# Geometric Approach Towards Complete Logarithmic Sobolev Inequalities

aa r X i v : . [ qu a n t - ph ] F e b GEOMETRIC APPROACH TOWARDS COMPLETE LOGARITHMIC SOBOLEVINEQUALITIES

LI GAO, MARIUS JUNGE, AND HAOJIAN LI

Abstract.

In this paper, we use the Carnot-Caratheodory distance from sub-Riemanian geometryto prove entropy decay estimates for all ﬁnite dimensional symmetric quantum Markov semigroups.This estimate is independent of the environment size and hence stable under tensorization. Ourapproach relies on the transference principle, the existence of t -designs, and the sub-Riemanniandiameter of compact Lie groups and implies estimates for the spectral gap. Introduction

Logarithmic Sobolev inequalities is a versatile tool in analysis and probability. It was ﬁrst in-troduced by Gross [Gro75b, Gro75a], and later found rich connections to geometry, graph theory,optimal transport as well as information theory. (See e.g. [BÉ85, OV00, BGL13] and the overview[Led04] by Ledoux and by Gross [Gro14]). The natural framework of logarithmic Sobolev inequalitiesis given by Markov semigroups, i.e. a semigroups of measure preserving maps on a measure space.Barky-Emery theory [BÉ85], however, indicates the importance of geometric data in obtaining goodestimates. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups have at-tracted a lot of attentions: see e.g. [Bar17, DR20, RD19, KT13a, CM17, CM20] for the connectionsto other functional and geometric inequalities; [DR20, DB14] for application in quantum informationtheory; [Wir18, WZ20, BGJ20b] for inﬁnite dimensional examples; [CRF20, BCL +

19] for quantumGibbs sampler on lattice spin systems. Quantum Markov semigroups model the Markovian evolu-tion of open quantum systems, which inevitably interact with the surrounding environment. Themotivation of this work is to study the entropy form of log-Sobolev inequlities, so-called modiﬁedlog-Sobolev inequality, for ﬁnite dimensional quantum systems and their tensorization property.A quantum Makrov semigroup on ﬁnite dimensional quantum system is described by a Lindbladgenerator. Let M n be the n × n matrix algebra and tr be the standard matrix trace. We considera (symmetric) Lindlabd generator (also called Lindbladian) on M n L ( x ) = k X j =1 a j x + xa j − a j xa j (1.1)where a j ∈ M n are self-adjoint operator. It was proved by Gorini, Kossakowski and Sudarshan[GKS76a] and Lindblad [Lin76] that L generates a semigroup T t = e − tL of complete positive tracepreserving maps, and conversely all such generators symmetric to the trace inner product has theform (1.1). The ﬁxed point algebra N := { x | T t ( x ) = x , ∀ t ≥ } is the commutant N = { a j | ≤ j ≤ k } ′ as a subalgebra. Let E N : M n → N be the conditional expectation onto N , which isthe projection onto the ﬁxed point space. We say the semigroup T t or its generator L satisﬁes λ -modiﬁed logarithmic Sobolev inequalities ( λ -MLSI) for λ > if for all positive operators ρ , λ tr( ρ log ρ − ρ log E N ( ρ )) ≤ tr( Lρ log ρ ) . (1.2)This inequality characterizes a strong convergence property in terms of entropy that D ( T t ρ || E N ( ρ )) ≤ e − λt D ( ρ || E N ( ρ )) , (1.3) HL and MJ are partially supported by NSF grants DMS 1800872 and Raise-TAG 1839177. where D ( ρ || σ ) = tr( ρ log ρ − ρ log σ ) is the quantum relative entropy. In contrast to classical Markovsemigroups, it is crucial to allow for environment system due to potential quantum entanglement.This leads us to consider the ampliﬁed semigroup T t ⊗ id M m over a noiseless (ﬁnite dimensional)auxiliary systems M m , which goes beyond the ergodic case. We say the semigroup T t satisﬁes λ -complete logarithmic Sobolev inequalities ( λ -CLSI) if for all m ≥ , T t ⊗ id M m satisﬁes λ -MLSI.The CLSI was ﬁrst introduced in [GJL20] and later studied in [BGJ20a, BGJ20b, WZ20]. Wewrite CLSI ( L ) for the optimal (largest) constant λ such that T t = e − Lt satisﬁes λ -CLSI. The CLSIconstant governs the convergence rate independently of the size of the environment system, and moreimportantly, satisﬁes the tensorization property CLSI ( L ⊗ id+id ⊗ L ) = min { CLSI (L ) , CLSI (L ) } .The tensorization property was used in [CRF20] as a key condition to obtain size independent MLSIfor quantum lattice systems. Therefore, it is desired to know whether all ﬁnite dimensional quantumMarkov semigroup T t = e − Lt admits CLSI( L ) > .It turns out that the above questions is closely related to matrix valued version of logarithmicSobolev inequalities for classical Matrix semigroup. Indeed, let G be a compact Lie group and itsLie algebra g . Given a generating family X = { X , · · · , X k } of the Lie algebra g via Lie bracket, X gives a hypoelliptic sub-Laplacian ∆ X = − s X j =1 X j X j Given a unitary representation u : G → M n , one can transfer the sub-Laplacian ∆ X to a Lindbladgenerator L X ( x ) = P j − [ a j , [ a j , x ]] where a j are self-adjoint elements such that π (exp( tX j )) = e ita j .Then L X generates a quantum Markov semigroup T t = e − tL X : M n → M n , and the conditionalexpectation onto the ﬁxed point subalgebra N is given by E N ( x ) = Z G π ( g ) ∗ xπ ( g ) dµ ( g ) . Here dµ is the Haar measure on G . L X is called a transferred Lindbladian of ∆ X via the rep-resentation u . Conversely, it was observed in [GJL20] that every ﬁnite dimensional self-adjointLindbladian can be realized as a transferred Lindbladian from a connected compact Lie group. (Werefer to Section 3 for more information on the transference principle.)Thanks to the above transference principle, it suﬃces to study sub-Laplacians on compact Liegroups for matrix-valued functions. Nevertheless, many classical tools assuming egordicity do notapply in this setting. One the technical diﬃculty is the fact that the generator L X is govern bya sub-Laplacian operator ∆ X . The impressive body of work by Baudoin, Thalmaier, and Grong[Bau14, GT19, BGKT19] indicates that a naive curvature identity ∇ H ∆ X = ˆ∆ X ∇ H + R ( ∇ H ) (1.4)for some ﬁrst order tensor R and generator ˆ∆ X may fail. In fact this does not appear to hold forthe basic example G = SU (2) and L = − X − Y given by two out of three directions. This meansthat entropy decay estimates from quantum information theory have to go beyond the standardBakry-Émry theory and circumvent the use of the famous Rothaus lemma, both are standard toolsin the ergodic case. We refer to [KT13b, LOZ10] for the Rothaus lemma in the ergodic quantumcase which no longer applies with additional environment.The main theorem of this work is a lower bound of the CLSI constant of a so-called transferredquantum Markov semigroup T t = e − L X t via the sub-Riemannian structure of X = { X , · · · , X s } on G . Theorem 1.1.

Let G be a connected compact Lie group and g be its Lie algebra. Let X = { X , · · · , X s } be a family of left invariant vector ﬁeld generating g . Suppose π : G → M n is EOMETRIC APPROACH TOWARDS

CLSI unitary representation such that (1.5) E N ( x ) := m X j =1 α j π ( g j ) ∗ xπ ( g j ) for a ﬁnite probability distribution P mj =1 α j = 1 , α j ≥ . Then the CLSI constant of the transferedLindbladian L X ( x ) = − P sj =1 [ a j , [ a j , x ]] satisﬁes CLSI( L X ) ≥ Csmd X ( d X + 1) . where C is an universal constant and d X is the diameter of G in the Carnot-Caratheodory distanceinduced by X . Here the Carnot-Caratheodory distance, also called sub-Riemannian distance, is deﬁned as d H ( p, q ) = inf γ (0)= p,γ (1)= q Z k γ ′ ( t ) k H dt (1.6)where the inﬁmum is taken over all piecewise smooth curves whose derivatives γ ′ ( t ) are a.e. inthe horizontal direction H = span { X k ( γ ( t )) } . This distance deﬁnes the same topology and hence G admits ﬁnite diameter d X with respect to this new metric. The equation (1.5) is an analog ofspherical design for G = SO ( n ) and of unitary design for G = U ( n ) , which are of interest fromcombinatorics and quantum computing. Thanks to Caratheodory theorem (see [Wat18]), we knowthat the design (1.5) always exists with m ≤ n + 4 n + 2 . Therefore, Theorem 1.1 shows that everyquantum Markov semigroup transferred from a sub-Laplacian on a compact Lie group satisﬁes CLSI.As a corollary, we obtain a positive solution to the existence of CLSI constants in ﬁnite dimensions. Corollary 1.2.

Every self-adjoint Lindbladian L on a ﬁnite dimensional matrix algebra satisﬁes CLSI( L ) > . The above results can be extended to Lindbladians L satisfying GNS-symmetry of states via thenoncommutative change of measure in [JLR19]. Very recently this result has been independentlyobtained in [GR21] using very diﬀerent techniques. These two results are complementary: while theproof presented here requires knowledge of the Carnot-Carathodory diameter of G and the size of adesign for the conditional expectation and implies spectral gap, the proof by Gao and Rouzé on theother hand relies on the spectral gap and the Popa-Pimnser index [PP86] of the inclusion N ⊂ M m .The lower bound in Theorem 1.1 does not depend much on the dimension of the representation π ,and holds uniformly for sub-representations of a given tensor product representations π ⊗ k ⊗ ¯ π ⊗ k . Incontrast, the Popa-Pimnser index for a direct sum of irreducible representations can become verylarge.The rest of paper is organized as follows. Section 2 discusses the complete logarithmic Sobolevconstant on the weighted interval. In Section 3, we use the interval result to prove Theorem 1.1.2. Complete Logarithmic Sobolev Inequalities on the Interval

In this section we discuss the complete logarithmic Sobolev inequalities (CLSI) for the weightedinterval. Let [0 , be the unit interval and µ be a probability measure on [0 , . We write L ∞ ([0 , , µ ) (resp. C ([0 , and C ∞ ([0 , ) as the space of L ∞ (resp. continuous and smooth) functions. Denote δ = i ddx as the derivative operator. We shall ﬁrst consider δ is a closable derivation on smoothfunctions f with periodic boundary conditions f (0) = f (1) . In this case, the underlying spaceis equivalent to unit circle T . We write δ ∗ as the adjoint operator on L ([0 , , µ ) and ∆ µ = δ ∗ ¯ δ as the weighted Laplacian operator. A matrix valued function f ∈ C ([0 , , M n ) is positive iffor every t ∈ [0 , , f ( t ) ≥ is a positive (semi-deﬁnite) matrix. We are interested in proving L. GAO, M. JUNGE, AND H. LI the following matrix-valued modiﬁed logarithmic Sobolev inequalities that for all smooth periodicpositive f ∈ C ∞ ([0 , , M n ) , λ Z tr( f ( x ) log f ( x ) − f ( x ) log E µ f ) dµ ( x ) ≤ Z tr((∆ µ f )( x ) log f ( x )) dµ ( x ) . (2.1)where E µ f = R f dµ is the weighted mean. The left hand side above is the relative entropy D ( f || E µ f ) for the matrix-valued f with respect to its mean E µ ( f ) , and the right hand side isthe Fisher information I ∆ µ ( f ) (also called entropy production). We denote CLSI([0 , , µ ) (resp. MLSI([0 , , µ ) ) for the optimal (largest) constant λ such that (2.1) is satisﬁed for n ≥ and periodicpositive f ∈ C ∞ ([0 , , M n ) (resp. for all periodic positive scalar valued function f ∈ C ∞ ([0 , ).We also denote CLSI((0 , , µ ) (resp. MLSI((0 , , µ )) as the CLSI (resp.

MLSI ) constant forfunctions f without periodic boundary conditions f (0) = f (1) .We emphasize that it is the constant CLSI([0 , , µ ) (or MLSI([0 , , µ ) ) that gives the expotentialdecay rate of relative entropy as in (1.3). On the other hand, the constants CLSI((0 , , µ ) and MLSI((0 , , µ )) are not associated with a semigroup because the derivation δ = i ddx are not closablewithout periodic boundary conditions. Nevertheless, the open interval constant CLSI((0 , , µ ) applyto more general functions and are more ﬂexible to use with semigroups. It follows from the standardsymmetrization and periodization argument in [BGL13, Proposition 4.5.5 & 5.7.5] that the CLSI constants of

CLSI([0 , , µ ) and CLSI((0 , , µ ) are related by a factor ,

14 CLSI([0 , , µ ) ≤ CLSI((0 , , µ ) ≤ CLSI(([0 , , µ ) . (2.2)It is clear that CLSI([0 , , µ ) ≤ MLSI([0 , , µ ) and CLSI((0 , , µ ) ≤ MLSI((0 , , µ ) but the otherdirection estimate is still unknown. The constant MLSI([0 , , µ ) and MLSI((0 , , µ ) for scalar-valued functions are discussed in [BGL13, Proposition 5.7.5]. Proposition 2.1.

Let n be a positive integer and dµ ( x ) = n x n − dx be a probability on [0 , . Then CLSI((0 , , µ ) ≥ CLSI([0 , , µ ) ≥ (2 e / ) − for all n ≥ .Proof. Denote dx as the Lebesgue measure. Consider the probability measure dν n ( x ) = a n x n − e − x dx on [0 , with a n = R x n − e − x dx . Since for periodic boundary funtions, the underlying space iscircle which has zero Ricci curvature. Then the Bakry-Émery’s weighted Ricci tensor isRic ( dν n ) = Hess ( x − ( n −

1) ln( x )) = 1 + n − x ≥ This implies

CLSI([0 , , ν n ) ≥ for funtions with periodic condition f (0) = f (1) . By comparingthe two measures na n e ≤ dν n ( x ) dµ ( x ) ≤ na n and the change of measure in [LJL20, Theorem 2.14], wehave CLSI([0 , , n x n − dx ) ≥ e − / .More generally, we have the following criterion. Corollary 2.2.

Let dµ ( x ) = f ( x ) dx be a probability measure [0 , with second diﬀerentiable densityfunction f . If there exists k > and a > such that kf ( x ) − f ′′ ( x ) f ( x ) − ( f ′ ( x )) ≥ a > , ∀ x ∈ (0 , . Then

CLSI((0 , , µ ) ≥ CLSI([0 , , µ ) ≥ (2 e k ) − .Proof. Let us consider the probability measure dγ = a f ( x ) e − kx with a = R f ( x ) e − kx dx . Theweighted Ricci tensor isRic ( dγ ) = Hess ( kx − ln( f ( x ))) = 2 kf ( x ) − f ′′ ( x ) f ( x ) − ( f ′ ( x )) f ( x ) ≥ a > . EOMETRIC APPROACH TOWARDS

CLSI The two measures dµ and dν are comparable ce − k a ≤ dνdµ ≤ ca . By the change of measure again, we have

CLSI([0 , , dµ )) ≥ e − k .The next estimate, despite of giving worse constants, applies to open interval constant and justdepends on the growth order. Proposition 2.3.

Let dµ ( x ) = a h ( x ) dx be a probablity measure on [0 , and a = R h ( x ) dx .Suppose c x α ≤ h ( x ) ≤ c βx β − for some c , c > and ≤ α < β with β ≥ . Then CLSI((0 , , dµ ) > .Proof. Let Φ( x ) = q π R ∞ x e − t / dt be the error function normalized. Let g : [0 , ∞ ) → [0 , be adecreasing function such that H ( g ( x )) = Φ( x ) and H ( y ) = Z y h ( x ) dx. Thus H ′ ( g ( x )) g ′ ( x ) = q π e − x and g ′ ( x ) = r π e − x / h ( H − (Φ( x ))) . (2.3)Write E f = ( R f dx )1 as the expectation to the uniform measure. Using the fact the Gaussianmeasure has Ricci curvature and Lemma 2.2, for positive matrix-valued function ρ ( t ) = f ( g ( t )) we have D ( ρ k E ( ρ )) ≤ r π Z ∞ τ ( ρ ′ ( t ) J log ρ ( t ) ρ ′ ( t )) e − t / dt = Z τ ( f ′ ( x ) J log f f ′ ( x )) | g ′ ( g − ( x )) | h ( x ) dx . where J log σ ( X ) = Z ∞ σ + r X σ + r dr is the double operator integral for log function. By thechange of variable we have D ( ρ k E ( ρ )) = D ( f k E ( f )) , then it suﬃces to ﬁnd an upper bound for g ′ ( x ) . Now, our assumption h ( x ) ≤ c βx β − implies H ( x ) ≤ c x β and hence by (2.3) h ( H − ( y )) ≥ c ( H − ( y )) α ≥ c ( yc ) α/β . (2.4)Now, we use the inequality Φ( x ) ≥ q π e − x / x (1+ x − ) and Φ( t ) ≥ Φ(1) for t ≤ . Together with (2.4),we obtain g ′ ( x ) ≤ c − c αβ (cid:18) π (cid:19) (1 − αβ ) (cid:18) x (1 + 1 x ) (cid:19) αβ e − x (1 − αβ ) . Note that for small t we may replace t (1 + 1 /t ) with a constant. Thus for ≤ α < β , this termis bounded, and hence its square is also bounded.For h ( x ) = n x n − , the CLSI constant from above is of the order n . Nevertheless, Lemma 2.3help us understand measures whose density functions h do not have desired smooth ( C ) propertiesas in Proposition 2.1. L. GAO, M. JUNGE, AND H. LI

Remark 2.4 (CLSI constant for uniform measure) . The above Proposition 2.1 gives a tight constantthen [LJL20, Example 4.7] that

CLSI((0 , , dx ) ≥

14 CLSI([0 , , dx ) ≥ , which was obtained by comparing the uniform measure with a another modiﬁed Gaussian distribu-tion. A sharper constant CLSI((0 , , dx ) ≥

14 CLSI([0 , , dx ) ≥ π ln 3 was obtained in [BGJ20a, Theorem 4.12] using heat kernel estimate and monotonicity of Fisherinformation. Note for the scalar case MLSI([0 , , dx ) = 4 π and MLSI((0 , , dx ) = π . Themethod in [BGJ20a] also applies to the weighted measure but the heat kernel estimate for theweighted Laplacian ∆ µ is less explicit. Remark 2.5 (Extension to piecewise smooth functions) . Here we discuss some subtlety about thedomain of ∆ µ and of the modiﬁed log-Sobolev inequality. On one hand, the semigroup T t = e − t ∆ µ is deﬁned for all functions f ∈ L ((0 , , µ ) . MLSI((0 , , µ ) ≥ λ is equivalent to that for any densityfunction ρ ∈ L ((0 , , µ ) , ρ ≥ and Z ρdµ = 1 , D ( T t ρ k ≤ e − λt D ( ρ k . where D ( f k

1) = Z f log f dµ is the entropy functional. For smooth ρ , we can take derivative at t = 0 and obtain the modiﬁed log-Sobolev inequality λD ( ρ k ≤ I ∆ µ ( ρ ) := Z ∆ µ ρ log ρdµ. This inequality can be extended to piecewise smooth ρ where the Fisher information I ∆ µ ( ρ ) has tobe interpreted as Dirichelet form I ∆ µ ( ρ ) = lim n →∞ E ( ρ, f n ( ρ )) := lim n →∞ Z ddx ( ρ ) ddx ( f n ( ρ )) dµ (2.5)where f n ( t ) = max { min { log( t ) , n } , − n } is the truncated logarithmic function (see [Wir18, Deﬁnition5.17]). Suppose ρ : [0 , → R is continuous piecewise smooth and its derivative ρ ′ is deﬁned andcontinuous except for ﬁnite points in [0 , . For our purpose, it suﬃces to consider ρ is boundedfrom below (also bounded from above by continuity). Thus I ( ρ ) = E ( ρ, f m ( ρ )) for some ﬁnite m .Let ǫ n be an approximation identity of smoothing kernels. Take ρ n = ρ ∗ ǫ n by convolution, and ρ ′ n = ρ ′ ∗ ǫ n be the derivative of ρ n . It is readily to see that k ρ n − ρ k ≤k ρ n − ρ k ∞ ≤k ρ ′ n − ρ k ≤k ρ ′ n − ρ k → , which means that both ρ n → ρ and ρ ′ n → ρ ′ in L . Hence ρ ∈ dom ( ddx ) = dom (∆ / µ ) by closableextension and by the Leibniz rule f m ( ρ ) also in dom ( ddx ) . Thus the Fisher information I ∆ µ ( ρ ) isalso well-deﬁned. Note that by data processing inequality ρ ρ ∗ ǫ n and lower-semicontinuity ofrelative entropy, lim sup n D ( ρ n k ≤ D ( ρ k ≤ lim inf n D ( ρ n k . For the Fisher information, I ∆ µ ( ρ ) = E ( ρ, f m ( ρ )) = lim n E ( ρ n , f m ( ρ n )) = lim n I ∆ µ ( ρ n ) . Here we use [DPWS02, Corollary 7.5] for the continuity k δ ( f m ( ρ n )) − δ ( f m ( ρ )) k → . Thus forcontinuous, piecewise smooth and strictly positive ρ , λD ( ρ || E ( ρ )) ≤ I ∆ µ ( ρ ) . EOMETRIC APPROACH TOWARDS

CLSI The same argument works for the matrix-valued functions. For smooth matrix-valued density ρ : [0 , → M n , I ∆ µ ( ρ ) = Z tr (cid:16) (∆ µ ⊗ id n ρ )(x) log ρ (x) (cid:17) dx = Z tr (cid:16) ρ ′ (x)J ln ρ (x) ρ ′ (x) (cid:17) dx . where J σ ( X ) = Z ∞ σ + r X σ + r dr is the double operator integral for f ( x, y ) = log x − log yx − y . Indeed,this is clear for smooth ρ n = ρ ∗ ( ǫ n M n ) (entry-wise molliﬁcation) and by limit I ∆ µ ( ρ ) = lim n I ∆ µ ( ρ n ) = lim n Z tr( ρ ′ n ( x ) J ln ρ n ( x ) ρ ′ n ( x )) dx = Z tr( ρ ′ ( x ) J ln ρ ( x ) ρ ′ ( x )) dx. Here we use the fact ρ ′ n → ρ ′ in L ([0 , , M n ) and ρ n → ρ in k · k ∞ (because ρ is continuous). Tosum up, our discussion above justiﬁes that modiﬁed log-Sobolev inequalites λD ( ρ || E µ ρ ) ≤ I ∆ µ ( ρ ) = Z τ ( ρ ′ ( x ) J ln ρ ( x ) ρ ′ ( x )) dx extends to piecewise smooth, strictly positive matrix-valued density function.3. Complete Logarithmic Sobolev Inequalities On Matrix Algebras

In this section we prove that every symmetric quantum Markov semigroup T t = e − tL : M n → M n on Matrix algebra satisﬁes complete logarithmic Sobolev inequality. A quantum Markov semigroup T t : M n → M n is a continuous family of maps satisfyingi) for each t ≥ , T t is completely positive and unital T t (1) = 1 .ii) for any t, s ≥ , T t ◦ T s = T t + s and T = id M n .where id M n is the identity map on M n . We denote L ( M n ) as the Hilbert-Schmidt space equippedwith the inner product h a, b i = tr( a ∗ b ) . We say a semigroup T t is symmetric if for each t ≥ , T t isa self-adjoint map on L ( M n ) . Namely, for any x, y ∈ M n , tr( T t ( x ) ∗ y ) = tr( x ∗ T t ( y )) . The generator of the semigroup (also called Lindbladian) is a operator on L ( M n ) deﬁned as Lx = lim t → t ( x − T t ( x )) , T t = e − tL , where L is a operator on L ( M n ) . In most of our discussion, we restrict ourselves to the symmetriccases. Thanks to [GKS76b, Lin76], the generator of symmetric semigroups is given by L ( x ) = − s X k =1 [ a k , [ a k , x ]] = s X k =1 ( a k x + xa k − a k xa k ) . (3.1)where a k ∈ M n are some self-adjoint operators. Then, L admits a ∗ -preserving derivation given by δ : M n → s M j =1 M n , δ ( x ) = i [ a , x ] ⊕ i [ a , x ] ⊕ · · · ⊕ i [ a s , x ] . Recall that δ is called a derivation because it satisﬁes the Leibniz rule δ ( xy ) = δ ( x ) y + xδ ( y ) . Inparticular, L = δ ∗ δ . The ﬁxed-point algebra is N := { x ∈ M n | T t ( x ) = x , ∀ t ≥ } = { x ∈ M n | Lx = 0 } = { a , . . . , a s } ′ . We denote by E N be the conditional expectation onto N ﬁx .For two states ρ and σ with tr( ρ ) = tr( σ ) , the relative entropy is deﬁned as D ( ρ k σ ) = ( tr( ρ ln ρ − ρ ln σ ) , if supp ( ρ ) ⊆ supp ( σ )+ ∞ , otherwise , L. GAO, M. JUNGE, AND H. LI where supp ( ρ ) (resp. supp ( σ ) ) is the support projection of ρ (resp. σ ). The Fisher information(also called entropy production) is I ( ρ ) = tr( Lρ log ρ ) . Deﬁnition 3.1.

We say T t satisﬁes λ -modiﬁed logarithmic Sobolev inequalities ( λ -MLSI) for λ > if for all state ρ λD ( ρ k E N ( ρ )) ≤ I ( ρ ) . We say T t satisﬁes λ -complete logarithmic Sobolev inequalities ( λ -CLSI) for λ > if for all m ≥ , T t ⊗ id M m satisﬁes λ -MLSI. Remark 3.2.

As a matter of simplicity the results in this work is stated for environments givenby matrix algebras. As the proof will show the auxiliary matrix algebra M m can be replaced by anyﬁnite von Neumann algebra M with a speciﬁed trace. It was proved in [GJL20] that a symmetric quantum Markov semigroup on matrix algebra isalways a transference of a classical Markov semigroup on a compact Lie group with sub-Laplacian asthe generator. Recall that for a Riemannian manifold M , a Hörmander system is a ﬁnite family ofvector ﬁelds X = { X , ..., X s } such that for some global constant l X , the set of iterated commutators(no commutator if k = 1 ) [ ≤ k ≤ l X { [ X j , [ X j , ..., [ X j k − , X j k ]]] | ≤ j , · · · , j k ≤ s } spans the tangent space T p M at every point p ∈ M . We denote ∆ X = P sj =1 X ∗ j X j as the sub-Laplacian where X ∗ j is the adjoint operator of X j with respect to L ( M, µ ) and µ is the volumeform of M . Lemma 3.3 ([GJL20]) . Let T t : M n → M n be a symmetric quantum Markov semigroup. Thereexists a connected compact Lie group G , a unitary representation u : G → M n and a Hörmandersystem X = { X , · · · , X d } of a compact connected Lie group G such that the following diagramcommute (3.2) L ∞ ( G, µ ; M n ) S t ⊗ id M n −→ L ∞ ( G, µ ; M n ) ↑ π ↑ π M n T t −→ M n . where L ∞ ( G, µ ; M n ) denotes the matrix-valued function on G and π : M n → L ∞ ( G, µ ; M n ) denotesthe transference map π ( x )( g ) = u ( g ) ∗ xu ( g ) . We brieﬂy describe the construction, as it will be used in later discussion (See [GJL20, Lemma4.10, 5.1] for detailed proof). Let { a , · · · , a r } be the self-adjoint elements in the (3.1). Denote u m = i ( M n ) s.a. as the Lie algebra of the unitary group U ( M m ) . Then X = { ia , · · · , ia r } generates aLie subalgebra g of u m which by basically Lie’s second theorem (see also [GJL20, Lemma 4.10]) is theLie algebra of connected compact Lie group G . Let u be the unitary representation induced by theLie algebra embedding g ⊂ u m . One can show that the ∗ -homomorphism π : M n → L ∞ ( G, µ ; M n ) satisﬁes that X j ⊗ id M n ( π ( x )) = − iπ ([ a j , x ]) , ∆ X ⊗ id M m ( π ( ρ )) = π ( L ( ρ )) , which yields the intertwining relation of the semigroups (3.2).From the above intertwining relation, we can view T t as a sub-semigroup for the matrix valuedsemigroup S t ⊗ id M n . In particular, when t → ∞ , we have the commutation relation for theconditional expectations ( E G ⊗ id M n ) ◦ π = π ◦ E N , EOMETRIC APPROACH TOWARDS

CLSI where E G f = ( Z G f dµ )1 G is the expectation on G and µ be the normalized Haar measure over G .In particular, the conditional expectation onto the ﬁxed-point subalgebra N = u ( G ) ′ is given by E N ( ρ ) = Z G u ( g ) ∗ ρu ( g ) dµ ( g ) . By Carathéodory’s Theorem (see e.g. Proposition 4.9 in [Wat18]), there exist ﬁnitely many elements { g j } mj =1 ⊂ G such that for every ρ ∈ M n , E N ( ρ ) = m X j =1 α j u ( g j ) ∗ ρu ( g j ) , (3.3)where { α j } is a ﬁnite probability distribution s.t. P α j = 1 and α j ≥ . Furthermore, we have CLSI( T t ) ≥ CLSI( S t ) . This transference does not apply to MLSI because T t is a restriction of thematrix-valued ampliﬁcation S t ⊗ id n . We are now ready to prove the main theorem of this paper. Theorem 3.4.

Let T t = e − tL : M n → M n be symmetric quantum Markov semigroup. Suppose L isa transferred Lindbladian of a sub-Laplacian ∆ X on a connected compact Lie group G given by theHörmander system on X = { X , · · · , X s } . Then CLSI( L X ) ≥ Csmd X ( d X + 1) . where C X is some constant depending on X and d X is the diameter of G in the Carnot-Caratheodorydistance induced by X .Proof. Recall d H be the horizontal distance deﬁned in (1.6) and denote d X := sup g ,g ∈ G d H ( g , g ) as the horizontal diameter. Without loss of generality, we can always assume that X = { X , ..., X s } forms orthonormal set with respect to the Riemmannian metric . Namely, for any point g ∈ G and λ k ∈ R , | X k λ k X k | g = X k | λ k | . Let { α j } be the probability given in (3.3). Deﬁne recursively β = α + d CC ( g , g ) and for ≤ j ≤ m − , β j +1 = β j + α j + d H ( g j , g j +1 ) . ( m + 1 is viewed as .) Then β m ≤ m X j =1 α j + X j d H ( g j , g j +1 ) ≤ md X . We split the interval I j = [ β j , β j +1 ] = [ β j , β j + α j ] ∪ [ β j + α j , β j +1 ] := I j (1) ∪ I j (2) into intervals of length | I j (1) | = α j and | I j (2) | = d H ( g j , g j +1 ) . Consider the new transference map π : M n → ℓ m ∞ ( M n ) deﬁned by π ( ρ )( j ) = u ( g j ) ∗ ρu ( g j ) . Let E µ ( f ) = P mj =1 α j f ( j ) be the expectation on ℓ m ∞ . Then we have E N ( ρ ) = E µ ( π ( ρ )) ,D ( ρ || E N ( ρ )) = tr( ρ log ρ ) − tr( ρ log E µ ( π ( ρ )) = D ( π ( ρ ) k E µ ( π ( ρ )) . Let γ j : [0 , d H ( g j , g j +1 )] → G be a piecewise smooth horizantal path such that γ j (0) = g j , γ j ( d H ( g j , g j +1 )) = g j +1 , γ ′ j ( t ) ∈ T γ j ( t ) H , for a.e. t ∈ (0 , d H ( g j , g j +1 ) and | γ ′ ( t ) | = 1 of unit length in the Riemmannian metric. Then wedeﬁne the piecewise smooth matrix-valued function ˜ ρ : M m → L ∞ ([0 , β m ] , M n ) , ˜ ρ ( t ) = ( u ( g j ) ∗ ρu ( g j ) , if t ∈ I j (1) u ( γ j ( t − β j − α j )) ∗ ρu ( γ j ( t − β j − α j )) , if t ∈ I j (2) . Denote E ρ = ( R ρdt )1 as the mean and take ˜ σ := E ˜ ρ . By the chain rule [JLR19, Lemma 3.4] andthe non-negativity of the relative entropy D ( π ( ρ ) k ˜ σ ) = D ( π ( ρ ) k E π ( ρ )) + D ( E π ( ρ ) k ˜ σ ) ≥ D ( π ( ρ ) k E π ( ρ )) (3.4)Moreover D ( π ( ρ ) k ˜ σ ) = m X j =1 α j D ( u ( g j ) ∗ ρu ( g j ) k ˜ σ ) ≤ D (˜ ρ k ˜ σ ) . (3.5)Let C := CLSI([0 , β m ] , dx ) be the CLSI constant on the interval [0 , β m ] . We obtain that D (˜ ρ k ˜ σ ) = D (˜ ρ k E ˜ ρ ) ≤ C I (˜ ρ ) = 1 C Z β m tr (cid:16) ˜ ρ ′ ( t ) J ln˜ ρ ( t ) (˜ ρ ′ ( t )) (cid:17) dx Let X t = γ ′ j ( t − β j ) ∈ T γ j ( t − β j ) H and g t = γ j ( t − β j ) . Denote φ u : g → i ( M n ) s.a. be the Lie algebrahomomorphism induced by u . For t ∈ I j (2) , we have ˜ ρ ′ ( t ) = dds ( u ( g t + s ) ∗ ρu ( g t + s )) | s =0 = u ( g t ) ∗ ( − φ u ( X t ) ρ + ρφ u ( X t )) u ( g t )= u ( g t ) ∗ ( − iY t ρ + iρY t ) u ( g t )= − u ( g t ) ∗ ( i [ Y t , ρ ]) u ( g t ) . where Y t = iφ u ( X t ) are self-adjoint operators. Then we observe that I (˜ ρ ) = Z S j I j (2) tr (cid:16) ˜ ρ ′ ( t ) J log˜ ρ ( t ) (˜ ρ ′ ( t )) (cid:17) dt . and for each j and t ∈ I j (2) , tr (cid:16) ˜ ρ ′ ( t ) J log˜ ρ ( t ) ˜ ρ ′ ( t ) (cid:17) = Z ∞ tr (cid:16) γ j ( t − β j ) ∗ i [ Y t , ρ ] γ j ( t − β j )( γ j ( t − β j ) ∗ ργ j ( t − β j ) + r ) − · γ j ( t − β j ) ∗ i [ Y t , ρ ] γ j ( t − β j )( γ j ( t − β j ) ∗ ργ j ( t − β j ) + r ) − (cid:17) dr , = Z ∞ tr (cid:0) i [ Y t , ρ ]( ρ + r ) − i [ Y t , ρ ]( ρ + r ) − (cid:1) dr Note that φ u ( X k ) = a k . Suppose that for each t ∈ I j (2) , Y ( t ) = P sk =1 λ k ( t ) a k and hence X t = P sk =1 λ k ( t ) X k | γ j ( t − β j ) . The horizontal path γ j has constant unit speed | X t | γ j ( t ) ≤ and hence P k λ k ( t ) ≤ . Take ω kr = ( ρ + r ) − i [ a k , ρ ]( ρ + r ) − . We have I (˜ ρ ) = m X j =1 Z I j (2) Z ∞ tr (cid:0) i [ Y t , ρ ]( ρ + r ) − i [ Y t , ρ ]( ρ + r ) − (cid:1) drdt = m X j =1 Z I j (2) Z ∞ s X k,l =1 λ k ( t ) λ l ( t ) tr (cid:0) i [ a k , ρ ]( ρ + r ) − i [ a l , ρ ]( ρ + r ) − (cid:1) drdt = m X j =1 Z I j (2) Z ∞ s X k,l =1 λ k ( t ) λ l ( t ) τ ( ω kr ω lr ) drdt EOMETRIC APPROACH TOWARDS

CLSI ≤ m X j =1 Z I j (2) Z ∞ s ( X k λ k ( t ) ) s X k =1 tr( ω kr ω kr ) drdt (by Cauchy-Schwarz inequality) = s | X j d CC ( g j , g j +1 ) | Z ∞ s X k =1 tr( ω kr ω kr ) dr = smd X I L ( ρ ) Combining the estimates above, we have D ( ρ k E ( ρ )) = D ( π ( ρ ) k E π ( ρ )) ≤ D ( π ( ρ ) k E ˜ ρ ) ≤ D (˜ ρ k E ˜ ρ ) ≤ C I (˜ ρ ) ≤ smd X C I L ( ρ ) . Note that the constant C = CLSI([0 , β m ] , dt ) = β − m CLSI([0 , , dt ) . Thus we prove the CLSIconstant CLSI( L X ) = CLSI([0 , ,dx ) smd X ( d X +1) . Finally, for non-orthonormal linearly independent X ′ k s, thechange of basis adds another multiplicative constant and completes the proof.Using the noncommutative change of measure in [JLR19, Theorem 4.1], we obtain the existenceof CLSI constant for all ﬁnite dimensional quantum Markov semigroup satisﬁes detailed balancecondition (see e.g. [JLR19] for detailed deﬁntion.). Corollary 3.5.

Let T t = e − Lt : M n → M n be a quantum Markov semigroup GNS-symmetric withrespect to a full rank state σ . Then T t satisﬁes the λ - CLSI constant for some λ > .Proof. By [JLR19, Theorem 4.1], we know the optimal CLSI constant of every GNS-symmetricsemigroup T t is comparable to the optimal CLSI constant of a trace symmetric semigroup ˜ T t .The argument in this section applies to complete Beckner inequalities C p SI , see [Li20] for deﬁnitionsof C p SI . Corollary 3.6.

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Zentrum Mathematik, Technische Universität München, 85748 Garching, Germany

Email address , Li Gao: [email protected]

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Email address , Marius Junge: [email protected]

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Email address , Haojian Li:, Haojian Li: