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.
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 - C p SI constant for some λ > . References [Bar17] Ivan Bardet. Estimating the decoherence time using non-commutative functional inequalities. arXivpreprint arXiv:1710.01039 , 2017.[Bau14] Fabrice Baudoin. Sub-laplacians and hypoelliptic operators on totally geodesic riemannian foliations. arXiv preprint arXiv:1410.3268 , 2014.[BCL +
19] Ivan Bardet, Angela Capel, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. On the modified log-arithmic sobolev inequality for the heat-bath dynamics for 1d systems. arXiv preprint arXiv:1908.09004 ,2019.[BÉ85] Dominique Bakry and Michel Émery. Diffusions hypercontractives. In
Seminaire de probabilités XIX1983/84 , pages 177–206. Springer, 1985.[BGJ20a] Michael Brannan, Li Gao, and Marius Junge. Complete logarithmic sobolev inequalities via ricci curvaturebounded below. arXiv preprint arXiv:2007.06138 , 2020.[BGJ20b] Michael Brannan, Li Gao, and Marius Junge. Complete logarithmic sobolev inequalities via ricci curvaturebounded below ii. arXiv e-prints , pages arXiv–2008, 2020.[BGKT19] Fabrice Baudoin, Erlend Grong, Kazumasa Kuwada, and Anton Thalmaier. Sub-laplacian comparisontheorems on totally geodesic riemannian foliations.
Calculus of Variations and Partial Differential Equa-tions , 58(4):1–38, 2019.[BGL13] Dominique Bakry, Ivan Gentil, and Michel Ledoux.
Analysis and geometry of Markov diffusion operators ,volume 348. Springer Science & Business Media, 2013.[CM17] Eric A Carlen and Jan Maas. Gradient flow and entropy inequalities for quantum markov semigroupswith detailed balance.
Journal of Functional Analysis , 273(5):1810–1869, 2017.[CM20] Eric A Carlen and Jan Maas. Non-commutative calculus, optimal transport and functional inequalities indissipative quantum systems.
Journal of Statistical Physics , 178(2):319–378, 2020. [CRF20] Ángela Capel, Cambyse Rouzé, and Daniel Stilck França. The modified logarithmic sobolev inequal-ity for quantum spin systems: classical and commuting nearest neighbour interactions. arXiv preprintarXiv:2009.11817 , 2020.[DB14] Payam Delgosha and Salman Beigi. Impossibility of local state transformation via hypercontractivity.
Communications in Mathematical Physics , 332(1):449–476, 2014.[DPWS02] B De Pagter, H Witvliet, and FA Sukochev. Double operator integrals.
Journal of Functional Analysis ,192(1):52–111, 2002.[DR20] Nilanjana Datta and Cambyse Rouzé. Relating relative entropy, optimal transport and fisher information:a quantum hwi inequality. In
Annales Henri Poincaré , pages 1–36. Springer, 2020.[GJL20] Li Gao, Marius Junge, and Nicholas LaRacuente. Fisher information and logarithmic sobolev inequalityfor matrix-valued functions. In
Annales Henri Poincaré , volume 21, pages 3409–3478. Springer, 2020.[GKS76a] Vittorio Gorini, Andrzej Kossakowski, and Ennackal Chandy George Sudarshan. Completely positivedynamical semigroups of n-level systems.
Journal of Mathematical Physics , 17(5):821–825, 1976.[GKS76b] Vittorio Gorini, Andrzej Kossakowski, and Ennackal Chandy George Sudarshan. Completely positivedynamical semigroups of n-level systems.
Journal of Mathematical Physics , 17(5):821–825, 1976.[GR21] Li Gao and Cambyse Rouzé. Spectral methods for entropy contraction coefficients. arXiv preprint , 2021.[Gro75a] Leonard Gross. Hypercontractivity and logarithmic sobolev inequalities for the clifford-dirichlet form.
Duke Mathematical Journal , 42(3):383–396, 1975.[Gro75b] Leonard Gross. Logarithmic sobolev inequalities.
American Journal of Mathematics , 97(4):1061–1083,1975.[Gro14] Leonard Gross. Hypercontractivity, logarithmic sobolev inequalities, and applications: a survey of surveys.
Diffusion, quantum theory, and radically elementary mathematics , 47:45–73, 2014.[GT19] Erlend Grong and Anton Thalmaier. Stochastic completeness and gradient representations for sub-riemannian manifolds.
Potential Analysis , 51(2):219–254, 2019.[JLR19] Marius Junge, Nicholas LaRacuente, and Cambyse Rouzé. Stability of logarithmic sobolev inequalitiesunder a noncommutative change of measure. arXiv preprint arXiv:1911.08533 , 2019.[KT13a] Michael J Kastoryano and Kristan Temme. Quantum logarithmic sobolev inequalities and rapid mixing.
Journal of Mathematical Physics , 54(5):052202, 2013.[KT13b] Michael J Kastoryano and Kristan Temme. Quantum logarithmic sobolev inequalities and rapid mixing.
Journal of Mathematical Physics , 54(5):052202, 2013.[Led04] Michel Ledoux. Spectral gap, logarithmic sobolev constant, and geometric bounds.
Surveys in differentialgeometry , 9(1):219–240, 2004.[Li20] Haojian Li. Complete sobolev type inequalities. arXiv, 2020.[Lin76] Goran Lindblad. On the generators of quantum dynamical semigroups.
Communications in MathematicalPhysics , 48(2):119–130, 1976.[LJL20] Haojian Li, Marius Junge, and Nicholas LaRacuente. Graph hörmander systems. arXiv preprintarXiv:2006.14578 , 2020.[LOZ10] P. Lugiewicz, R. Olkiewicz, and B. Zegarlinski. Ergodic properties of diffusion-type quantum dynamicalsemigroups.
J. Phys. A , 43(42):425207, 14, 2010.[OV00] Felix Otto and Cédric Villani. Generalization of an inequality by talagrand and links with the logarithmicsobolev inequality.
Journal of Functional Analysis , 173(2):361–400, 2000.[PP86] Mihai Pimsner and Sorin Popa. Entropy and index for subfactors. In
Annales scientifiques de l’Ecolenormale supérieure , volume 19, pages 57–106, 1986.[RD19] Cambyse Rouzé and Nilanjana Datta. Concentration of quantum states from quantum functional andtransportation cost inequalities.
Journal of Mathematical Physics , 60(1):012202, 2019.[Wat18] John Watrous.
The theory of quantum information . Cambridge University Press, 2018.[Wir18] Melchior Wirth. A noncommutative transport metric and symmetric quantum markov semigroups asgradient flows of the entropy. arXiv preprint arXiv:1808.05419 , 2018.[WZ20] Melchior Wirth and Haonan Zhang. Complete gradient estimates of quantum markov semigroups. arXivpreprint arXiv:2007.13506 , 2020.
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: