Characterisation of gradient flows for a given functional

Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\alpha \beta }$$\end{document}gαβ on M such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_\beta = g_{\alpha \beta } X^\alpha $$\end{document}Yβ=gαβXα? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds.


Introduction
This paper deals with the following general question: Let X α P ΓpT M q be a vector field and Y β P ΓpT ˚M q be a co-vector field on a smooth manifold M .Does there exist a smooth Riemannian metric g αβ on M such that Y β " g αβ X α ?1 Clearly, this is not always true: X α and Y β will have to satisfy some compatibility conditions.Firstly, X α and Y β need to have the same set of zeroes (critical points).Secondly, at all other points m P M , they need to satisfy X α Y α | m ą 0. A third (and slightly less obvious) compatibility condition is obtained by differentiating the equation Y β " g αβ X α : at each critical point m P M there should exist a scalar product ḡαβ P T mM b S T mM such that ∇ α Y γ | m " ḡβγ ∇ α X β | m for some (equivalently, any) connection ∇ α .This condition does not hold automatically: it represents a compatibility constraint on X α and Y β with a natural interpretation in some examples below.
While these three conditions are clearly necessary, it is not obvious that they are also sufficient.The main result of this paper shows that this is indeed the case, under mild smoothness and non-degeneracy assumptions; namely, at all critical points, we require non-degeneracy of the derivative of Y β and we assume that X α and Y β are real analytic in suitable local coordinates; cf.Section 2 for the details.
Theorem 1.1 (Main result).Let X α P ΓpT M q and Y β P ΓpT ˚M q satisfy Assumption 2.1 below.Then there exists a metric g αβ P ΓpT ˚M b T ˚M q satisfying Y β " g αβ X α if and only if the following conditions hold: piq For all m P M with Y β | m ‰ 0 we have X α Y α | m ą 0; piiq For all m P M with Y β | m " 0 we have X α | m " 0; piiiq For all m P M with Y β | m " 0 there exists a scalar product ḡαβ P T mM b S T mM such that The choice of the connection ∇ in (iii) is arbitrary.
We shall also prove a variant of this result where X α and Y β are of class C k`1 for some k P N. In this case, the metric g αβ is of class C k ; see Theorem 2.6 below.
While Theorem 1.1 is of independent interest, our motivation comes from an open question on gradient flow structures for dissipative quantum systems, that will be discussed below.
Let us first briefly sketch the structure of the proof.To prove the sufficiency of conditions (i)-(iii), it suffices to construct a local metric around every point of M .The global metric can then be constructed using a partition of unity.Around non-critical points the construction is straightforward: in local coordinates, it corresponds to constructing a positive definite matrix that maps one given vector to another one.However, it is not trivial to construct a smooth metric satisfying Y β " g αβ X α in a neighbourhood of a critical point.
To solve this problem, we assume that the sought metric has a power series expansion in a suitable chart around the critical point.We then derive an infinite hierarchy of tensor equations, which express power series coefficients of degree N in terms of coefficients of degree at most N ´1 for N ě 1. Solvability of the lowest order equation is guaranteed by compatibility condition (iii).We then prove that higher order equations can be solved iteratively.Moreover, the norms of the solutions are exponentially bounded in the degree, which allows us to construct a convergent power series that satisfies the desired equation in a neighbourhood of the critical point.
Application to gradient structures.Consider now the special case where Y P ΓpT ˚M q is the derivative of a smooth function f P C 8 , i.e, Y β " ∇ β f .Then our question becomes: Does there exist a smooth Riemannian metric g αβ such that X is the gradient of f with respect to the metric g, i.e., X α " g αβ ∇ β f ?In other words, the question is whether the ODE 9 u " ´Xpuq on M can be formulated as a gradient flow equation 9 uptq " ´∇f `uptq ˘for a suitable Riemannian metric.Our main result yields necessary and suffcient conditions.
Gradient flows describe motion in the direction of steepest descent of the function f in the geometry defined by the metric g.The identification of an ODE as a gradient flow equation is often fruitful, as there are powerful techniques available for the analysis of gradient flows [1].
As an application of our main result, we address an open question on the gradient flow structure of finite-dimensional dissipative quantum systems.To put this result into context, let us first discuss the corresponding classical setting.
Classical Markov semigroups.Consider an irreducible continuous-time Markov chain on a finite set X with transition rates q xy ě 0 for x, y P X with x ‰ y.The associated Markov semigroup pP t q tě0 is a C 0 -semigroup of positive operators on R X that preserves the constant functions.Its infinitesimal generator L : R X Ñ R X is given by `Lψ ˘pxq :" ÿ yPX q xy `ψpyq ´ψpxq ˘.
As time evolves, the marginal law of the Markov chain describes a curve pµ t q tą0 in P ˚pX q, the simplex of probability densities with positive density.It evolves according to the Kolmogorov forward equation (KFE) where `L˚µ ˘pxq " ÿ y‰x µpyqq yx ´µpxqq xy for µ P PpX q.Let π P P ˚pX q be the unique stationary distribution.It is well known and easy to verify that the relative entropy Ent π pµq :" ÿ xPX µpxq log ´µpxq πpxq decreases along trajectories of the KFE.Much more is true if the Markov chain is reversible, i.e., the detailed balance condition π x q xy " π y q yx holds for all x ‰ y.Equivalently, this means that the generator L is selfadjoint in the Hilbert space L 2 pX , πq.In this case, it was shown in [17,18] that the KFE can be written as the gradient flow equation of Ent π with respect to a Riemannian metric on P ˚pX q.The associated Riemannian distance is given by a discrete dynamical optimal transport problem, in the spirit of the Benamou-Brenier formulation for the Wasserstein distance [4].This gradient flow structure is a discrete version of the Wasserstein gradient flow structure for the Fokker-Planck equation discovered by Jordan, Kinderlehrer, and Otto [15].This construction has been the starting point for the development of discrete Ricci curvature based on geodesic convexity with applications to functional inequalities [11,20,12,13,10] It was shown by Dietert [9] that the reversibility assumption is also necessary: if the KFE can be written as gradient flow equation for Ent π with respect to some Riemannian metric on P ˚pX q, then the underlying Markov chain is necessarily reversible.Combined with the results from [17,18], this result characterises reversible Markov chains as exactly those that admit a gradient flow structure for the relative entropy Ent π .
In this paper we provide a noncommutative analogue of this result.
Quantum Markov semigroups.Let pP t q tě0 be a quantum Markov semigroup on a finitedimensional C ˚-algebra A, i.e., pP t q tě0 is a C 0 -semigroup of linear operators on A such that P t 1 " 1 and the operators P t are completely positive, i.e., P t b I n is a positive operator on A b M n pCq for all n ě 1. (Here, 1 P A denotes the unit element, and I n denotes the identity operator on the algebra of n ˆn-matrices M n pCq.)The infinitesimal generator of pP t q tě0 will be denoted by L .Let pP : t q tě0 be the adjoint semigroup with respect to the duality pairing xA, By " TrrA ˚Bs.This is a C 0 -semigroup of completely positive and trace-preserving linear operators with generator L : .In particular, the operators P : t map the set of density matrices P :" tρ P A : ρ ě 0 and Trrρs " 1u into itself.Here we restrict our attention to the ergodic setting: we assume that there exists a unique stationary state, i.e., a unique density matrix σ P P satisfying L : σ " 0. We shall assume that σ is invertible.
The non-commutative analogue of the KFE is the Lindblad equation B t ρ t " L : ρ t .It is well known [22,23] that the von Neumann relative entropy H σ pρq :" Trrρplog ρ ´log σqs decreases along solutions to this equation.Moreover, following the earlier works [6,19], it was shown in [7,21] that the Lindblad equation B t ρ " L : ρ can be written as gradient flow equation for H σ under the condition of gns-detailed balance.This condition means that the generator L is selfadjoint with respect to the weighted L 2 -type scalar product xA, By gns σ :" TrrσA ˚Bs named after Gelfand, Naimark, and Segal.As in the discrete setting above, the associated Riemannian metric is related to a dynamical optimal transport problem.
It is now natural to ask whether the condition of gns-detailed balance is also necessary for the existence of a gradient flow structure for the von Neumann relative entropy.However, it was shown in [8] that a different symmetry condition is necessary, namely the condition of bkm-detailed balance.This condition corresponds to the selfadjointness of L with respect to another weighted L 2 -type scalar product xA, By bkm σ :" Trrσ 1´s A ˚σs Bs ds, named after Bogoliubov, Kubo, and Mori.As the condition of bkm-detailed balance is strictly weaker than gns-detailed balance [8], there was a gap between the known necessary and sufficient conditions.As an application of Theorem 1.1 we prove the following result, which closes this gap.Theorem 1.2.Let L be the generator of an ergodic quantum Markov semigroup on a finite dimensional C ˚-algebra A, and let σ P P `be its stationary state.The following statements are equivalent: (1) The operator L is selfadjoint with respect to the bkm scalar product x¨, ¨ybkm σ .(2) There exists a Riemannian metric on the interior of P for which the Lindblad equation 9 ρ t " L : ρ t is the gradient flow equation of the von Neumann relative entropy H σ .
Structure of the paper.Section 2 contains the main result and a reformulation of the result in the gradient case.The proof of the main result is contained in Section 3, except for the construction of the local metric, which is presented in Section 4. Section 5 deals with the construction of a metric of class C k under the assumption that the fields X α and Y β are of class C k`1 .The application to quantum Markov semigroups is contained in Section 6.

Main results
Let X α P ΓpT M q be a vector field and Y β P ΓpT ˚M q be a co-vector field on a smooth manifold M .Let N Y :" tm P M : Y | m " 0u be the set of critical points of Y .
In the sequel we impose the following mild assumptions on the fields X α and Y β .
Assumption Using the notation introduced above, we restate our main result (Theorem 1.1) for the convenience of the reader.
Theorem 2.3 (Main result).Let X α P ΓpT M q and Y β P ΓpT ˚M q satisfy Assumption 2.1.Then there exists a smooth metric g αβ P ΓpT ˚M b T ˚M q satisfying Y β " g αβ X α , if and only if the following conditions hold: piiiq For all m P N Y there exists a scalar product ḡαβ P T mM b S T mM , such that where ∇ α is an arbitrary connection.
Remark 2.4.As the necessity of the three conditions has been discussed above, it remains to prove their sufficiency.This will be done in Section 3 below.
In the special case where the co-vector field Y α :" ∇ α F P ΓpT ˚M q is the derivative of a scalar function f : M Ñ R, the above result admits a convenient reformulation.Assuming that f attains its minimum at a unique critical point m P M , the next results shows that property (iii) above is equivalent to the symmetry and positivity of the linearised map Λ : The relevant scalar product is given by the Hessian of f .Corollary 2.5 (Gradient case).Let f P C 8 pM q be a function and X α P ΓpT M q be a vector field, such that X α and Y α :" ∇ α f satisfy Assumption 2.1.Suppose that Y has a unique zero, m P M , at which f attains its minimum.Then there exists a Riemannian metric if and only if the following conditions hold: (iii) The linear map Λ :" ∇ α X β | m : T mM Ñ T mM is positive and symmetric with respect to the Hessian scalar product h αβ :" Proof.It is clear that the conditions (i) and (ii) match the corresponding conditions in Theorem 2.3.Suppose now that condition (iii) from Theorem 2.3 holds, for some scalar product ḡαβ P T mM b S T mM .We have to show that for all Z α , W α P T mM, and To show this, note that pΛZq α " Z γ ∇ γ X α " Z γ ḡαδ h δγ for Z α P T mM .Hence, for W α P T mM , we see that the expression is invariant under interchanging Z and W , which proves the desired symmetry.Moreover, this expression implies that Conversely, suppose that condition (iii) of the corollary holds.For all Z α , W α P T mM it follows that h αβ pΛZq α W β " r g αβ Z α W β for a positive and symmetric tensor r Since r g αβ is positive and symmetric and h αδ is invertible, ḡαβ defines a scalar product.Moreover, we have the desired identity ∇ α X β | m " ḡβγ h αγ , which completes the proof.
In the special case were Y β is the derivative of a scalar function f , the existence of a metric satisfying ∇ β f " g αβ X α was proved in [3] on the complement of the set of critical points.The existence of a metric with the desired property on the whole manifold was stated as an open question [3, Question 1].Subsequently, under an additional assumption, which corresponds to piiiq in Theorem 2.3, the existence of a continuous extension of g αβ to all of M was obtained in [5]; cf.Section 5 below for more details.However, the metric constructed [5] is in general not differentiable, even if the fields X α and Y β are smooth; see Example 5.2 below.
Here we show that C k -regularity of the metric can be obtained if the fields X α and Y β are assumed to be of class C k`1 .
Theorem 2.6 (Existence of a metric of class C k ).Let X α and Y β be of class C k`1 on M for some k P N and assume that ∇ α Y β | m is non-degenerate for all m P N Y for some (equivalently, any) connection ∇.Then there exists a metric g αβ of class C k on M satisfying Y β " g αβ X α if and only if conditions piq, piiq, and piiiq of Theorem 2.3 hold.
The proof of this result will be given in Section 5 below.It relies on the construction based on tensor equations that we develop in the proof of Theorem 2.3.

Proof of the main result
Our main result (Theorem 2.3) relies on two local versions of this result.First we construct a local solution around any non-critical point m P M zN Y .In the special case were Y β is the derivative of a scalar function, a different construction of a metric away from critical points was carried out in [3]; see Section 5 below.Theorem 3.1 (Local solutions around non-critical points).Suppose that X α P ΓpT M q and Y β P ΓpT ˚M q satisfy X α Y α | m ą 0 for some m P M .Then there exists a neighbourhood U of m and a smooth local metric for all m P U .
Proof.Since X α Y α | m ą 0, we have Y α | m ‰ 0. Therefore, we can complete the co-vector field e 1 α :" Y α P T ˚M to a dual frame E :" pe 1 α , . . ., e n α q in a neighbourhood V of m, i.e., pe 1  α | m , . . ., e n α | m q is a basis of T mM for all m P V .The coordinates of X α with respect to this frame are given by X j :" X α e j α : V Ñ R for j " 1, . . ., n.Since X 1 | m ą 0, the set U :" V X tX 1 ą 0u is still a neighbourhood of m.Let us define X : U Ñ R n´1 and f : U Ñ R by X :" pX 2 , . . ., X n q, f :" We then define the bilinear form g αβ in coordinates G " pg ij q n i,j"1 as where I n is the identity matrix.Since the matrix G is symmetric, the bilinear form g is symmetric as well.To verify that G ą 0, we write as desired.To complete the proof, note that the coordinates of Y α are given by Y 1 " 1 and Y j " 0 for j ‰ 1.Consequently, The second local version of Theorem 2.3 concerns the construction of a smooth local metric in a neighbourhood of a critical point.Theorem 3.2 (Local solutions around critical points).Let X α P ΓpT M q and Y β P ΓpT ˚M q satisfy Assumption 2.1.Suppose that X α | m " Y α | m " 0 for some m P M , and suppose that there exists a scalar product ḡ P T mM b S T mM , such that Then there exists a neighbourhood U of m and a smooth local metric The proof of Theorem 3.2 is the main challenge of this paper and will be carried out in section 4.
We now show that the main result (Theorem 2.3) follows readily from the local Theorems 3.1 and 3.2 using a partition of unity argument; see, e.g., [14,Theorem 1.131] for the existence of a partition of unity.
Proof of Theorem 2.3.The local results Theorems 3.1 and 3.2 guarantee that for any m P M there exists a neighbourhood U m and a local metric g αβ defined on U m , such that the desired identity Let tf k u kPN be a partition of unity subordinated to the cover tU m : m P M u of the manifold M , i.e., there exists a locally finite open covering tV k u kPN of M , such that each V k is contained in U m k for some m k P M , each function f k : M Ñ R is nonnegative and smooth and its support is contained in V k , and we have ř kPN f k pmq " 1 for all m P M (where the sum is finite for each m).We then define As g αβ is a finite convex combination of the scalar products g αβ m k , it is a scalar product.By linearity, g αβ satisfies the desired equation X α " g αβ Y β .

Local solutions around critical points
In this section we give the proof of Theorem 3.2, which deals with the construction of the metric around critical points.
Fix m P M and let ϕ : U Ñ Ω be a coordinate chart which maps a neighbourhood U of m onto an open set Ω Ď R n .Using this chart we can identify the vector field X α P ΓpT M q defined on U Ď M with the function r X α : Ω Ñ V :" R n , where r X α :" X α ˝ϕ´1 .Similarly, the co-vector field Y β P ΓpT ˚M q defined on U Ď M can be identified with a function r and the metric g αβ P ΓpT ˚M b S T ˚M q can be identified with a function r In the remainder of this section, we will work on a fixed chart and remove the tildes to lighten notation.
4.1.Motivation of the tensor equations.Let x P Ω be such that Y β | x " 0, and suppose that the identity X α " g αβ Y β holds in a neighbourhood of x.For N P N and all indices c 1 , . . ., c N P t1, . . ., nu we will derive a system of equations that the partial derivatives Taking partial differentives B c for c P t1, . . ., nu yields Taking second order derivatives, we find, for c 1 , c 2 P t1, . . ., nu, As Y b | x " 0, the first term on the right-hand side vanishes, and we infer that the tensor of first-order derivatives T ab c :" B c g ab is a solution to the system where we use the shorthand notation Since Y b " 0, the term with |S| " N vanishes.Thus, the derivatives of order pN ´1q, given by T ab c 1 ¨¨¨c N´1 :" B c 1 ¨¨¨B c N´1 g ab solve the system where U cb :" B c Y b , and rNszS Y b depends on (derivatives of) X and Y , and on derivatives of g of order at most N ´2.The notation T ab c 1 ¨¨¨q c i ¨¨¨c N means that the index c i is removed.
The identity (4.1) suggests an iterative scheme to construct a local solution g αβ to the equation X α " g αβ Y β around a critical point x P U as a power series The idea is to define, for N " 0, T ab :" ḡab , where ḡ P T x M b S T x M is the scalar product satisfying which exists by assumption.Higher order Taylor coefficients T ab c 1 ...c N are then constructed by iteratively solving a system of tensor equations of the form (4.1).
Section 4.2 deals with the existence of a solution to these equations.The construction and the convergence of the iterative scheme is contained in Section 4.3.

4.2.
Solving the tensor equations.We start by formulating an explicit solution to the tensor equation (4.1) of order N " 2.
Lemma 4.1.Let V be a finite-dimensional vector space, and let R α γδ P V b pV ˚bS V ˚q and U αβ P V ˚b V ˚be given.We assume that U aβ is invertible with inverse U αβ P V b V .Then the tensor T αβ γ P pV b V q b V ˚defined by Proof.The fact that T αβ γ " T βα γ follows readily from the definition.To show that (4.2) holds, note that by definition of T , Relabeling indices on the right-hand side and using the symmetry of R, we observe that the second term in (4.3) equals the third term in (4.4), and the second term in (4.4) equals the third term in (4.3).Summing these identities, we thus obtain (4.2).
We also need the following multilinear generalisation.
Lemma 4.2.Fix N ě 2. Let V be a finite-dimensional vector space, and let R α γ 1 ¨¨¨γ N P V b pV ˚qbsN and U αβ P V ˚b V ˚be given.We assume that U αβ is invertible with inverse U αβ P V b V .Then the tensor T αβ γ 1 ¨¨¨γ N´1 P V bs2 b pV ˚qbspN ´1q defined by Proof.The fact that T belongs to V bs2 b pV ˚qbspN ´1q follows readily from the definition.To show that (4.6) holds, note that This yields the result, as the first term has the desired form, and the second term cancels against the third term, as can be seen by renaming indices pα 1 , γ 1 j q into pδ, βq.

4.3.
Iterative construction of the power series & Proof of Theorem 3.2.We now place ourselves in the setting of Theorem 3.2.Thus, let X α P ΓpT M q and Y β P ΓpT ˚M q satisfy Assumption 2.1, and suppose that X α | m " Y α | m " 0 for some fixed m P M .We assume that there exists a scalar product ḡ P T mM b S T mM satisfying Our goal is to construct the local metric g αβ around m as a convergent power series centered at x " ϕp mq.We now present the definition of its coeffients T ab c 1 ¨¨¨c N , which is motivated by the equations (4.1).Our computations will be performed in a fixed chart ϕ : U Ñ Ω around m which satisfies Assumption 2.1.and then define T αβ γ P pV b S V q b V ˚as the solution to the system Here we use the shorthand notation T c S :" T c i 1 ¨¨¨c i k for S :" ti 1 , . . ., i k u with i µ ‰ i ν for µ ‰ ν.Then we define the tensor T αβ γ 1 ¨¨¨γ N´1 P V b S pN ´1q b pV ˚qb S 2 as the solution to the system Remark 4.4.The nondegeneracy assumption on the derivative ∇ α Y β | m is crucially used in this construction, as the application of Lemmas 4.1 and 4.2 requires the invertibility of U αβ .
Our next aim is to show that the power series converges and defines a Riemannian metric in a neigbourhood of x.For this purpose we equip the spaces V bk b pV ˚qbℓ with the norm where W b 1 ¨¨¨b ℓ a 1 ¨¨¨a k are the coordinates of W β 1 ...β ℓ α 1 ¨¨¨α k in the standard basis of R n .For brevity, let us write r N :" }R α γ 1 ¨¨¨γ N } 8 and t N :" }T αβ γ 1 ¨¨¨γ N } 8 .We then obtain the following crucial growth bound on the power series coefficients.Lemma 4.5.There exist constants C, p ă 8 such that t N ď CN !p N for all N ě 1.
Proof.Recall that we work in a chart for which Assumption 2.1 holds.Therefore, the real analyticity assumption implies that there exist constants C 1 , q ă 8 such that Using the bounds on the power series coefficients from (4.8) and the definitions of T and R from (4.5) and (4.7), we obtain the following relations between the norms r k and t k : where r K ă 8 depends on K and n.Using these estimates we shall now prove the desired result by induction.
We thus assume, for some N ě 0, that the desired inequality t k {k! ď Cp k holds for all k ď N , with suitable constants C, p ă 8.We will now show that t N `1{pN `1q! ď Cp N `1.Indeed, using the inequalities above and the induction assumption, we obtain Assuming, without loss of generality, that C ě 1 and p ą q, this yields By choosing p sufficiently large, the last term in brackets can be made smaller than pC 1 r Kqq ´1.This yields the result.Corollary 4.6.There exists a neigbourhood U Q x, such that the power series converges for all x P U , its inverse defines a Riemannian metric, and the equality X α | x " g αβ Y β | x holds for all x P U .
To verify that g αβ defines a metric, note first that g ab " g ba by construction.To show that g αβ is positive definite when x is close enough to x, it suffices to note that g αβ | x " ḡαβ is positive definite and the map x Þ Ñ g αβ | x is continuous.
Since the tensor fields X α , Y β , and g αβ are given by convergent power series, and since by assumption, it is enough to verify that all derivatives at x coincide, i.e., for all N P N and all c 1 , . . ., c N P t1, . . ., nu.To prove this identity, we use the notation from Definition 4.3, to obtain at x " x, To obtain the third equality, we use that x is a critical point, together with the definitions of R, T , and U in Definition 4.3.In the final step we use the tensor equation (4.6).
The proof of Theorem 3.2 is now complete, as the metric g αβ constructed above can be pushed back to M using the chart ϕ.

Construction of a metric of class C k
Let X α be a vector field and Y β be a co-vector field on a smooth manifold M .As before, let N Y :" tm P M : Y | m " 0u be the set of critical points of Y .In this section we weaken the regularity assumptions on X and Y .In Proposition 5.1 these fields are assumed to be merely differentiable.Subsequently we provide the proof of Theorem 2.6, which deals with fields of class C k`1 for k P N.
The following result, which does not require an iterative scheme, is known in the special case where Y β is the derivative of a scalar function [3,5].In this setting, the existence of a metric with the desired property away from critical points is proved in [3].The construction of the metric below is taken from there.It relies on the unique decomposition of vector fields into a component parallel to X and a component annihilating Y , which only works away from critical points.The proof of the existence of a continuous extension to all of M is adapted from [5].
Proposition 5.1 (Existence of a continuous metric).Let X α and Y β be differentiable fields on M and suppose that the bilinear form ∇ α Y β | m is non-degenerate for all m P N Y for some (equivalently, any) connection ∇.Suppose that the following conditions hold: piiiq For all m P N Y there exists a scalar product ḡm P T m M b S T m M , such that where ∇ α is an arbitrary connection.
Then there exists a continuous metric g αβ on M satisfying Y β " g αβ X α .
Proof.Let m P M zN Y be a non-critical point, hence X| m ‰ 0 and Y | m ‰ 0 by piiq.The assumption piq implies that we have the direct sum decomposition T m M " Y K m ' spantX m u, hence every vector Z P T m M can be uniquely decomposed as Let g " g αβ be an arbitrary continuous metric on M satisfying g| m " ḡm at all critical points m P N Y .Following [3], we construct a perturbation of r g as follows: for Z, W P ΓpT M q.In view of piq, it readily follows that g defines a continuous metric on M zN Y .It remains to show that r g can be continuously extended to all of M .It will be convenient to use abstract index notation.Taking into account that xZ p1q , Y y " xZ, Y y and xW p1q , Y y " xW, Y y, it follows from the definition that Introducing the deficit R β :" Y β ´gαβ X α , we can write Fix a critical point m P N Y .Using assumptions piiq and piiiq we shall show that r g| m Ñ g| m as m Ñ m, following the arguments in [5].Using the notation from Section 4, we shall perform a Taylor expansion of the terms in (5.2) in a fixed chart, where m P M corresponds to x P R n .As X and Y are differentiable, and x is a critical point, it follows from piiq that Since ḡab pxq is a scalar product, there exists κ ą 0 such that ḡab pxqv a v b ě κ|v| 2 for all v P R n .Furthermore, ∇ b X a is non-degenerate by assumption piiiq and the non-degeneracy assumption on ∇ b Y a .Therefore, |∇ b X a v| 2 ě r κ|v| 2 for some constant r κ ą 0. Using these inequalities, together with piiiq, yields which bounds the denominator in (5.2) from below.As for the terms in the numerator, we first note that X a pxq " O `|x ´x| ˘and Y b pxq " O `|x ´x| ˘.These bounds trivially imply that R b pxq " O `|x ´x| ˘as well, but this is not sufficient.The key point of the proof is that this bound can be improved.Indeed, using piiiq and the continuity of g at x, we obtain R b pxq " `Yb ´gab X a ˘pxq " ∇ c Y b pxqpx ´xq c ´gab pxq∇ c X a pxqpx ´xq c `op|x ´x|q " `ḡ ab pxq ´gab pxq ˘∇c X a pxqpx ´xq c `op|x ´x|q " o `|x ´x| ˘. (5.5) It now follows from (5.4) and (5.5) together with the bounds on X and Y , that the fractions in (5.2) vanish as x Ñ x.This shows that r g can be continuously extended to M by setting r g ab pxq :" ḡab pxq.
While the metric r g constructed in the proof of Proposition 5.1 is continuous, it is not in general differentiable, even if the background metric g αβ and the vector fields X α and Y β are smooth.Here is an explicit counterexample.
Example 5.2.Let M be the open unit ball in R 2 .We work in cartesian coordinates.Set Xpxq " Y pxq " x for x P M , and consider the background metric g αβ defined by g ab pxq :" " 1 `x2 0 0 1  for x " px 1 , x 2 q P M .Since g is smooth and g| 0 " I, it is a valid background metric.An explicit computation yields The latter is a non-constant homogeneous function and as such discontinuous at x " 0, thus r g αβ does not belong to C 1 .Theorem 2.6 shows that better regularity properties can be obtained by a careful choice of the background metric g αβ .In the following proof we define g αβ by making use of the construction in Section 4, which yields improved bounds on the deficit R β :" Y β ´gαβ X α around critical points.This allows us to construct a metric r g αβ of class C k whenever X α and Y β are of class C k`1 .
Proof of Theorem 2.6.First we note that the necessity of conditions piq and piiq was already observed in the introduction.The necessity of piiiq follows, even when g is assumed to be merely continuous, from the expansions for X and Y in (5.3) and the expansion gpxq " gpxq`o `|x´x| ȋn local coordinates around a critical point x.Therefore it remains to show that these three conditions are also sufficient.
As in Proposition 5.1, we construct a metric of the form (5.2) on the non-critical set M zN Y : where R β :" Y β ´gαβ X α denotes the deficit, and g αβ is a background metric on M that will be carefully chosen below.As noted before, it is immediate to verify that the desired identity Y β " r g αβ X α holds on M zN Y .Construction of the background metric.Fix m P N Y .As in Section 4 we work in a fixed coordinate chart where m corresponds to x P R n .In these local coordinates we then define the background metric by g ab m pxq :" for x in a small neigbourhood around x.It is crucial that we use the tensors T αβ γ 1 ¨¨¨γ N that were constructed in Definition 4.3.Note that T αβ γ 1 ¨¨¨γ N is indeed well defined for N ď k due to our assumption that X α and Y β are k `1 times continuously differentiable.As T αβ is positive definite, it follows that pg mq αβ defines a metric in a neighbourhood of x.
For each cricitical point m, this construction yields a Riemannian metric in an open neighbourhood V m of m.By the non-degeneracy assumption, we may assume that the sets tV mu mPN Y are pairwise disjoint.Let U m be an open neighbourhood of m satisfying U m Ď V m and let f m : M Ñ r0, 1s be a smooth function on M satisfying f m| U m " 1 and f m| M zV m " 0. Using an arbitrary metric pg ˚qαβ on M and the function r f :" 1 ´ř mPN Y f m, we define The crucial property of this background metric g, which will be used below, is that the deficit for all m P N Y and p ď k `1.This follows from the definition of the tensors T ab c 1 ¨¨¨c N using the computation (4.10).
Differentiability of the metric.To verify that r g αβ is k times continuously differentiable, we will show that the partial derivatives can be continuously extended from M zN Y to all of M for p ď k.In view of (5.6) this yields the desired result.We use the notation from Definition 4.3, thus B c S " B c i 1 ¨¨¨B c iq for S " ti 1 , . . ., i q u Ď t1, . . ., pu with i µ ‰ i ν for µ ‰ ν.With this notation we have where X p is the collection of all possible partitions of t1, . . ., pu.
Let us fix a critical point m P N Y and let x be the corresponding point in R n .Recall from (5.4) Furthermore, since X α | x " 0 and Y α | x " 0, Taylor's formula yields, for any S Ď t1, . . ., pu, To estimate B c S R α pxq we use the crucial point, observed in (5.8), that our background metric is constructed so that B c S R β pxq " 0 when |S| ď k `1.This ensures that Combining these bounds, we estimate the right-hand sides of U αβ c 1 ...cp and V αβ c 1 ...cp as follows: 1 where the exponents u and v satisfy Since |S 1 | `¨¨¨`|S ℓ | `|A| `|B| " p for tS 1 , . . ., S ℓ , A, Bu P X p , we obtain u ě k ´p `1 ě 1 and v ě k ´p `1 ě 1, which shows that Therefore U αβ c 1 ...cp and V αβ c 1 ...cp can be extended continuously to all of M by assigning the value zero for m P N Y .
6. Application to Quantum Markov Semigroups (QMS) In this section prove Theorem 1.2 by an application of Corollary 2.5.As in Section 1, let L be the generator of an ergodic quantum Markov semigroup pP t q tě0 on a finite dimensional C ˚-algebra A with stationary state σ P P `.The manifold under consideration is the set of strictly positive density matrices P `" tρ P P : ρ ą 0u.
Note that P `is a relatively open subset of the affine space σ `T Ď A, where T :" tA P A : A " A ˚, TrrAs " 0u.
Therefore, the tangent space of P `can be naturally identified with T .We will apply Corollary 2.5 to the triple pM, f, Xq where M :" P `and f : P `Ñ R, f pσq :" H σ pρq " Trrρplog ρ ´log σqs, X : P `Ñ T, Xpρq :" L : ρ.
The functional H σ is everywhere strictly positive, except at its global minimum σ.Moreover, a standard computation shows that, for ρ P P `and A P T , B ε ˇˇε"0 H σ pρ `εAq " Trrplog ρ ´log σqAs, (6.1) Therefore, the differential of H σ is everywhere non-zero except at σ, so that we are in a position to apply Corollary 2.5.
Recall that we are interested in the bkm-scalar product on A given by xA, By bkm σ :" TrrA ˚Mσ pBqs, where M σ pBq :" for A, B P A. We refer to [2] for a recent study of this scalar product.It is natural to also consider the inner product on A defined in terms of the inverse operator M ´1 σ : A Ñ A given by xA, By Ć We will use the following simple result.
Lemma 6.1.For a linear operator K : A Ñ A the following assertions are equivalent: (1) K is selfadjoint with respect to the inner product x¨, ¨ybkm σ .(2) K : is selfadjoint with respect to the inner product x¨, ¨yĆ bkm σ .Proof.It is readily seen that both assertions are equivalent to M σ K " K : M σ .
The entropy production functional I σ : P `Ñ R is defined by I σ pρq " ´Trrplog ρ ´log σqL : ρs for ρ P P `.Note that indeed d dt H σ pP : t ρq " ´Iσ pP : t ρq.The functional I σ is nonnegative and convex [22,23].The following result shows the strict positivity of the entropy production (except at stationarity) under the assumption of bkm-detailed balance.Proposition 6.2.Let L be the generator of an ergodic quantum Markov semigroup on a finite dimensional C ˚-algebra A, with invariant state σ P P `.If bkm-detailed balance holds, then I σ pρq ą 0 for all ρ P P `with ρ ‰ σ.
Proof.As remarked above, I σ is nonnegative and convex.Therefore, it suffices to show that I σ is strictly convex at its minimum σ.Take A P T with A ‰ 0.
Proof of Theorem 1.2.First we will translate condition piiiq of Corollary 2.5, namely the selfadjointness of the linearised operator Λ with respect to the Hessian scalar product h.We claim that this is exactly the assumption of bkm-detailed balance in our setting.
Indeed, since L : is a linear operator, its linearisation Λ : T Ñ T appearing in condition piiiq is simply given by Λ :" L : .Moreover, the Hessian of ρ Þ Ñ H σ pρq at ρ " σ is given by hpA, Bq :" B ε ˇˇε"0 B η ˇˇη"0 H σ `σ `εA `ηB ˘" ż for A, B P T .Hence the Hessian scalar product in condition piiiq is the Ć bkm-scalar product.Thus, condition piiiq is the Ć bkm-selfadjointness of L : .By Lemma 6.1 this corresponds to the bkm-selfadjointness of L , which is the assumption of bkm-detailed balance.
This argument shows that the necessity of bkm-detailed balance for the gradient flow structure follows from Corollary 2.5.To show that bkm-detailed balance is also sufficient, we note first that condition piiq of Corollary 2.5 is simply the stationarity condition L : σ " 0, which holds by assumption.Thus, it remains to show that condition piq of Corollary 2.5 is implied by the assumption of bkm-detailed balance.Then the existence of the gradient flow structure follows by applying Corollary 2.5 in the opposite direction.
Hence, condition piq is the strict positivity of the entropy production I σ pρq or ρ ‰ σ, which follows from the assumption of bkm-detailed balance by Proposition 6.2.

Definition 4 . 3 (
The power series coeffients T ab c 1 ¨¨¨c N ).Write U αβ :" ∇ α Y β | m for brevity.‚ Initialisation: We define the initial tensor T αβ P V b S V of our iteration as T ab :" ḡab .‚ Iterative step (special case N " 2): We first define R α γδ P V b pV ˚bS V ˚q by R a cd :" B c B d X a ´T ab B c B d Y b ) which yields a C k metric g αβ on M satisfying g αβ | m " pg mq αβ | m for all m P N Y and m P U m.
2.1.piq (Non-degeneracy) The bilinear form ∇ α Y β | m is non-degenerate for all m P N Y for some (equivalently, any) connection ∇. piiq (Real analyticity) For all m P N Y there exists a neighbourhood U m Q m, an open set Ω Ă R n , and a coordinate chart ϕ m : U m Ñ Ω, such that the fields r Y β " ∇ α Y β for m P N Y .For the same reason, the choice of the connection is irrelevant in (iii) in the following result.