that sup i, j∈J |q i j | < 1. This is to simplify the definition of Fock space; our main results still apply to the general sup i, j∈J |q i j | ≤ 1 case.
Put P (0) = id H . For any k ≥ 1, denote by P (k) the linear operator on H ⊗k such that where u = e j 1 ⊗ · · · ⊗ e j k is any k-basis atom and is well-defined, though such representation of σ is not unique. When all the entries of Q are the same, that is, q i j ≡ q, the operator P (k) reduces to Under the condition that sup i, j∈J |q i j | < 1, the operator P (k) is strictly positive [6,Theorem 2.3].
Let F finite Q be the subspace of finite sums of the spaces H ⊗k , k ≥ 0, where H ⊗0 = RΩ and Ω is the vacuum vector. Then F finite Q is a dense subset of ⊕ k≥0 H ⊗k , and we define an inner product ·, · Q on F finite Q as: where ·, · 0 is the usual inner product. The Fock space F Q (H ) is the completion of F finite Q with respect to the inner product ·, · Q . When q i j ≡ q, the Fock space F Q (H ) is also denoted by F q (H ) for short. Notice that if we only have sup i, j∈J |q i j | ≤ 1, then each P (k) is only positive. One should divide F finite Q by the kernel of ·, · Q before taking the completion. The definition of Fock space here is actually the same as the one in [6] associated to the Yang-Baxter operator See [30, Part I] for a detailed proof for this when dim H < ∞.
Now we recall the mixed q-Gaussian algebra Γ Q (H ). For any i ∈ J , the left creation operator l i is defined by Its adjoint with respect to ·, · Q , the left annihilation operator l * i , is given by The left annihilation operators and left creation operators satisfy the deformed communication relations on F Q (H ): The mixed q-Gaussian algebra Γ Q (H ) is defined as the von Neumann subalgebra of B(F Q (H )) generated by self-adjoint operators It is equipped with a normal faithful tracial state τ Q given by for any k-atom f 1 ⊗ · · · ⊗ f k and any k ≥ 1. Moreover, there exists a unique unital and completely positive map for two contractions S, T on H . If q i j ≡ q ∈ [−1, 1], then we write the functor Γ Q as Γ q for short. It interpolates between the bosonic and the fermionic functors by taking q = +1 and q = −1 respectively. When q = 0, it becomes the free functor by Voiculescu [44]. For more examples, see [30,Introduction].
It extends to a semigroup of contractions on L 2 (Γ Q (H ), τ Q ) and is τ Q -symmetric. Note that the generator of P t is L where Then the generator L of P t takes the form L = ∂ † ∂ and ∂ is a derivation [30,Proposition 3.2]: CGE(1, ∞). For this let us first take a look of the semigroup T t = e −t N on F Q (H ). By definition, it equals e −kt id on its eigenspace H ⊗k . For each t ≥ 0, consider the map where S t is a contraction on H given by Then by definition, we have the intertwining condition In fact, it is obvious when acting on RΩ. If u is a k-atom on H , k ≥ 1, then Remark that if one chooses T t = F Q (e −t id H ), then we can only obtain CGE(0, ∞).

thus by (4) we have the intertwining condition
This, together with the definitions of h H (3) and P t , yields By Theorem 2, to show that P t satisfies GE(1, ∞), it remains to check (ii) and (iii) with P t as above and the left and right action of To prove (ii) we need to show that for any ρ ∈ Γ Q (H ) + and a ∈ Γ Q (H ): where the inner product is induced by τ Q . To see this, note that P t is completely positive and P t (1) = e −t 1 [30, Lemma 3.1]. By the Kadison-Schwarz inequality and (5), we have which finishes the proof of (ii). The proof of (iii) is similar. So P t satisfies GE(1, ∞).
Applying the same argument to P t ⊗ id N , we obtain CGE(1, ∞).

Remark 8.
As mentioned in [28,Section 4.4], the previous example can also be deduced from the complete gradient estimate for the classical Ornstein-Uhlenbeck semigroup using the ultraproduct methods from [26]. However, in contrast to this approach we do not need to use the Ricci curvature bound for the classical Ornstein-Uhlenbeck semigroup, but get it as a special case (with minor modifications accounting for |q| = 1 in this case).
In the commutative case θ = 0, A θ = C(T 2 ) is the C*-algebra of all continuous functions on flat 2 torus T 2 and the semigroup (P t ) is the heat semigroup generated by the Laplace-Beltrami operator on the flat 2-torus, which has vanishing Ricci curvature. Thus the constant 0 in the gradient estimate is optimal.
In fact, for any θ, θ ∈ [0, 1), the semigroup P θ t on L ∞ (A θ , τ θ ) satisfies CGE(K , ∞) if and only if the semigroup (P θ t ) on L ∞ (A θ , τ θ ) satisfies CGE(K , ∞). Thus the gradient estimate CGE(0, ∞) is optimal for any θ ∈ [0, 1). To see this, note first that by standard approximation arguments it suffices to show GE(K , ∞) for ρ ∈ (A θ ) + and a ∈ D(L 1/2 2 ) ∩ A θ . By universal property of A θ+θ , there exists a * -homomorphism π : Clearly π is trace preserving and satisfies So if P θ t satisfies CGE(K , ∞), then so does P θ+θ t . Since θ and θ are arbitrary, we finish the proof of the assertion. This idea of transference was used in [39] to give a simple proof that the completely bounded Fourier multipliers on noncommutative L p -spaces associated with quantum tori A θ do not depend on the parameter θ . The transference technique has been used in [21,28] to study complete logarithmic Sobolev inequality.
The same conclusion goes for d-dimensional quantum torus A θ with θ being a d-by-d real skew-symmetric matrix.
Example 4 (Quantum groups). A compact quantum group is a pair G = (A, Δ) consisting of a unital C*-algebra A and a unital * -homomorphism Δ : A → A ⊗ A such that Here A⊗ A is the minimal C*-algebra tensor product. The homomorphism Δ is called the comultiplication on A. We denote A = C(G). Any compact quantum group G = (A, Δ) admits a unique Haar state, i.e. a state h on A such that Denote Pol(G) = span u α i j : 1 ≤ i, j ≤ n α , α ∈ Irr(G) . This is a dense subalgebra of A. On Pol(G) the Haar state h is faithful. It is well-known that (Pol(G), Δ) is equipped with the Hopf*-algebra structure, that is, there exist a linear antihomormophism S on Pol(G), called the antipode, and a unital * -homomorphism : Pol(G) → C, called the counit, such that Here m denotes the multiplication map m : Pol(G)⊗ alg Pol(G) → Pol(G), a⊗b → ab. Indeed, the antipode and the counit are uniquely determined by Since the Haar state h is faithful on Pol(G), one may consider the corresponding GNS construction (π h , H h , ξ h ) such that h(x) = ξ h , π h (x)ξ h H h for all x ∈ Pol(G). The reduced C * -algebra C r (G) is the norm completion of π h (Pol(G)) in B(H h ). Then the restriction of comultiplication Δ to Pol(G), extends to a unital * -homomorphism on C r (G), which we still denote by Δ. The pair (C r (G), Δ) forms a compact quantum group, and in the following we always consider this reduced version (instead of the universal one, since the Haar state h is always faithful on C r (G)). Denote by L ∞ (G) = C r (G) the von Neumann subalgebra of B(H h ) generated by C r (G), and we can define the noncommutative L p -spaces associated with (L ∞ (G), h). In particular, we identify L 2 (G) with H h . We refer to [32] and [49] for more details about compact quantum groups.
A compact quantum group G is of Kac type if the Haar state is tracial. In the following G is always a compact quantum group of Kac type, which is the case for later examples O + N and S + N . Given a Lévy process ( j t ) t≥0 [14, Definition 2.4] on Pol(G) one can associate it to a semigroup P t = (id ⊗ φ t )Δ on C r (G), where φ t is the marginal distribution of j t . This (P t ) is a strongly continuous semigroup of unital completely positive maps on C r (G) that are symmetric with respect to the Haar state h [14, Theorem 3.2]. Then (P t ) extends to a h-symmetric QMS on L ∞ (G).
The corresponding first-order differential calculus can be described in terms of a Schürmann triple ((H, π), η, ϕ) [ Note that the QMS (P t ) is always right translation invariant: (id ⊗ P t )Δ = ΔP t for all t ≥ 0. In fact, any right translation invariant QMS must arise in this way [14,Theorem 3.4]. Here we are interested in semigroups (P t ) that are not only right translation invariant but also left translation invariant, or translation bi-invariant: for all t ≥ 0 In this case, let P t = P t ⊗ id H , and we have It is not hard to check that P t is J -real. We will show that it also satisfies the condition (iii) from Theorem 2 for K = 0. For ξ 1 , . . . , ξ n ∈ H and x 1 , . . . , x n ∈ A we have Clearly, the matrix [ ξ k , ξ l ] k,l is positive semi-definite. By Kadison-Schwarz inequality, Since the Hadamard product of positive semi-definite matrices is positive semi-definite, it follows that and we get the desired result. Thus (P t ) satisfies GE(0, ∞). Applying the same argument to (P t ⊗ id N ), we get CGE(0, ∞).
The convolution semigroup of states φ t = + n≥1 t n n! ψ n is generated by ψ, called the generating functional, where ψ is hermitian, conditionally positive and vanishes on 1 (see [14,Section 2.5] for details). Then once the generating functional ψ is central, the QMS P t = (id ⊗ φ t )Δ = e tT ψ is translation-bi-invariant, and thus satisfies CGE(0, ∞), For the geometric Ricci curvature condition this result was independently proven in [7,Lemma 4.6].
In the next few examples we collect some specific instances of QMS on quantum groups which are translation-bi-invariant. Firstly we give some commutative examples. G, (C(G), Δ) forms a compact quantum group, where C(G) is the C*-algebra of all continuous functions on G and the comultiplication Δ :

Example 5. (Compact Lie groups) For any compact group
The Haar state h is nothing but · dμ, with μ being the Haar (probability) measure. Consider the QMS Then (P t ) is translation bi-invariant if and only if the kernel K t is bi-invariant under G: K t (gr, gs) = K t (r, s) = K t (rg, sg) for all g, r, s ∈ G, or equivalently, (P t ) is a convolution semigroup with the kernelK t (s) = K (e, s) being conjugate-invariant: Let G be a compact Lie group with a bi-invariant Riemann metric g. If (P t ) is the heat semigroup generated by the Laplace-Beltrami operator, then a direct computation shows that the bi-invariance of the metric implies the translation-bi-invariance of (P t ). Thus we recover the well-known fact from Riemannian geometry that the Ricci curvature of a compact Lie group with bi-invariant metric is always nonnegative (see e.g. [34,Section 7]).
Secondly, we give co-commutative examples. By saying co-commutative we mean Δ = Π • Δ, where Π is the tensor flip, i.e., Π(a ⊗ b) = b ⊗ a. Example 6. (Group von Neumann algebras) Let G be a countable discrete group with unit e, C * r (G) the reduced C * -algebra generated by the left regular representation λ of G on 2 (G) and L(G) the group von Neumann algebra L(G) = C * r (G) ⊂ B( 2 (G)). Then G = (C * r (G), Δ) is a quantum group with comultiplication given by Δ(λ g ) = λ g ⊗ λ g . The Haar state on G is given by τ (x) = xδ e , δ e , which is tracial and faithful. Here and in what follows, δ g always denotes the function on G that takes value 1 at g and vanishes elsewhere.
By Schoenberg's Theorem (see for example [9,Theorem D.11]), to every cnd function one can associate a τ -symmetric QMS on L(G) given by It is easy to check that (P t ) satisfies the translation-bi-invariant condition (6). Thus it satisfies CGE(0, ∞).
is the universal C*-algebra generated by N 2 self-adjoint operators u i j , and the comultiplication Δ is given by    N ) is the universal C*algebra generated by N 2 self-adjoint operators p i j , 1 ≤ i, j ≤ N , such that and the comultiplication Δ is given by for s ∈ Irr(S + N ) = N, 1 ≤ i, j ≤ n s , where U s denotes the s-th Chebyshev polynomial of second kind, b > 0, and ν is a finite measure on [0, N ]. Similarly, given (b, ν), the central functional ψ defined as above induces a QMS P ψ t = e tT ψ satisfying (6), where T ψ = (id ⊗ ψ)Δ. Hence it satisfies CGE(0, ∞).

Remark 9.
Although many interesting functional inequalities like the Poincaré and the modified logarithmic Sobolev inequality only follow directly from GE(K , ∞) for K > 0, the gradient estimate with constant K ≤ 0 can still be helpful in conjunction with additional assumptions to prove such functional inequalities (see [7,17]).

Stability Under Tensor Products and Free Products
In this section we prove that the complete gradient estimate CGE(K , ∞) is stable under taking tensor products and free products of quantum Markov semigroups. We refer to [45] and [2] for more information on free products of von Neumann algebras and to [3] for free products of completely positive maps. Proof. Let H j and ∂ j denote the tangent bimodule and derivation for (P j t ) and let The tangent module H for P t = j P j t is a submodule of H = jH j with the natural left and right action and derivation ∂ = (∂ 1 , . . . ,∂ n ).
For j ∈ {1, . . . , n}, putP Together with the previous estimate, we obtain So (P t ) satisfies GE(K , ∞). The same argument can be applied to (P t ⊗ id N ), so that we obtain CGE(K , ∞).
where L 2 0 denotes the orthogonal complement of C1 in L 2 . Then H can be identified with a submodule of with the natural left and right action on each direct summand and ∂ acts as 0 on C1 and as ∂(a 1 ⊗ · · · ⊗ a n ) = (∂ j 1 (a 1 ) ⊗ a 2 · · · ⊗ a n , . . . , a 1 ⊗ a 2 ⊗ · · · ⊗ ∂ j n (a n )) on the direct summand j 1 =··· = j n L 2 (M j l , τ j l ). Since ∂ and (P t ) restrict nicely to the direct summand of L 2 (M, τ ), the rest of the proof is similar to the one of Theorem 3.
Remark 10. The same argument applies to free products with amalgamation if the common subalgebra over which one amalgates is contained in the fixed-point algebra of (P j t ) for all j ∈ {1, . . . , n} (compare with the results from [25, Section 6.2] for the Γ 2 condition).

Quantum Markov Semigroups Generated by Commuting Projections
In this section we move beyond applications of the intertwining result Theorem 2 and obtain complete gradient estimate for quantum Markov semigroups whose generators take special Lindblad forms.
In particular, L j is of the form L j = I − Φ j with I = id M and the conditional expectation Φ j (x) = p j x p j + (1 − p j )x(1 − p j ). Thus the QMS (P j t ) generated by L j is given by A first-order differential calculus for (P t ) is given by H = n j=1 L 2 (M, τ ) as bimodules, L = (L j ) j , R = (R j ) j with L j and R j being the usual left and right multiplications of M on L 2 (M, τ ) respectively, and Moreover, ∂ j P j t x = e −t ∂ j x and consequently On the other hand, by the concavity of operator means [27, Theorem 3.5] we have Since Recall that L j and R j are respectively the usual left and right multiplications of M on L 2 (M, τ ) and denote by E j the projection onto ran ∂ j in L 2 (M, τ ). It follows that The analogous identity for the left multiplication follows similarly. Note that both the left and right multiplication by Φ j (x) = p j x p j +(1− p j )x(1− p j ) leave ran ∂ j invariant. In fact, for any x, y ∈ M one has and a similar equation holds for the right multiplication.
Therefore we have This, together with the conditions (a) and (b) in the definition of operator means, implies In other words, Together with (9) we conclude In view of (8), we have proved Now let us come back to our original semigroup (P t ). Let Since the p j 's commute, so do the generators L j 's and the semigroups P Combined with the estimate (10) for (P j t ), we obtain So (P t ) satisfies GE(1, ∞). To prove CGE(1, ∞), it suffices to note that the generator of (P t ⊗ id N ) is given by and the elements ( p j ⊗ 1) are again commuting projections.
Remark 11. Since L 2 j = L j , the spectrum of L j is contained in {0, 1} with equality unless L j = 0. Thus the gradient estimate for the individual semigroups (P j t ) is optimal (unless v j = 0). It should also be noted that it is better than the gradient estimate one would get from Example 1.

Remark 12.
Inspection of the proof shows that the same result holds if the generator of (P t ) is of the form L = 1 2 n j=1 (x − u j xu j ) with commuting self-adjoint unitaries u j . Example 9. Let X = {0, 1} n and j : X → X the map that swaps the j-th coordinate and leaves the other coordinates fixed. Let v j = x | j (x) x| ∈ B( 2 (X )). By the previous remark, the QMS on B( 2 (X )) with generator satisfies CGE(1, ∞). The restriction of this semigroup to the diagonal algebra is (up to rescaling of the time parameter, depending on the normalization) the Markov semigroup associated with the simple random walk on the discrete hypercube (see [19,Example 5.7]).
To apply the theorem above to group von Neumann algebras, we will use the following Lindblad form for QMS generated by cnd length functions. Recall that for a countable discrete group G, a 1-cocycle is a triple (H, π, b), where H is a real Hilbert space, π : G → O(H ) is an orthogonal representation, and b : G → H satisfies the cocycle To any cnd function ψ on a countable discrete group G, one can associate with a 1-cocycle (H, π, Appendix D] for more information. where v j is a linear operator on 2 (G) given by v j δ g = b(g), e j δ g .
Proof. By definition we have This is nothing but Remark 13. The elements v j are not contained in the group von Neumann algebra L(G) so that Theorem 5 is not directly applicable (even if the v j are projections). However, if G is finite, then the operator generates a tracially symmetric QMS on B( 2 (G)) and we can apply Theorem 5 to that semigroup instead. It is an interesting open question how to treat infinite groups for which the generator has such a Lindblad form.
Example 10. The cyclic group Z n = {0, 1, . . . , n − 1}; see [25,Example 5.9]: Its group (von Neumann) algebra is spanned by λ k , 0 ≤ k ≤ n − 1. One can embed Z n to Z 2n , so let us assume that n is even. The word length of k ∈ Z n is given by ψ(k) = min{k, n −k}.
To extend the last example to the infinite symmetric group S ∞ , we need the following approximation result.

Lemma 3.
Let (M n ) be an ascending sequence of von Neumann subalgebras such that n M n is σ -weakly dense in M. Further let (P t ) be a Γ -regular QMS on M and assume that P t (M n ) ⊂ M n . Let (P n t ) denote the restriction of (P t ) to M n . If (P n t ) satisfies GE(K , ∞) for all n ∈ N, then (P t ) also satisfies GE(K , ∞). The same is true for CGE.
Proof. It is not hard to see that n M n is dense in L 2 (M, τ ). Since P t (M n ) ⊂ M n and P t maps into the domain of its L 2 generator L 2 , the space V = D(L 1/2 2 ) ∩ n M n is also dense in L 2 (M, τ ) and invariant under (P t ). Using a standard result in semigroup theory, this implies that V is a form core for L . Thus it suffices to prove for a ∈ V and ρ ∈ M + . Moreover, by Kaplansky's density theorem and the strong continuity of functional calculus, checking it for ρ ∈ ( n M n ) + is enough. But for a ∈ D(L 1/2 2 ) ∩ M n and ρ ∈ (M n ) + , this is simply the gradient estimate for (P n t ), which holds by assumption.
The argument for CGE is similar.

Corollary 3.
If G is the ascending union of subgroups G n and ψ is a cnd length function on G such that for every n the QMS associated with ψ| G n satisfies GE(K , ∞), then the QMS associated with ψ satisfies GE(K , ∞). The same is true for CGE. Proof. To ease notation, we will only deal with GE(1, ∞). The proof of complete gradient estimate is similar, embedding L(G) ⊗ N into a suitable ultraproduct. Let (F n ) be a Følner sequence for G and ω ∈ βN \ N. Endow B( 2 (F n )) with the normalized trace τ n and let p n denote the projection from 2 (G) onto 2 (F n ). Then we have a trace-preserving embedding For each j, let v j be the linear operator on 2 (G) given by v j (δ g ) = b(g), e j δ g , and denote its restriction to 2 (F n ) by the same symbol. Note that for every fixed n ∈ N, there are only finitely many indices j ∈ J such that v j is non-zero on 2 (F n ).
Let For x = g x g λ g with g ψ(g)|x g | 2 < ∞, we have v j p n x p n (δ g ) = 1 F n (g)v j p n x(δ g ) = h∈G 1 F n (g)x h v j p n (δ hg ) = h∈G 1 F n (g)1 F n (hg)x h v j (δ hg ) = h∈G 1 F n (g)1 F n (hg)x h b(hg), e j δ hg , and p n x p n v j (δ g ) = b(g), e j p n x p n (δ g ) = 1 F n (g) b(g), e j p n x(δ g ) = h∈G 1 F n (g)x h b(g), e j p n (δ hg ) = h∈G 1 F n (g)1 F n (hg)x h b(g), e j δ hg .

Hence
[v j , p n x p n ](δ g ) = (v j p n x p n − p n x p n v j )(δ g ) = h∈G and we get ∂ n ( p n x p n ) 2 x h x h b(hg) − b(g), e j b(h g) − b(g), e j 1 F n (hg)1 F n (h g) δ hg , δ h g = 1 |F n | g∈F n j∈J h∈G |x h | 2 b(hg) − b(g), e j | 2 1 F n (hg) which converges to h∈G ψ(h)|x h | 2 = ∂ x 2 H as n → ω by the Følner property of (F n ) after an application of the dominated convergence theorem. Thus the tangent bimodule H for (P t ) can be viewed as a submodule of ω H n with the obvious left and right action and ∂ = (∂ n ) • . Let (P n t ) be the QMS on B( 2 (F n )) generated by ∂ † n ∂ n . Since b(·), e j takes values in {0, 1}, the operators v j 's are projections. Clearly all the v j 's commute. Hence by Theorem 5 and Remark 13, (P n t ) satisfies GE(1, ∞). From the ultraproduct structure of H and ∂ we deduce for (x n ) • ∈ L(G) and (ρ n ) • ∈ L(G) + . In other words, (P t ) satisfies GE(1, ∞).

Remark 14.
The group von Neumann algebra embeds into an ultraproduct of matrix algebras if and only if the underlying group is hyperlinear, so it might be possible to extend the previous proposition beyond amenable groups.
Example 13. (Amenable groups acting on trees) Let T be a tree (viewed as unoriented graph) and G an amenable subgroup of Aut(T ). For fixed x 0 ∈ T define the length function ψ on G by ψ(g) = d(x 0 , gx 0 ), where d is the combinatorial graph distance.
As in the case of free groups, one sees that ψ is conditionally negative definite and the associated 1-cocycle can be described as follows (see [ For example this is the case for G = Z with ψ(k) = |k|. Here the tree is the Cayley graph of Z and the action is given by the left-regular representation. This QMS on L(Z) corresponds, under the Fourier transform, to the Poisson semigroup on L ∞ (S 1 ).
More generally, the Cayley graph of a group is a tree if and only if it is of the form Z * k * Z * l 2 for k, l ≥ 0. This group is not amenable unless k + l ≤ 1, but the free product structure allows us to obtain the same bound. Theorem 6. If G is a group whose Cayley graph is a tree and the cnd function ψ is given by ψ(g) = d(g, e), where d is the combinatorial metric on the Cayley graph, then the QMS associated with ψ satisfies CGE(1, ∞) and CLSI(2) and the constants in both inequalities are optimal.
Proof. As previously mentioned, G is of the form Z * k * Z * l 2 with k, l ≥ 0. It is not hard to see that the QMS associated with ψ decomposes as free product of the QMS associated with the word length functions on the factors. Thus it satisfies CGE(1, ∞) by Theorem 4 and CLSI(2) by Corollary 1. Since both the gradient estimate and the modified logarithmic Sobolev inequality imply that the generator has a spectral gap of 1, the constants are optimal.
Example 14. If G is a free group and ψ the word length function, then the associated QMS satisfies CGE(1, ∞) and CLSI (2). Note that the usual logarithmic Sobolev inequality, which is equivalent to the optimal hypercontractivity, is still open. Some partial results have been obtained in [24,40]. Our optimal modified LSI supports the validity of optimal LSI from another perspective. article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.