# Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

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## Abstract

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional \(C^*\)-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

## Keywords

Non-commutative optimal transport Functional inequalities Lindblad equation Gradient flow## 1 Introduction

- (1)
The gradient flow of the Dirichlet energy in \(L^2\);

- (2)
The gradient flow of the Boltzmann entropy in the space of probability measures endowed with the 2-Kantorovich metric.

Our focus in this paper is on developing the relations between (1) and (2) in the non-commutative setting with the aim of proving functional inequalities relevant to the study of the rate of approach to equilibrium for quantum Markov semigroups, in close analogy with what has been accomplished along these lines in the classical setting in recent years.

In order not to obscure the main ideas we shall work in a finite-dimensional setting and postpone the infinite-dimensional extension to a future work. The finite-dimensional case is of direct interest in quantum information theory, and the essential aspects of our new results are interesting even in this setting where they can be explained to a wider audience that is not thoroughly familiar with the Tomita–Takesaki theory. We now briefly describe the content of the paper. Any unfamiliar terminology is explained in the next subsection, but hopefully many readers will not need to look ahead.

The central object of study in this paper is a quantum Markov semigroup (QMS) \((\mathscr {P}_t)_{t>0}\) on \(\mathcal {A}\), a finite-dimensional \(C^*\)-algebra containing the identity \({\mathbf{1}}\). That is, for each *t*, \(\mathscr {P}_t{\mathbf{1}}= {\mathbf{1}}\) and \(\mathscr {P}_t\) is completely positive. The generators \({\mathscr {L}}\) of such semigroups have been characterized in [24, 31].

We are concerned with the case in which there is a unique faithful invariant state \(\sigma \) for the dual semigroup; i.e., \(\mathscr {P}_t^\dagger \sigma = \sigma \) for all *t*. The paper [47] is an excellent source for the physical context and makes it clear that assuming that the invariant state \(\sigma \) is tracial, which we do not do, would preclude a great many physical applications. Let \({{\mathfrak {P}}}_+\) denote the space of faithful states. We would like to know, for instance, when there is a Riemannian metric *g* on \({{\mathfrak {P}}}_+\) such that the flow on \({{\mathfrak {P}}}_+\) given by the dual semigroup \((\mathscr {P}_t^\dagger )_{t>0}\) is the gradient flow driven by the relative entropy functional \({{\,\mathrm{Ent}\,}}_\sigma (\rho ) = {{\,\mathrm{Tr}\,}}[\rho (\log \rho - \log \sigma )]\) with respect to the Riemannian metric. In [10, 36], it is shown that when each \(\mathscr {P}_t\) is self-adjoint with respect to the Gelfand-Naimark-Segal (GNS) inner product induced on \(\mathcal {A}\) by \(\sigma \), this is the case. We constructed the metric using ideas from optimal mass transport, and showed that, as in the classical case, the framework provided an efficient means for proving functional inequalities. This has been taken up and further developed by other authors, in particular Rouzé and Datta [45, 46]. As in the classical case, Ricci curvature bounds are essential for the framework to be used to obtain functional inequalities. As shown in [10, 46], once one has Ricci curvature bounds, a host of functional inequalities follow. A central problem then is to prove such bounds. A main contribution of the present paper is a flexible framework for doing this. It turns out that there are many ways to write a given QMS generator \(\mathscr {L}\) (that is self-adjoint in the GNS sense) in “divergence form” for non-commutative derivatives. Each of the different ways of doing this can be associated to a Riemannian metric on \({{\mathfrak {P}}}_+\). Different ways of writing \(\mathscr {L}\) in divergence form may have advantages over others, for example in proving Ricci curvature bounds. Hence it is important to have as much flexibility here as possible. We shall use this flexibility to give new examples in which we can obtain sharp Ricci curvature bounds. The machinery is useful for other functionals and other flows; the methods of this paper are not by any means restricted to gradient flow for relative entropy, despite our focus on this example here in the introduction.

An interesting problem remains: For each way of writing \(\mathscr {L}\) in divergence form, we have a Riemannian metric. The formulas are different, but in principle, all of the metrics might be the same. That is, they might all be determined by \(\mathscr {L}\), and not the particular way of writing in divergence form, even though doing this one way or another may facilitate certain computations.

The problem of writing QMS as gradient flow for the relative entropy was also taken up independently by Mittnenzweig and Mielke [36], and although their framework is somewhat different, their approach also works in the case that each \(\mathscr {P}_t\) is self-adjoint with respect to the GNS inner product induced on \(\mathcal {A}\) by \(\sigma \). Here, we shall show that if \((\mathscr {P}_t)_{t\ge 0}\) can be written as gradient flow for \({{\,\mathrm{Ent}\,}}_\sigma \) with respect to some continuously differentiable Riemannian metric, then each \(\mathscr {P}_t\) is necessarily self-adjoint with respect to another inner product associated to \(\sigma \), the Boguliobov-Kubo-Mori (BKM) inner product. As we show, the class of QMS with this self-adjointness property is strictly larger than the class of QMS with the GNS self-adjointness property. Thus, there is at present an interesting gap between the known necessary condition for the construction of the Riemannian metric, and the known sufficient condition. Of course, in the classical setting, the two notions of self-adjointness coincide, and one has a pleasing characterization of reversible Markov chains in terms of gradient flow [15].

### 1.1 Notation

Let \(\mathcal {A}\) be finite-dimensional \(C^*\)-algebra containing the identity \({\mathbf{1}}\). In the finite-dimensional setting, all topologies one might impose on \(\mathcal {A}\) are equivalent, and \(\mathcal {A}\) is also a von Neumann algebra. In particular, it is generated by the projections it contains. We may regard any such algebra as a \(*\)-subalgebra of \({{\mathbb {M}}}_n({{\mathbb {C}}})\), the set of all complex \(n \times n\) matrices. Let \(\mathcal {A}_h\) be the subset of hermitian elements in \(\mathcal {A}\), and let \(\mathcal {A}_+ \subseteq \mathcal {A}\) denote the class of elements that are positive definite (i.e., \({{\,\mathrm{sp}\,}}(A) \subseteq (0,\infty )\) for \(A \in \mathcal {A}_+\). For \(\mathcal {A}= {{\mathbb {M}}}_n({{\mathbb {C}}})\) we write \(\mathcal {A}_+ = {{\mathbb {M}}}_n^+({{\mathbb {C}}})\).

Throughout this section we fix a positive linear functional \(\tau \) on \(\mathcal {A}\) that is *tracial* (i.e., \(\tau [AB] = \tau [BA]\) for all \(A, B \in \mathcal {A}\)) and *faithful* (i.e., \(A=0\) whenever \(\tau [A^*A] =0\)). Under these assumptions, \(\tau \) induces a scalar product on \(\mathcal {A}\) given by \(\langle {A,B}\rangle _{L^2(\mathcal {A},\tau )} = \tau [A^* B]\) for \(A, B \in \mathcal {A}\). In our applications, \(\tau \) will often be the usual trace \({{\,\mathrm{Tr}\,}}\) on \({{\mathbb {M}}}_n({{\mathbb {C}}})\) in which case the scalar product is the Hilbert–Schmidt scalar product, but it will be useful to include different situations, e.g., the trace induced by a non-uniform probability measure on a finite set.

A *state* on \(\mathcal {A}\) is a positive linear functional \(\varphi \) on \(\mathcal {A}\) such that \(\varphi ({\mathbf{1}}) =1\). If \(\varphi \) is a state, there is a uniquely determined \(\sigma \in \mathcal {A}\) such that \(\varphi (A) = \tau [\sigma A]\) for all \(A \in \mathcal {A}\). Note that \(\sigma \) is a *density matrix*; i.e., it is positive semidefinite and \(\tau [\sigma ] =1\). Let \({{\mathfrak {P}}}(\mathcal {A})\) denote the set of density matrices. We write \({{\mathfrak {P}}}_+(\mathcal {A}) = \{ \rho \in {{\mathfrak {P}}}(\mathcal {A}) : \rho \text { is positive definite} \}\). We will simply write \({{\mathfrak {P}}}= {{\mathfrak {P}}}(\mathcal {A})\) and \({{\mathfrak {P}}}_+ = {{\mathfrak {P}}}_+(\mathcal {A})\) if the algebra \(\mathcal {A}\) is clear from the context.

### Definition 1.1

*The relative modular operator*) Let \(\sigma ,\rho \in {{\mathfrak {P}}}_+\). The corresponding

*relative modular operator*\(\Delta _{\sigma ,\rho }\) is the linear transformation on \(\mathcal {A}\) defined by

*modular operator*corresponding to \(\sigma \), \(\Delta _\sigma \), is defined by \(\Delta _{\sigma } := \Delta _{\sigma ,\sigma }\).

Since \(\langle B, \Delta _{\sigma ,\rho } A\rangle _{L^2(\mathcal {A},\tau )} = \tau [(\sigma ^{1/2}B\rho ^{-1/2})^*(\sigma ^{1/2}A\rho ^{-1/2})]\) for all \(A,B\in \mathcal {A}\), the operator \(\Delta _{\sigma ,\rho }\) is positive definite on \(L^2(\mathcal {A},\tau )\). In case that \(\tau \) is the restriction of the usual trace \({{\,\mathrm{Tr}\,}}\) to \(\mathcal {A}\subseteq {{\mathbb {M}}}_n({{\mathbb {C}}})\), the operators \(\sigma \) and \(\rho \) are also positive density matrices in \({{\mathbb {M}}}_n({{\mathbb {C}}})\), and the same computations are valid for all \(A,B\in {{\mathbb {M}}}_n({{\mathbb {C}}})\). We may regard \(\Delta _\sigma \) as an operator on \({{\mathbb {M}}}_n({{\mathbb {C}}})\), equipped with the Hilbert–Schmidt inner product, and then, so extended, it is still positive definite.

We are interested in evolution equations on \({{\mathfrak {P}}}_+(\mathcal {A})\) that correspond to forward Kolmogorov equations for ergodic Markov processes satisfying a detailed balance condition, or in other words a reversibility condition, with respect to their unique invariant probability measure. Before presenting our results, we introduce the class of quantum Markov semigroups satisfying a detailed balance condition that are the focus of our investigation.

## 2 Quantum Markov Semigroups with Detailed Balance

*quantum Markov semigroup*on \(\mathcal {A}\) is a \(C_0\)-semigroup of operators \((\mathscr {P}_t)_{t \ge 0}\) acting on \(\mathcal {A}\), satisfying

- (1)
\(\mathscr {P}_t {\mathbf{1}}= {\mathbf{1}}\);

- (2)
\(\mathscr {P}_t\) is

*completely positive*, i.e., \(\mathscr {P}_t \otimes I_{{{\mathbb {M}}}_n({{\mathbb {C}}})}\) is a positivity preserving operator on \(\mathcal {A}\otimes {{\mathbb {M}}}_n({{\mathbb {C}}})\) for all \(n \in {{\mathbb {N}}}\).

*real*, i.e., \( (\mathscr {P}_t A)^* = \mathscr {P}_t A^*\) for all \(A \in \mathcal {A}\). Let \(\mathscr {P}_t^\dagger \) be the Hilbert–Schmidt adjoint of \(\mathscr {P}_t\) satisfying \(\tau [A^* \mathscr {P}_t^\dagger B] = \tau [(\mathscr {P}_t A)^*B]\) for all \(A, B \in \mathcal {A}\). It follows that \(\mathscr {P}_t^\dagger \) is trace-preserving and completely positive.

*Lindblad form*

### 2.1 Detailed Balance

The starting point of our investigations is the assumption that \((\mathscr {P}_t)_{t \ge 0}\) satisfies the condition of *detailed balance*.

In the commutative setting, if \(P = (P_{ij})\) is the transition matrix of a Markov chain on \(\{1,\ldots , n\}\) with invariant probability vector \(\sigma \), we say that detailed balance holds if \(\sigma _i P_{ij} = \sigma _j P_{ji}\) for all *i*, *j*. An analytic way to formulate this condition is that *P* is self-adjoint with respect to the weighted inner product on \({{\mathbb {C}}}^n\) given by \(\langle {f,g}\rangle _\sigma = \sum _{j=1}^n \sigma _j \overline{f_j}g_j\).

The following lemma of Alicki [1] relates some of the possible definitions of detailed balance; a proof may be found in [10].

### Lemma 2.1

Let \(\mathscr {K}\) be a real linear transformation on \(\mathcal {A}\). If \(\mathscr {K}\) is self-adjoint with respect to the \(\langle \cdot , \cdot \rangle _s\) inner product for some \(s\in [0,1/2)\cup (1/2,1]\), then \(\mathscr {K}\) commutes with \(\Delta _\sigma \), and \(\mathscr {K}\) is self-adjoint with respect to \(\langle \cdot , \cdot \rangle _s\) for all \(s\in [0,1]\), including \(s=1/2\).

As we have remarked, for a QMS \((\mathscr {P}_t)_{t\ge 0}\), each \(\mathscr {P}_t\) is real, and so \(\mathscr {P}_t\) is self-adjoint with respect to the GNS inner product if and only if it is self-adjoint with respect to the \(\langle \cdot , \cdot \rangle _s\) inner product for all \(s\in [0,1]\). However, if each \(\mathscr {P}_t\) is self-adjoint with respect to the KMS inner product, then it need not be self-adjoint with respect to the GNS inner product: There exist QMS for which each \(\mathscr {P}_t\) is self-adjoint with respect to the KMS inner product, but for which \(\mathscr {P}_t\) does not commute with \(\Delta _\sigma \), and therefore cannot be self-adjoint with respect to the GNS inner product. A simple example is provided in appendix B of [10]. The generators of QMS such that \(\mathscr {P}_t\) is self-adjoint with respect to the KMS inner product have been investigated by Fagnola and Umanita [20]. However, there is a third notion of detailed balance that is natural in the present context, namely the requirement that each \(\mathscr {P}_t\) be self-adjoint with respect to the Boguliobov–Kubo–Mori inner product:

### Definition 2.2

*BKM inner product*) The

*BKM inner product*is defined by

By what we have remarked above, if each \(\mathscr {P}_t\) is self-adjoint with respect to the GNS inner product, then each \(\mathscr {P}_t\) is self-adjoint with respect to the BKM inner product. However, as will be discussed at the end of this section, the converse is not in general true. The relevance of the BKM version of detailed balance is due to the following result that we show in Theorem 2.9: If the forward Kolmogorov equation for an ergodic QMS \((\mathscr {P}_t)_{t\ge 0}\) with invariant state \(\sigma \in {{\mathfrak {P}}}_+\) is gradient flow for the quantum relative entropy \({{\,\mathrm{Ent}\,}}_{\sigma }(\rho ) := \tau [\rho ( \log \rho - \log \sigma ) ]\) with respect to some continuously differentiable Riemannian metric on \({{\mathfrak {P}}}_+\), then each \(\mathscr {P}_t\) is self-adjoint with respect to the BKM inner product. The BKM inner product is closely connected to the relative entropy functional, and for this reason it appears in some of the functional inequalities that we consider in Sect. 11.

On the other hand, only when each \(\mathscr {P}_t\) is self-adjoint with respect to the GNS inner product do we have a construction of such a Riemannian metric. The same is true for other constructions of Riemannian metrics on \({{\mathfrak {P}}}_+\) for which QMS become gradient flow for \({{\,\mathrm{Ent}\,}}_{\sigma }(\rho )\), in particular see [36]. Since most of this paper is concerned with our construction and its consequences, we make the following definition:

### Definition 2.3

*Detailed balance*) Let \(\sigma \in \mathcal {A}\) be non-negative. We say that a quantum Markov semigroup \((\mathscr {P}_t)_{t\ge 0}\) satisfies the

*detailed balance condition*with respect to \(\sigma \) if for each \(t>0\), \(\mathscr {P}_t\) is self-adjoint with respect to the GNS inner product on \(\mathcal {A}\) induced by \(\sigma \), i.e.,

The following result gives the general form of the generator of quantum Markov semigroups on \(B(\mathscr {H})\) satisfying detailed balance. This result is due to Alicki [1, Theorem 3]; see [10] for a detailed proof.

### Theorem 2.4

### 2.2 Gradient Flow Structure for the Non-commutative Dirichlet Energy

*t*, \(\mathscr {P}_t\) is self-adjoint with respect to

*both*the GNS and the KMS inner products induced by \(\sigma \). Therefore, we may define a

*Dirichlet form*\(\mathscr {E}\) on \(\mathcal {A}\) by

*Kolmogorov backward equation*\(\partial _t A = \mathscr {L}A\) is a gradient flow equation for the energy \(\mathscr {E}(A,A)\) with respect to the chosen \(L^2\) metric.

The class of bilinear forms \(\mathscr {E}\) defined in terms of a self-adjoint QMS \((\mathscr {P}_t)_{t\ge 0}\) through (2.9) is, by definition, the class of *conservative completely Dirichlet forms* on \(\mathcal {A}\) in the specified inner product. The abstract Beurling–Deny Theorem, discussed in the next section, provides an intrinsic characterization of such bilinear forms.

Although Definition 2.3 might seem to suggest that the natural choice of the \(L^2\) metric is the one given by the GNS inner product, we shall show that in some sense it is the KMS inner product that is more natural: The Dirichlet form defined by (2.9) using the KMS inner product induced by \(\sigma \) can be expressed in terms of a “squared gradient”, and the associated non-commutative differential calculus will turn out to be very useful for investigating properties of the flow specified by \(\partial _t A = \mathscr {L}A\). A somewhat different construction leading to the representation of Dirichlet forms with respect to the KMS metric in terms of derivations has been given by Cipriani and Sauvageot [13]. Our “derivatives” are not always derivations, and this more general structure is suited to applications. Indeed, one of the first non-commutative Dirichlet forms to be investigated in mathematical physics, the *Clifford Dirichlet form* of Gross, is most naturally expressed in terms of a sum of squares of *skew derivations*. The flexibility of our framework will be essential to our later applications. In this part of the introduction, we present only some of the key computations in a simple setting involving derivations to explain the roles of the KMS inner product. Our more general framework will be presented in Sect. 4.

*partial derivative operators on*\(\mathcal {A}\):

### Proposition 2.5

### Proof

In the next section we show how the non-commutative differential calculus associated to the Dirichlet from \(\mathscr {E}\) allows us to write the corresponding *forward equation* as gradient flow for the relative entropy with respect to a Riemannian metric constructed in terms of this differential calculus.

### 2.3 A Gradient Flow Structure for the Quantum Relative Entropy

*quantum relative entropy functionals*\({{\,\mathrm{Ent}\,}}_{\sigma }: {{\mathfrak {P}}}_{+} \rightarrow {{\mathbb {R}}}\) given by

*forward equation*, can be formulated as the gradient flow equation for \({{\,\mathrm{Ent}\,}}_{\sigma }\) with respect to a suitable Riemannian metric on \({{\mathfrak {P}}}_{+}\). The construction of the Riemannian metric will make use of the “quantum directional derivatives” \(\partial _j\) introduced in the last subsection.

Since \({{\mathfrak {P}}}_{+}\) is a relatively open subset of the \({{\mathbb {R}}}\)-affine subspace \(\{ A \in \mathcal {A}_{h} : \tau [A] = 1 \}\), we may identify, at each point in \(\rho \in {{\mathfrak {P}}}_{+}\), its tangent space \(T_{\rho } {{\mathfrak {P}}}_{+}\) with \(\mathcal {A}_{0} := \{ A \in \mathcal {A}_{h} : \tau [A] = 0 \}\). The cotangent space \(T_{\rho }^\dagger {{\mathfrak {P}}}_{+}\) may also be identified with \(\mathcal {A}_{0}\) through the duality pairing \(\langle {A,B}\rangle = \tau [A B]\) for \(A, B \in \mathcal {A}_{0}\).

*differential*\(\mathrm {D}\mathcal {F}(\rho ) \in T_{\rho }^{\dagger }{{\mathfrak {P}}}_{+}\) is defined by \(\lim _{\varepsilon \rightarrow 0} \varepsilon ^{-1}(\mathcal {F}(\rho + \varepsilon A) - \mathcal {F}(\rho ) ) = \langle {A,\mathrm {D}\mathcal {F}(\rho )}\rangle \) for \(A \in T_{\rho }{{\mathfrak {P}}}_{+}\) (independently of the Riemannian metric \(g_{\rho }\)). Its

*gradient*\(\nabla _g \mathcal {F}(\rho ) \in T_{\rho }{{\mathfrak {P}}}_{+}\) depends on the Riemannian metric through the duality formula \(g_{\rho }(A, \nabla _g \mathcal {F}(\rho )) = \langle {A, \mathrm {D}\mathcal {F}(\rho )}\rangle \) for \(A \in T_{\rho }{{\mathfrak {P}}}_{+}\). It follows that \(\mathscr {G}_{\rho }\nabla _g \mathcal {F}(\rho ) = \mathrm {D}\mathcal {F}(\rho )\), or equivalently

*contraction operator*\(\# : (\mathcal {A}\otimes \mathcal {A}) \times \mathcal {A}\rightarrow \mathcal {A}\) defined by

### Lemma 2.6

### Proof

*i*, and \(\{E_\ell \}_\ell \) are the spectral projections, so that \(E_\ell E_m = \delta _{\ell m} E_\ell \) and \(\sum _\ell E_\ell = {\mathbf{1}}\). Observe that

The following result shows that the Kolmogorov forward equation \(\partial _t \rho = \mathscr {L}^\dagger \rho \) can be formulated as the gradient flow equation for \({{\,\mathrm{Ent}\,}}_{\sigma }\).

### Proposition 2.7

### Proof

In this paper we extend this result into various directions: we consider more general entropy functionals, more general Riemannian metrics, and nonlinear evolution equations.

### Remark 2.8

*Kolmogorov backward equation*

### 2.4 The Necessity of BKM-Detailed Balance

In the classical setting of irreducible finite Markov chain, Dietert [15] has proven that if the Kolmogorov forward equation for a Markov semigroup can be written as gradient flow for the relative entropy with respect to the unique invariant measure for some continuously differentiable Riemannian metric, then the Markov chain is necessarily reversible. That is, it satisfies the classical detailed balance condition.

### Theorem 2.9

Let \((\mathscr {P}_t)_{t\ge 0}\) be an ergodic QMS with generator \(\mathscr {L}\) and invariant state \(\sigma \in {{\mathfrak {P}}}_+\). If there exists a continuously differentiable Riemannian metric \((g_\rho )\) on \({{\mathfrak {P}}}_+\) such that the quantum master equation \(\partial \rho = \mathscr {L}^\dagger \rho \) is the gradient flow equation for \({{\,\mathrm{Ent}\,}}_\sigma \) with respect to \((g_\rho )\), then each \(\mathscr {P}_t\) is self-adjoint with respect to the BKM inner product associated to \(\sigma \).

### Proof of Theorem 2.9

We are unaware of any investigation of the nature of the class of QMS generators that are self-adjoint for the BKM inner product associated to their invariant state \(\sigma \). Therefore we briefly demonstrate that this class *strictly* includes the class of QMS generators that are self-adjoint for the GNS inner product associated to their invariant state \(\sigma \).

## 3 Beurling–Deny Theory in Finite-Dimensional von Neumann Algebras

In this section we recall some key results of Beurling–Deny theory that will be used in our construction of Dirichlet forms in Sect. 4. We present some proofs of known results for the reader’s convenience, especially when available references suppose a familiarity with the Tomita–Takesaki theory. However, Theorem 3.8, which singles out the KMS inner product, is new.

### 3.1 Abstract Beurling–Deny Theory

In this subsection, \(\mathcal {H}\) always denotes a *real* Hilbert space with inner product \(\langle \cdot ,\cdot \rangle \). Let \(\mathscr {P}\) be a cone in \(\mathcal {H}\). That is, \(\mathscr {P}\) is a convex subset of \(\mathcal {H}\) such that if \(\varphi \in \mathscr {P}\), then \(\lambda \varphi \in \mathscr {P}\) for all \(\lambda >0\). The cone \(\mathscr {P}\) is *pointed* in case \(\varphi \in \mathscr {P}\) and \(-\varphi \in \mathscr {P}\) together imply that \(\varphi =0\). In particular, a subspace of \(\mathcal {H}\) is a cone, but it is not a pointed cone.

### Definition 3.1

*Dual cone*) The

*dual cone*\(\mathscr {P}^{\circ }\) of a cone \(\mathscr {P}\) is the set

*self-dual*in case \(\mathscr {P}^\circ = \mathscr {P}\).

### Theorem 3.2

### Proof

To see that \(\varphi _+\) and \(\varphi _-\) are orthogonal, let \(\epsilon \in (-1,1)\), so that \((1+\epsilon )\varphi _+ \in \mathscr {P}\). It follows that \( \Vert \varphi _-\Vert ^2 = \Vert \varphi - \varphi _+\Vert ^2 \le \Vert \varphi - (1+\epsilon )\varphi _+\Vert ^2 = \Vert \varphi _-\Vert ^2 + 2\epsilon \langle \varphi _-,\varphi _+\rangle + \epsilon ^2 \Vert \varphi _+\Vert ^2\) which yields a contradiction for negative \(\epsilon \) sufficiently close to zero, unless \(\langle \varphi _-,\varphi _+\rangle = 0\). This proves existence of the decomposition. Now the fact that \(\varphi _- = \mathsf {P}_{\mathscr {P}}(-\varphi )\) follows from a theorem of Moreau [37], as does the uniqueness of the decomposition, though both points can be proved directly by variations on the arguments just provided. \(\square \)

### Definition 3.3

Let \(\mathcal {H}\) be a real Hilbert space with a non-empty self-dual cone \(\mathscr {P}\). For \(\varphi \) in \(\mathcal {H}\), define \(\varphi _+\) and \(\varphi _-\) as in Theorem 3.2. Then \(\varphi _+\) is the *positive part of*\(\varphi \), \(\varphi _-\) is the *negative part of*\(\varphi \), and \(|\varphi |:= \varphi _+ + \varphi _-\) is the *absolute value of*\(\varphi \). If \(\varphi _- = 0\), we write \(\varphi \ge 0\).

We next recall some elements of the abstract theory of symmetric Dirichlet forms. A *bilinear form* on a real Hilbert space \(\mathcal {H}\) is a bilinear mapping \(\mathscr {E}: \mathcal {D}\times \mathcal {D}\rightarrow {{\mathbb {R}}}\) where \(\mathcal {D}\subseteq \mathcal {H}\) is a linear subspace (called the *domain* of \(\mathscr {E}\)). We say that \(\mathscr {E}\) is *non-negative* if \(\mathscr {E}(\varphi ,\varphi )\ge 0\) for all \(\varphi \in \mathcal {D}\); *symmetric* if \(\mathscr {E}(\varphi ,\psi ) = \mathscr {E}(\psi ,\varphi )\) for all \(\psi , \psi \in \mathcal {D}\); *closed* if \(\mathcal {D}\) is complete when endowed with the norm \(\Vert \varphi \Vert _\mathscr {E}= (\Vert \varphi \Vert ^2 + \mathscr {E}(\varphi ,\varphi ))^{1/2}\); and *densely defined* if \(\mathcal {D}\) is dense in \(\mathcal {H}\).

### Definition 3.4

*Dirichlet form*) Let \(\mathcal {H}\) be a real Hilbert space with a non-empty self-dual cone \(\mathscr {P}\). A non-negative, symmetric, closed bilinear form \(\mathscr {E}\) on \(\mathcal {H}\) with dense domain \(\mathcal {D}\) is a

*Dirichlet form*in case \(|\varphi | \in {{\mathcal {D}}}\) for all \(\varphi \in {{\mathcal {D}}}\) , and

*operator*\(\mathscr {L}: \mathcal {D}_\mathscr {L}\subseteq \mathcal {H}\rightarrow \mathcal {H}\)

*associated to*\(\mathscr {E}\) is defined by

The following abstract result by Ouhabaz [40] characterizes the invariance of closed convex sets under the associated semigroup (in a more general setting that includes nonsymmetric Dirichlet forms).

### Theorem 3.5

- (1)
\(e^{t\mathscr {L}} \varphi \in \mathcal {C}\) for all \(\varphi \in \mathcal {C}\) and all \(t \ge 0\);

- (2)
\(\mathsf {P}_\mathcal {C}\varphi \in \mathcal {D}\) and \(\mathscr {E}(\mathsf {P}_\mathcal {C}\varphi , \varphi - \mathsf {P}_\mathcal {C}\varphi ) \le 0\) for all \(\varphi \in \mathcal {D}\).

Combining Theorems 3.2 and 3.5 we obtain the following result.

### Corollary 3.6

(Abstract Beurling–Deny Theorem) Let \(\mathcal {H}\) be a real Hilbert space with a non-empty self-dual cone \(\mathscr {P}\). Let \(\mathscr {E}\) be a non-negative, symmetric, closed bilinear form with domain \(\mathcal {D}\). Then, \(\mathscr {E}\) is a Dirichlet form if and only if \(e^{t\mathscr {L}}\varphi \ge 0\) for all \(t \ge 0\) and all \(\varphi \ge 0\).

### 3.2 Completely Dirichlet Forms

Let \(\mathscr {E}\) be a Dirichlet form on \(\big (\mathcal {A}, \langle {\cdot , \cdot }\rangle _{L^2_\mathrm{KMS}(\sigma )}\big )\) with the KMS inner product specified by a faithful state \(\sigma \). Here, the notion of Dirichlet form is understood with respect to the self-dual cone consisting of all positive semidefinite matrices belonging to \(\mathcal {A}\); see Lemma 3.10 below. Let \(\mathscr {P}_t = e^{t\mathscr {L}}\) where \(\mathscr {L}\) is the semigroup generator associated to \(\mathscr {E}\). Recall that the Dirichlet form \(\mathscr {E}\) is said to be *completely Dirichlet* in case for each *t*, \(\mathscr {P}_t\) is completely positive.

The condition that \(\mathscr {E}\) be completely Dirichlet may be expressed in terms of \(\mathscr {E}\) itself, permitting one to check the property directly from a specification of \(\mathscr {E}\).

*i*,

*j*)-entry is 1, with all other entries being 0. Alternatively, \(E_{ij}\) represents the linear transformation taking \(\mathbf{e}_j\) to \(\mathbf{e}_i\), while annihilating \(\mathbf{e}_k\) for \(k\ne j\). (Here \(\{\mathbf{e}_1,\dots ,\mathbf{e}_m\}\) is the standard orthonormal basis of \({{\mathbb {C}}}^m\).) It follows that \(E_{ij}E_{k\ell } = \delta _{jk}E_{i\ell }\). The general element of \(\mathcal {A}\otimes {{\mathbb {M}}}_m({{\mathbb {C}}})\) can be written as

A QMS \((\mathscr {P}_t)_t\) is not only completely positive; it also satisfies \(\mathscr {P}_t{\mathbf{1}}= {\mathbf{1}}\) for all *t*. This too may be expressed in terms of the Dirichlet form \(\mathscr {E}\): A Dirichlet form \(\mathscr {E}\) is *conservative* in case \(\mathscr {E}(A, {\mathbf{1}}) = 0\) for all \(A \in \mathcal {A}\), and one readily sees that this is equivalent to the condition that \(\mathscr {P}_t{\mathbf{1}}= {\mathbf{1}}\) for all *t*.

### 3.3 Moreau Decomposition with Respect to the Cone of Positive Matrices

Let \({{\mathbb {H}}}_n({{\mathbb {C}}})\) denote the set of self-adjoint \(n \times n\) matrices, which contains a distinguished pointed cone \(\mathscr {P}\), namely the cone of positive semidefinite matrices *A*. If we equip \({{\mathbb {H}}}_n({{\mathbb {C}}})\) with the Hilbert–Schmidt inner product \(\langle X,Y\rangle = {{\,\mathrm{Tr}\,}}[X Y]\), then \(\mathscr {P}\) is self-dual: for \(X\in {{\mathbb {H}}}_n({{\mathbb {C}}})\), \(\langle X,A\rangle \ge 0\) for all \(A\in \mathscr {P}\) if and only if \(\langle v,Xv\rangle \ge 0\) for all \(v \in {{\mathbb {C}}}^n\), as one sees by considering rank one projections and using the spectral theorem.

*spectral decomposition*\(X = X_{(+)} - X_{(-)}\) where

### Theorem 3.7

(Moreau decomposition for Hilbert–Schmidt) Let \(\mathcal {H}\) be \({{\mathbb {H}}}_n({{\mathbb {C}}})\) equipped with the Hilbert–Schmidt inner product, and let \(\mathscr {P}\) be the cone of positive semidefinite matrices. Then the spectral decomposition of \(X\in \mathcal {H}\) coincides with the decomposition of *X* into its positive and negative parts with respect to \(\mathscr {P}\).

### Proof

*v*in the range of \(X_+\), we have \(X_+ - \epsilon |v\rangle \langle v| \in \mathscr {P}\) for all sufficiently small \(\epsilon > 0\). Therefore,

*X*. Thus, the projectors onto the ranges of \(X_+\) and \(X_-\) are both spectral projectors of

*X*. Since \(X = X_+ - X_-\) it follows that \(X_+ = X_{(+)}\) and \(X_- = X_{(-)}\). \(\square \)

The situation is more interesting for other inner products on \({{\mathbb {H}}}_n({{\mathbb {C}}})\). Let \(\sigma \) be an invertible density matrix. For \(s\in [0,1]\), let \(\langle \cdot , \cdot \rangle _s\) be the inner product on \({{\mathbb {M}}}_n({{\mathbb {C}}})\) given by \(\langle A, B\rangle _s = {{\,\mathrm{Tr}\,}}[ A^*\sigma ^s B \sigma ^{1-s}]\).

### Theorem 3.8

Let \(\sigma \) be an invertible \(n\times n\) density matrix that is not a multiple of the identity. Then the cone \(\mathscr {P}\) of positive matrices in \({{\mathbb {H}}}_n({{\mathbb {C}}})\) is self-dual with respect to the inner product \(\langle \cdot , \cdot \rangle _s\) determined by \(\sigma \) if and only if \(s=\frac{1}{2}\).

### Proof

Let \(X\in {{\mathbb {H}}}_n({{\mathbb {C}}})\) and \(A\in \mathscr {P}\). Then \(\langle X,A\rangle _s = {{\,\mathrm{Tr}\,}}[ X\sigma ^s A \sigma ^{1-s}] = {{\,\mathrm{Tr}\,}}[ (\sigma ^{1-s} X\sigma ^s) A ]\). Therefore, \(\langle X,A\rangle _s \ge 0\) for all \(A\in \mathscr {P}\) if and only if \(\sigma ^{1-s} X\sigma ^s \in \mathscr {P}\). If \(\sigma ^{1-s} X\sigma ^s \in \mathscr {P}\), then \(\sigma ^{1-s} X\sigma ^s\) is self-adjoint, and hence \(\sigma ^{1-s} X\sigma ^s = \sigma ^{s} X\sigma ^{1-s}\), or, what is the same, \([\sigma ^{1-2s},X] = 0\). Let \(X := |v \rangle \langle v|\) with *v* chosen *not* to be an eigenvector of \(\sigma \). Then for \(s\ne \frac{1}{2}\), \([\sigma ^{1-2s},X] \ne 0\). Therefore, \(X\in \mathscr {P}\), but \(X\notin \mathscr {P}^\circ \). Hence, \(\mathscr {P}\) is not self-dual when \(\mathcal {H}\) is equipped with the inner product \(\langle \cdot , \cdot \rangle _s\) for \(s\ne \frac{1}{2}\).

*A*ranges over \(\mathscr {P}\), \(\sigma ^{1/4}A\sigma ^{1/4}\) ranges over \(\mathscr {P}\), and so \(\langle X,A\rangle _{1/2}\ge 0\) for all \(A\in \mathscr {P}\) if and only if \(\sigma ^{1/4} X\sigma ^{1/4} \in \mathscr {P}\). Again, since \(\sigma \) is invertible, this is the case if and only if \(X\in \mathscr {P}\). Hence, \(\mathscr {P}\) is self-dual for \(\langle \cdot , \cdot \rangle _{1/2}\), the KMS inner product. \(\square \)

The Moreau decomposition for the KMS scalar product can easily be obtained from Theorem 3.7 by a unitary transformation.

### Theorem 3.9

*X*in the decomposition according to \(\mathscr {P}\), \(X_+\), is given by

### Proof

We conclude the section by extending the results above to an arbitrary \(*\)-subalgebra \(\mathcal {A}\) of \({{\mathbb {M}}}_n({{\mathbb {C}}})\). Let \(\sigma \) be an invertible \(n\times n\) density matrix belonging to \(\mathcal {A}\).

### Lemma 3.10

Let \(\mathcal {H}\) be \(\mathcal {A}_{h}\) equipped with the KMS inner product induced by \(\sigma \), and let \(\mathscr {P}\) be the positive matrices in \({{\mathbb {M}}}_n({{\mathbb {C}}})\), and let \(\mathscr {P}_\mathcal {A}:= \mathscr {P}\cap \mathcal {A}\). Then \(\mathscr {P}_\mathcal {A}\) is self-dual in \(\mathcal {H}\).

### Proof

Let \(X \in \mathscr {P}_\mathcal {A}\). For any \(A \in \mathscr {P}_\mathcal {A}\) we have \(\sigma ^{1/2}A\sigma ^{1/2} \ge 0\), hence \( \langle {X,A}\rangle _{L^2_\mathrm{KMS}(\sigma )} = {{\,\mathrm{Tr}\,}}[X \sigma ^{1/2}A\sigma ^{1/2}] \ge 0 \), which shows that \(X \in \mathscr {P}_\mathcal {A}^\circ \).

Conversely, suppose that \(X \in \mathcal {A}_h\) belongs to \(\mathscr {P}_\mathcal {A}^\circ \). For every \(A \in \mathscr {P}_\mathcal {A}\) we then have \( {{\,\mathrm{Tr}\,}}[X \sigma ^{1/2} A \sigma ^{1/2}] = \langle {X,A}\rangle _{L^2_\mathrm{KMS}(\sigma )} \ge 0\). Since \(\sigma \) is invertible, it follows that \({{\,\mathrm{Tr}\,}}[X B] \ge 0\) for every \(B \in \mathscr {P}_\mathcal {A}\). Therefore, the spectrum of *X* is non-negative, which implies that *X* belongs to \(\mathscr {P}\) and hence to \(\mathscr {P}_\mathcal {A}\). \(\square \)

### Lemma 3.11

*X*be a self-adjoint element of \(\mathcal {A}\). Then the decomposition of

*X*with respect to \(\mathscr {P}_\mathcal {A}\) is given by \(X = X_+ - X_-\) where

### Proof

Let *X* be a self-adjoint element of \(\mathcal {A}\). Then by Theorem 3.9, \(\min \{\Vert X - A\Vert _{L^2_\mathrm{KMS}(\sigma )}\ : A \in \mathscr {P}\}\) is achieved at \(A = \sigma ^{-1/4}(\sigma ^{1/4}X\sigma ^{1/4})_{(+)}\sigma ^{-1/4}\), and since this belongs to \(\mathcal {A}\), this same choice of *A* also achieves the minimum in \(\min \{\Vert X - A\Vert _{L^2_\mathrm{KMS}(\sigma )}\ : A \in \mathscr {P}_\mathcal {A}\}\). \(\square \)

## 4 Construction of Dirichlet Forms on a Finite-Dimensional von Neumann Algebra

Motivated by the results in Sects. 2 and 3 we introduce a general framework in which various gradient flow structures can be studied naturally. This setting unifies and extends several previous approaches to gradient flows, in particular for reversible Markov chains on finite spaces [32, 35], the fermionic Fokker-Planck equation [8], and Lindblad equations with detailed balance [10, 36]

While the results in Sect. 2 show that the general QMS satisfying the \(\sigma \)-DBC can be represented in terms of a Dirichlet form specified in terms of derivations, our applications require us to work with representations for the generator \(\mathscr {L}\) in terms of “partial derivative operators” \(\partial _j\) that are not simply derivations. The reason is that, to obtain functional inequalities and sharp rates of convergence to equilibrium, it will be important to obtain commutation relations of the form \([\partial _j, \mathscr {L}] = -a \partial _j\) for \(a \in {{\mathbb {R}}}\). We shall demonstrate that such commutation relations may hold for the general class of representations introduced in this section, but not for the simpler representation in terms of derivations discussed in Sect. 2.

Our starting point is a finite-dimensional von Neumann algebra \(\mathcal {A}\) which we may regard as a subalgebra of \({{\mathbb {M}}}_n({{\mathbb {C}}})\) for some \(n\in {{\mathbb {N}}}\). On account of the finite-dimensionality of \(\mathcal {A}\), there is always a tracial positive linear functional \(\tau \) on \(\mathcal {A}\): One choice is the normalized trace \(\tau [A] = n^{-1}{{\,\mathrm{Tr}\,}}[A]\). However, if \(\mathcal {A}\) is commutative (hence isomorphic to \(\ell _n^\infty \)), there will be many other tracial positive linear functionals — any positive measure on \(\{1,\ldots , n\}\) specifies such a positive linear functional. In what follows, \(\tau \) will denote any faithful positive linear functional on \(\mathcal {A}\) that is tracial; i.e., such that \(\tau [AB] = \tau [BA]\) for all \(A,B\in \mathcal {A}\). Since \(\tau \) is faithful, every state \(\sigma \) on \(\mathcal {A}\) can be represented as \(\sigma (A) = \tau [\sigma A]\), where on the right side \(\sigma \in \mathcal {A}\subseteq {{\mathbb {M}}}_n({{\mathbb {C}}})\) is the \(n\times n\) density matrix belonging to \(\mathcal {A}\) determined by the state \(\sigma \).

The basic operation in terms of which we shall construct completely Dirichlet forms on \(\mathcal {A}\) has several components.

*compatible*in case for all \(A\in \mathcal {A}\),

*r*be a pair of \((\tau ,\tau _\mathcal {B})\)-compatible unital \(*\)-homomorphisms from \(\mathcal {A}\) into \(\mathcal {B}\). Then define the operator \(\partial _V: \mathcal {A}\rightarrow \mathcal {B}\) by

*r*are the identity, this reduces to (2.11). The following Leibniz rule shows that \(\partial _V\) is an \((\ell , {r})\)-skew derivation.

### Lemma 4.1

### Proof

### Remark 4.2

*r*are \((\tau ,\tau _\mathcal {B})\)-compatible, \(\ell (\sigma )\) and \(r(\sigma )\) are density matrices (with respect to \(\tau _\mathcal {B}\) on \(\mathcal {B}\)). The inner product that we use on \(\mathcal {B}\) is a KMS inner product based on both \(\ell (\sigma )\) and \(r(\sigma )\) defined in terms of the

*relative modular operator*\(\Delta _{\ell (\sigma ),r(\sigma )}\):

*two*pairs \((\ell ,{r})\) and \((\ell _*,{r}_*)\) of (\(\tau ,\tau _\mathcal {B}\))-compatible \(*\)-homomorphisms of \(\mathcal {A}\) into \(\mathcal {B}\), define \(\partial _V\) by (4.2), and define

*V*, \((\ell ,{r})\) and \((\ell _*,{r}_*)\) under which \(\mathscr {E}\) is a conservative completely Dirichlet form on \(\mathcal {A}\) equipped with the KMS inner product induced by \(\sigma \).

It is first of all necessary that the operator \(\mathscr {L}\) determined by \(\mathscr {E}\) through \(\mathscr {E}(A_1,A_2) = -\langle B,\mathscr {L}A\rangle _{L^2_\mathrm{KMS}(\sigma )}\) be real; i.e., \((\mathscr {L}(A))^* = \mathscr {L}A^*\). Since \(\langle A_1, A_2\rangle _{L^2_\mathrm{KMS}(\sigma )} = \langle A_2^*,A_1^*\rangle _{L^2_\mathrm{KMS}(\sigma )}\) for all \(A_1,A_2\in \mathcal {A}\), it is easily seen that \(\mathscr {L}\) is real if and only if \(\mathscr {E}(A_1,A_2) = \mathscr {E}(A_2^*,A_1^*)\) for all \(A_1,A_2\in \mathcal {A}\).

### Lemma 4.3

### Remark 4.4

One can satisfy (4.7) in a trivial way by taking \(\ell \), *r*, \(\ell _*\) and \({r}_*\) each to be the identity. Almost as trivially, one may take \(\ell _* = {r}\) and \({r}_* = \ell \). However, we shall see that one can also satisfy (4.7) with \(\ell _* = \ell \) and \({r}_* = {r}= I_\mathcal {B}\) with a non-trivial \(*\)-homomorphism \(\ell \); see the discussion in the next section on the Clifford Dirichlet form. Other non-trivial realizations of (4.7) arise in practice.

### Proof of Lemma 4.3

*V*by \(V^*\), \(A_1\) by \(A_2^*\), and \(A_2\) by \(A_1^*\). Similar computations then yield the identity

Thus, the condition (4.7) suffices to ensure that the sesquilinear form \(\mathscr {E}\) defined in (4.6) is real. In the rest of this section, we suppose that this condition is satisfied, and then since \(\mathscr {E}\) is real, it suffices to consider its bilinear restriction to \(\mathcal {A}_h\).

*V*(resp. \(V^*\)) is an eigenvector of the relative modular operator \(\Delta _{\ell (\sigma ),r(\sigma )}\) (resp. \(\Delta _{\ell _*(\sigma ),r_*(\sigma )}\)). Since the relative modular operator is positive, there exist \(\omega , \omega _*\in {{\mathbb {R}}}\) such that

### Lemma 4.5

### Proof

Note that \(( \Delta ^t_{\ell (\sigma ),r(\sigma )})_{t\in {{\mathbb {R}}}}\) is a group of linear operators on \(\mathcal {B}\), and the generator \(\mathscr {G}\) of this group is given by \(\mathscr {G}B = \ell (\log \sigma ) B - B {r}(\log \sigma )\), thus \(\mathscr {G}V = - \partial _V \log \sigma \). The equivalences thus follow from basic spectral theory.

We are now ready to state the main result of this section.

### Theorem 4.6

Let \(\sigma \) be a faithful state on \(\mathcal {A}\). Let \(V\in \mathcal {B}\) and two pairs \((\ell ,{r})\) and \((\ell _*,{r}_*)\) of \((\tau ,\tau _\mathcal {B})\)-compatible \(*\)-homomorphisms be given. Suppose also that (4.7) is satisfied, and suppose that *V* (resp. \(V^*\)) is an eigenvector of the relative modular operator \(\Delta _{\ell (\sigma ),r(\sigma )}\) (resp. \(\Delta _{\ell _*(\sigma ),r_*(\sigma )}\)) satisfying (4.10). Then the sesquilinear form \(\mathscr {E}: \mathcal {A}\times \mathcal {A}\rightarrow {{\mathbb {C}}}\) given by (4.6) defines a conservative completely Dirichlet form on \(L_\mathrm{KMS}^2(\mathcal {A}_h,\sigma )\).

### Proof

*V*is an eigenvector of the relative modular operator, so that (4.10) is satisfied, we fix \(V, W \in \mathcal {B}\) and (temporarily) define the operators \(\partial , \partial _*: \mathcal {A}\rightarrow \mathcal {B}\) by \(\partial A := V {r}(A) - \ell (A) W\) and \(\partial _* A := V^* {r}_*(A) - \ell _*(A) W^*\), and set

- (1)
If \(W = e^{\omega /2} \Delta ^{1/2}_{\ell (\sigma ),r(\sigma )} V\) and \(W^* = e^{\omega _*/2} \Delta ^{1/2}_{\ell _*(\sigma ),r_*(\sigma )} V^*\) for some \(\omega , \omega _* \in {{\mathbb {R}}}\), then \(\mathscr {E}\) defines a Dirichlet form on \(L_\mathrm{{KMS}}^2(\mathcal {A}_h,\sigma )\).

- (2)
If, in addition, (4.10) holds, then \(\mathscr {E}({\mathbf{1}},A) = 0\) for all \(A \in \mathcal {A}_h\), hence \(\mathscr {E}\) is conservative.

An entirely analogous argument shows that \(\langle \partial _* A_+, \partial _* A_- \rangle _{L^2_\mathrm{KMS}(\mathcal {B},\ell _*(\sigma ),{r}_*(\sigma ))} \le 0\), and this proves that \(\mathscr {E}(A_+,A_-)\) is a Dirichlet form.

Observe now that \(\partial {\mathbf{1}}= V - W\) and \(\partial _* {\mathbf{1}}= V^* - W^*\). Thus, to conclude that \(\partial {\mathbf{1}}= \partial _* {\mathbf{1}}= 0\), we need to assume that *V* is an eigenvector of \(\Delta _{\ell (\sigma ),r(\sigma )}\) with eigenvalue \(e^{-\omega }\), and that \(V^*\) is an eigenvector of \(\Delta _{\ell _*(\sigma ),r_*(\sigma )}\) with eigenvalue \(e^{-\omega _*}\). It immediately follows that \(\mathscr {E}({\mathbf{1}},A) = 0\) for all \(A \in \mathcal {A}_h\), hence \(\mathscr {E}\) is conservative.

It remains to prove that under the given conditions, \(\mathscr {E}\) is completely Dirichlet. Let \({{\,\mathrm{Tr}\,}}\) be the standard trace on \({{\mathbb {M}}}_m({{\mathbb {C}}})\). Let \(\mathbf{H}\) be a self-adjoint element of \(\mathcal {A}\otimes {{\mathbb {M}}}_m({{\mathbb {C}}})\), and let \(\mathbf{H}_+\) and \(\mathbf{H}_-\) be the elements of its decomposition \(\mathbf{H} = \mathbf{H}_+ - \mathbf{H}_-\) in \(L^2_\mathrm{KMS}(\sigma \otimes {{\,\mathrm{Tr}\,}})\), where \(\mathbf{H}_+\) and \(\mathbf{H}_-\) are positive and such that \( \langle \mathbf{H}_+,\mathbf{H}_-\rangle _{L^2_\mathrm{KMS}(\sigma \otimes {{\,\mathrm{Tr}\,}})} =0\).

Evidently, the sum of a finite set of conservative completely Dirichlet forms on \(\mathcal {A}\) is a conservative completely Dirichlet form. Thus, we may construct a large class of conservative completely Dirichlet forms by taking sums of forms of the type considered in Theorem 4.6. In the remainder of this section, we consider such a conservative, completely Dirichlet form and the associated QMS \(\mathscr {P}_t\).

It will be convenient going forward to streamline our notation. In the rest of this section we are working in the framework specified as follows:

### Definition 4.7

*differential structure*on \(\mathcal {A}\) consists of the following:

- (1)
A finite index set \(\mathcal {J}\), and for each \(j\in \mathcal {J}\), a finite dimensional von Neumann algebra \(\mathcal {B}_j\) endowed with a faithful tracial positive linear functional \(\tau _j\).

- (2)
For each \(j\in \mathcal {J}\), a pair \((\ell _j,{r}_j)\) of unital \(*\)-homomorphisms from \(\mathcal {A}\) to \(\mathcal {B}_j\) such that for each \(A\in \mathcal {A}\) and each \(j\in \mathcal {J}\), \(\tau _j(\ell _j(A)) = \tau _j({r}_j(A)) = \tau (A)\), and a non-zero \(V_j\in \mathcal {B}_j\).

- (3)It is further required that for each \(j\in \mathcal {J}\), there is a unique \(j^*\) such that \(V_j^* = V_{j^*}\), hence \(\{ V_j \}_{j \in \mathcal {J}} = \{ V_j^* \}_{j \in \mathcal {J}}\) and \(\mathcal {B}_{j^*} = \mathcal {B}_j\). Moreover, for \(j\in \mathcal {J}\) and \(A_1, A_2 \in \mathcal {A}\),$$\begin{aligned} \tau _j[ V_j^* \ell _j (A_1) V_j {r}_j(A_2) ] = \tau _j[ V_j^* {r}_{j^*}(A_1) V_j \ell _{j^*} (A_2) ] \ . \end{aligned}$$(4.17)
- (4)An invertible density matrix \(\sigma \in {{\mathfrak {P}}}_+\), such that, for each \(j\in \mathcal {J}\), \(V_j\) is an eigenvector of the relative modular operator \(\Delta _{\ell _j(\sigma ),{r}_j(\sigma )}\) on \(\mathcal {B}_j\) withfor some \(\omega _j \in {{\mathbb {R}}}\).$$\begin{aligned} \Delta _{\ell _j(\sigma ),{r}_j(\sigma )}(V_j) = e^{-\omega _j}V_j \end{aligned}$$(4.18)

*A*, or derivative of

*A*, with respect to the differential structure on \(\mathcal {A}\) defined above. We will denote the differential structure by the triple \((\mathcal {A}, \nabla , \sigma )\).

### Remark 4.8

As we have seen earlier in this section, *(3)* ensures that the sesquilinear form \(\mathscr {E}\) defined by (4.20) is real and leads to the symmetry condition (4.13), and then *(4)* ensures that \(\mathscr {E}\) is completely Dirichlet.

### Proposition 4.9

### Proof

The following result provides an explicit expression for \(\mathscr {L}\).

### Proposition 4.10

### Proof

The following result is an immediate consequence.

### Proposition 4.11

### Proof

The identity \(\mathscr {L}A = - \sum _{j \in \mathcal {J}} \partial _{j,\sigma }^\dagger \partial _j A \) implies that \({{\,\mathrm{\mathsf {Ker}}\,}}(\nabla ) \subseteq {{\,\mathrm{\mathsf {Ker}}\,}}(\mathscr {L})\). The reverse inclusion follows from the identity \(- \langle {\mathscr {L}A, A}\rangle _{L_\mathrm{KMS}^{2}(\sigma )} = \sum _{j \in \mathcal {J}} \langle {\partial _j A, \partial _j A}\rangle _{L_{\mathrm{KMS}, j}^{2}(\sigma )}\). The identification of the ranges is a consequence of duality. \(\square \)

### Proposition 4.12

### Proof

This follows from a direct computation using (4.22). \(\square \)

## 5 Examples

We provide a number of examples of conservative completely Dirichlet forms defined in the context of a differential structure on a finite-dimensional von Neumann algebra \(\mathcal {A}\) equipped with a faithful state \(\sigma \).

### 5.1 Generators of Quantum Markov Semigroups in Lindblad Form

We have seen in Sect. 2 that generators of quantum Markov semigroups satisfying detailed balance (see Theorem 2.4) naturally fit into the framework of Sect. 4 by taking \(\mathcal {A}= \mathcal {B}_j = B(\mathscr {H})\) and \(\ell _j = {r}_j = I_\mathcal {A}\).

The framework also includes quantum Markov semigroups on subalgebras \(\mathcal {A}\) of \(B(\mathscr {H})\). In this case we set \(\mathcal {B}_j = B(\mathscr {H})\), so that the situation in which \(V_j \notin \mathcal {A}\) is covered. Such a situation also arises naturally in the following example.

### 5.2 Classical Reversible Markov Chains in the Lindblad Framework

*k*,

*p*, we set \(\mathcal {B}_{k p} = {{\mathbb {M}}}_n({{\mathbb {C}}})\), and we endow \(\mathcal {A}\) and \(\mathcal {B}_{k p}\) with the usual normalized trace given by \(\tau (B) = \frac{1}{n} \sum _i \langle {B e_i, e_i}\rangle \). Let \(\ell _{k p} = {r}_{k p}\) be the canonical embedding from \(\mathcal {A}\) into \(\mathcal {B}_{k p}\). It then follows that \(\ell _{k p}^\dagger (B) = {r}_{k p}^\dagger (B) = \sum _{i} \langle {Be_i,e_i}\rangle E_{ii}\).

For \(k \ne p\), let \(q_{k p} \ge 0\) be the transition rate of a continuous-time Markov chain on \(\{1, \ldots , n\}\). We set \(V_{k p} = 2^{-1/2}(q_{k p} q_{p k})^{1/4} E_{k p}\) so that \(V_{k p}^* = V_{p k}\). Moreover, it is immediate to see that the identity in (4.17) holds. Fix positive weights \(\pi _1, \ldots , \pi _n\). It then follows that \(\sigma = \sum _i \pi _i E_{ii}\) satisfies (4.18) with \(\omega _{k p} = \log (\pi _p /\pi _k)\).

*k*,

*p*. Then we have

*k*to

*p*given by \(q_{kp}\).

### 5.3 Another Approach to Reversible Markov Chains

*k*’th unit vector in \(\ell _n^\infty \). Therefore,

### 5.4 The Discrete Hypercube

For a given Markov chain generator, there are different ways to write the generator in the framework of this paper, and it is often useful to represent \(\mathscr {L}\) using set \(\mathcal {J}\) that is smaller than in Example 5.3; see also [21]. We illustrate this for the simple random walk on the discrete hypercube \(\mathcal {Q}^n = \{-1, 1\}^n\). Set \(\mathcal {J}= \{1, \ldots , n\}\), and let \(s_j:\mathcal {Q}^n \rightarrow \mathcal {Q}^n\) define the *j*-th coordinate swap defined by \(s_j (x_1, \ldots , x_n) = (x_1, \ldots , - x_j, \ldots , x_n)\).

### 5.5 The Fermionic Ornstein–Uhlenbeck Equation

*canonical anti-commutation relations*(CAR):

*Clifford algebra*\(\mathfrak {C}^n\) is the \(2^n\)-dimensional algebra generated by \(\{Q_j\}_{j=1}^n\). Let \(\Gamma : \mathfrak {C}^n\rightarrow \mathfrak {C}^n\) be the principle automorphism on \(\mathfrak {C}^n\), i.e., the unique algebra homomorphism satisfying \(\Gamma (Q_j) = - Q_j\) for all

*j*. Let \(\tau \) be the canonical trace on \(\mathfrak {C}^n\), determined by \(\tau (Q_1^{\alpha _1} \cdots Q_n^{\alpha _n}) := \delta _{0,|{\varvec{\alpha }}|}\) for all \(\mathbf {A}= (\alpha _j)_j \in \{0,1\}^n\), where \(|{\varvec{\alpha }}| := \sum _j \alpha _j\). We then set \(\mathcal {J}= \{1, \ldots , n\}\), \(\mathcal {A}:= \mathcal {B}_{j} := \mathfrak {C}^n\), and \(\tau _{j} := \tau \). Furthermore we set \(V_{j} = Q_j\), \(\ell _{j} = \Gamma \), and \({r}_{j} = I\). Then \(\ell _j^\dagger = \Gamma \), and the operators \(\partial _j\) and \(\partial _j^\dagger \) are skew-derivations given by

*fermionic number operator*(see [8, 9] for more details).

### 5.6 The Depolarizing Channel

## 6 Non-commutative Functional Calculus

*double operator sum*

### Remark 6.1

A systematic theory of infinite-dimensional generalizations of \(\theta (A,B)\) has been developed under the name of *double operator integrals*, see, e.g., [5, 43].

*discrete derivative*of a differentiable function \(f : \mathcal {I}\rightarrow {{\mathbb {R}}}\), defined by

### Proposition 6.2

### Proof

*j*. Therefore,

### Remark 6.3

Note that the function *f* is not required to be differentiable in Proposition 6.2. In this case, \(\delta f\) is not defined on the diagonal, but the second line in (6.8) shows that its diagonal value is irrelevant.

The following well-known chain rule can also be formulated in terms of \(\delta f\).

### Proposition 6.4

*f*be a real-valued function on an interval containing \({{\,\mathrm{sp}\,}}(A(t))\) for all \(t \in \mathcal {I}\). Then:

### Proof

The first assertion follows by passing to the limit in (6.6). The second identity follows easily using the definition of \(\delta f\) and the cyclicity of the trace. \(\square \)

### Example 6.5

We finish this subsection with some useful properties of the sesquilinear form \((A,B)\mapsto \langle { A, \varphi (R,S) \# B }\rangle _{L^2(\tau )}\) on \(\mathcal {A}\).

### Lemma 6.6

### Proof

### Proposition 6.7

### Proof

### 6.1 Higher Order Expressions

*x*and

*y*in \(\theta (x,y)\) respectively. More complicated trees are then constructed by iteratively replacing one of the children \(\bullet \) by Open image in new window. This will correspond to discrete differentiation with respect to the respective variables, e.g.,

*multiple operator sum*

### Proposition 6.8

### Proof

We have \(\partial _t \theta (A_t, B_t) = \partial _s|_{s = t} \theta (A_s, B_t) + \partial _s|_{s = t} \theta (A_t, B_s)\). Since we can write \(\theta (A_t,B_s) = \sum _{k} \theta (A_t,\mu _{s,k})\otimes F_{s,k}\), where \(B_s = \sum _{k} \mu _{s,k} F_{s,k}\) denotes the spectral decomposition of \(B_s\), the result follows by applying (6.9) from Proposition 6.4 twice. \(\square \)

Higher order derivatives can also be naturally expressed in terms of trees, but since this will not be needed in the sequel, we will not go into details here.

## 7 Riemannian Structures on the Space of Density Matrices

In this section we shall analyze a large class of Riemannian metrics on the space of density matrices. Throughout the section we fix a differentiable structure \((\mathcal {A}, \nabla , \sigma )\) in the sense of Definition 4.7. The generator of the associated quantum Markov semigroup \((\mathscr {P}_t)_t\) will be denoted by \(\mathscr {L}\).

### 7.1 Riemannian Structures on Density Matrices

### Remark 7.1

Of special interest is the *ergodic* case, i.e., the case where \({{\,\mathrm{\mathsf {Ker}}\,}}(\mathscr {L}) = {{\,\mathrm{lin}\,}}\{{\mathbf{1}}\}\). In this case we have \(\mathcal {A}_0 = \{ A \in \mathcal {A}_h : \tau [A] = 0 \}\), and therefore \(\mathscr {M}_\rho = {{\mathfrak {P}}}_+\) for all \(\rho \in {{\mathfrak {P}}}_+\).

In order to define a Riemannian structure, we shall fix for each \(j \in \mathcal {J}\) a function \(\theta _j : [0,\infty ) \times [0,\infty ) \rightarrow {{\mathbb {R}}}\) satisfying the following properties:

### Assumption 7.2

### Lemma 7.3

### Proof

*j*by \(j^*\) and using that \(\theta _j^{km} = \theta _{j^*}^{mk}\) by Assumption 7.2, we obtain

*m*and

*k*. \(\square \)

The following result expressing the unique solvability of the continuity equation is now an immediate consequence.

### Corollary 7.4

For \(\rho \in {{\mathfrak {P}}}_+\), the linear mapping \(\mathscr {K}_\rho \) is a bijection on \(\mathcal {A}_0\) that depends smoothly \((C^\infty )\) on \(\rho \).

### Proof

It follows from Lemma 7.3 that \(\mathscr {K}_\rho \) maps \(\mathcal {A}_0\) into itself. Since the restriction of a self-adjoint operator to its range is injective, the result follows. Smooth dependence on \(\rho \) follows from the smoothness of \(\theta \). \(\square \)

The following elementary variational characterization is of interest.

### Proposition 7.5

### Proof

Existence of a gradient vector \(\mathbf {B}\) field solving (7.7) follows from Corollary 7.4. To prove uniqueness, suppose that \({{\,\mathrm{div}\,}}({\widehat{\rho }}\# \nabla A) = - \nu = {{\,\mathrm{div}\,}}({\widehat{\rho }}\# \nabla {\widetilde{A}})\) for some \(A, {\widetilde{A}} \in \mathcal {A}\). This means that \(\mathscr {K}_\rho A = \mathscr {K}_\rho {\widetilde{A}}\), hence Lemma 7.3 yields \(\nabla A = \nabla {\widetilde{A}}\). The remaining part follows along the lines of the proof of [8, Theorem 3.17]. \(\square \)

We are now ready to define a class of Riemannian metrics that are the main object of study in this paper.

### Definition 7.6

*Quantum transport metric*) Fix \(\rho \in {{\mathfrak {P}}}_+\) and let \(\theta = (\theta _j)_j\) satisfy Assumption 7.2. The associated

*quantum transport metric*is the Riemannian metric on \(\mathscr {M}_\rho \) induced by the operator \(\mathscr {K}_\rho \), i.e., for \({{\dot{\rho }}}_1, {\dot{\rho }}_2 \in \mathcal {A}_0\),

It follows from Lemma 7.3 and Corollary 7.4 that \(\mathscr {K}_\rho \) indeed induces a Riemannian metric on \(\mathscr {M}_\rho \).

### 7.2 Gradient Flows of Entropy Functionals

In this section we shall show that various evolution equations of interest can be interpreted as gradient flow equations with respect to suitable quantum transport metrics introduced in Sect. 7.1.

### Theorem 7.7

This result generalises the gradient flow structure from [10, 36] as described in Sect. 2. The proof relies on the following version of the chain rule.

### Lemma 7.8

### Proof

### Proof of Theorem 7.7

### Theorem 7.9

### Proof

### Remark 7.10

The result remains true if *f* is required to be strictly concave and \(\varphi \) is required to be strictly decreasing. Note that \(\theta \) is positive in this case, so that \((\mathscr {K}_\rho )_\rho \) induces a Riemannian metric.

### Remark 7.11

This result contains various known results as special cases. Take \(f(\lambda ) = \lambda \log \lambda \) and \(\varphi (r) = r\). Then the functional \(\mathcal {F}\) is the von Neumann entropy \(\mathcal {F}(\rho ) = \tau [\rho \log \rho ]\), and we recover the special case of Theorem 7.7 with \(\sigma = {\mathbf{1}}\). It also contains the gradient flow structure for the fermionic Fokker-Planck equation from [8]. In the special case where \(\mathscr {L}\) is the generator of a reversible Markov chain, we recover the gradient flow structure for discrete porous medium equations obtained in [19].

### Remark 7.12

*logarithmic mean*. The integral representation \(\theta (\lambda ,\mu ) = \int _0^1 \lambda ^{1-s}\mu ^s \; \mathrm {d}s\) allows one to express \({\widehat{\rho }}_j\) in terms of the functional calculus for \(\ell _j(\rho )\) and \({r}_j(\rho )\):

*power difference means*defined by

Another special case is obtained by taking \(\varphi (\lambda ) = \lambda \) and \(f(\lambda ) = \lambda ^2/2\), which yields \(\theta (\lambda , \mu ) \equiv 1\), so that \(\mathscr {K}_\rho = -\mathscr {L}\) for all \(\rho \), and \(\mathcal {F}(\rho ) = \frac{1}{2}\tau [\rho ^2] = \frac{1}{2} \Vert \rho \Vert _{L^2(\tau )}^2\). In this case, the distance associated to \(\mathscr {K}_\rho \) may be regarded as a non-commutative analogue of the Sobolev \(H^{-1}\)-metric.

### 7.3 Geodesics

### Proposition 7.13

### Remark 7.14

### Remark 7.15

If \(\theta _j(r,s) := \Lambda (e^{\omega _j/2}r, e^{-\omega _j/2}s)\) where \(\Lambda \) is the logarithmic mean, the expression above can be simplified. In this case we have the integral representation

so that

### Proof of Proposition 7.13

*A*is self-adjoint, it follows using (7.1) and (4.17) thatThis implies the equality of the two sums in (7.16), and it also follows that

### Proposition 7.16

### Proof

### Remark 7.17

## 8 Preliminaries on Quasi-entropies

In this section we collect some known results on trace functionals that will be useful in the study of quantum transport metrics. Special cases of the results in this section already played a key role in the proof of functional inequalities in [10].

In this section we shall assume that the function \(\theta \) is 1*-homogeneous*, i.e., \(\theta (\lambda r,\lambda s) = \lambda \theta (r,s)\) for all \(\lambda , r, s > 0\). Clearly, this assumption is satisfied if and only if there exists a function \(f : (0,\infty ) \rightarrow (0,\infty )\) such that \(\theta (r,s) = s f(r/s)\) for all \(r,s > 0\), in which case we have \(f(r) = \theta (r,1)\). To simplify notation, we write \(k(r) = 1/f(r)\).

### Remark 8.1

*R*(resp.

*S*), let \(\{\lambda _k\}\) (resp. \(\{\mu _\ell \}\)) be the corresponding eigenvalues, and set \(E_{k\ell } := \,|\xi _k\rangle \langle \eta _\ell |\,\). It follows that \(\Delta _{R,S}(E_{k \ell }) = \frac{\lambda _k}{\mu _\ell } E_{k \ell }\), hence the \(E_{k\ell }\)’s form a complete basis of eigenvectors of \(\Delta _{R,S}\). Moreover, the \(E_{k\ell }\)’s are orthonormal with respect to the Hilbert–Schmidt inner product \(\langle {A,B}\rangle _{L^2({{\,\mathrm{Tr}\,}})} = {{\,\mathrm{Tr}\,}}[A^* B]\) on \({{\mathbb {M}}}_n({{\mathbb {C}}})\). Consequently, the spectral decomposition of \(\Delta _{R,S}\) is given by

### Example 8.2

*tilted logarithmic mean*\(\theta _{1,\beta }\) given by

*f*and

*m*. For this purpose we recall that a function \(f : (0,\infty ) \rightarrow (0,\infty )\) is said to be

*operator monotone*, whenever \(f(A) \le f(B)\) for all positive matrices \(A \le B\) in all dimensions. Each operator monotone function is continuous, non-decreasing and concave. We set \(f(0) := \inf _{t > 0} f(t)\).

The following result has been obtained in [27, Theorem 2.1]. The implication “\((2) \Rightarrow (1)\)”, as well as the reverse implication for fixed \(p =1\) had already been proved in [26].

### Theorem 8.3

- (1)
The function \(\Upsilon _{f,p}\) is jointly convex in its three variables;

- (2)
The function

*f*is operator monotone and \(p \in (0,1]\).

Applying this result to the functions \(f = f_{m,\beta }\), we obtain the following result.

### Corollary 8.4

### Proof

### Remark 8.5

In the case where \(\theta = \theta _{1,\beta }\), the operator monotonicity of \(f_{1,\beta }\) can be checked elementarily, by writing \(f_{1,\beta }(r) = \int _0^1 e^{-\beta (1/2-\alpha )} r^\alpha \; \mathrm {d}\alpha \), and applying the Löwner-Heinz Theorem (e.g., [7, Theorem 2.6]), which asserts that the function \(r \mapsto r^\alpha \) is operator monotone for \(\alpha \in [0,1]\).

The following result is proved in [26, Theorem 5].

### Theorem 8.6

In the case where \(f = f_{m, \beta }\) as in Example 8.2, we obtain the following result.

### Corollary 8.7

## 9 The Riemannian Distance

Fix a differentiable structure \((\mathcal {A}, \nabla , \sigma )\) in the sense of Definition 4.7 and a collection of functions \((\theta _j)_j\) satisfying Assumption 7.2. For simplicity we restrict ourselves to the ergodic case, so that \(\mathscr {M}_\rho = {{\mathfrak {P}}}_+\) for all \(\rho \in {{\mathfrak {P}}}_+\).

### Lemma 9.1

### Proof

Any admissible curve \((A_t)\) in (9.1) yields an admissible curve \((\mathbf {B}_t)\) in (9.3) given by \(\mathbf {B}_t = {\widehat{\rho }}_t \nabla A_t\), that satisfies \(\Vert \nabla A_t\Vert _{\rho _t} = \Vert \mathbf {B}_t \Vert _{-1, \rho _t}\). This implies the inequality “\(\ge \)” in (9.3).

### Proposition 9.2

(Extension of the distance to the boundary) Suppose that \(\theta _j(a,b) \ge C\min \{a,b\}^p\) for some \(C > 0\) and \(p<2\). Then the distance function \(\mathscr {W}: {{\mathfrak {P}}}_+ \times {{\mathfrak {P}}}_+ \rightarrow {{\mathbb {R}}}\) extends continuously to a metric on \({{\mathfrak {P}}}\).

### Proof

Let \(\rho _0, \rho _1 \in {{\mathfrak {P}}}\) and let \(\{\rho _0^n\}_n, \{\rho _1^n\}_n\) be sequences in \({{\mathfrak {P}}}_+\) satisfying \(\tau \big [ |\rho _i^n -\rho _i|^2 \big ] \rightarrow 0\) as \(n \rightarrow \infty \) for \(i = 0, 1\). We claim that the sequence \(\{ \mathscr {W}(\rho _0^n, \rho _1^n) \}_n\) is Cauchy.

We can thus extend \(\mathscr {W}\) to \({{\mathfrak {P}}}\) by setting \(\mathscr {W}(\rho _0, \rho _1) = \lim _{n \rightarrow \infty }\mathscr {W}(\rho _0^n, \rho _1^n)\). It immediately follows that \(\mathscr {W}\) is symmetric and the triangle inequality extends to \({{\mathfrak {P}}}\). The fact that \(\mathscr {W}(\rho _0, \rho _1) \ne 0\) whenever \(\rho _0\) and \(\rho _1\) are distinct, follows from Proposition 9.4 below. \(\square \)

### Lemma 9.3

### Proof

### Proposition 9.4

*M*be as in Lemma 9.3 and set \(N := \sup \{ \Vert \nabla A\Vert _{\mathcal {B},2} \ : \ \Vert A\Vert _\mathcal {A}\le 1 \}\). Then, for \(\rho _0, \rho _1 \in {{\mathfrak {P}}}\) we have

### Proof

The first inequality follows from the definitions, since \(\tau [|B|] = \sup _{\Vert A\Vert _\mathcal {A}\le 1} \tau [AB]\) for \(B \in \mathcal {A}\).

In the remainder of this section we impose the following natural additional conditions in addition to Assumption 7.2.

### Assumption 9.5

The functions \(\theta _j: [0,\infty ) \times [0,\infty ) \rightarrow [0,\infty )\) are 1-homogeneous (which implies that \(\theta _j(r,s) = s f_j(r/s)\) for some function \(f_j\)). The functions \(f_j\) are assumed to be operator monotone.

Under this assumption, we will prove some crucial convexity properties for the action functional and the squared distance.

### Proposition 9.6

### Theorem 9.7

### Proof

Using these convexity properties, the existence of constant speed geodesics for the metric \(\mathscr {W}\) follows by standard arguments; cf. [18, Theorem 3.2]) for a proof in the commutative setting and [46] for a proof in a non-commutative context.

### Theorem 9.8

(Existence of \(\mathscr {W}\)-geodesics) For any \({\bar{\rho }}_0, {\bar{\rho }}_1 \in {{\mathfrak {P}}}\) there exists a curve \(\rho : [0,1] \rightarrow {{\mathfrak {P}}}\) satisfying \(\rho _0 = {\bar{\rho }}_0\), \(\rho _1 = {\bar{\rho }}_1\), and \(\mathscr {W}(\rho _s, \rho _t) = | s-t | \mathscr {W}(\rho _0, \rho _1)\) for all \(s, t \in [0,1]\).

## 10 Geodesic Convexity of the Entropy

In this section we will analyse geodesic convexity of the relative entropy functional \({{\,\mathrm{Ent}\,}}_\sigma \). Throughout this section we fix a differential structure \((\mathcal {A}, \nabla , \sigma )\) and assume that the associated quantum Markov semigroup \((\mathscr {P}_t)\) is ergodic. We consider the transport metric \(\mathscr {W}\) defined in Theorem 7.7 using the functions \(\theta _j\) given by \(\theta _j(r,s) := \Lambda (e^{\omega _j/2}r, e^{-\omega _j/2}s)\), so that the Kolmogorov forward equation \(\partial _t \rho = \mathscr {L}^\dagger \rho \) is the gradient flow of the relative von Neumann entropy \({{\,\mathrm{Ent}\,}}_\sigma \) with respect to the Riemannian metric induced by \((\mathscr {K}_\rho )_\rho \).

The following terminology will be useful.

### Definition 10.1

*weakly geodesically*\(\lambda \)*-convex*if any pair \(x_0, x_1 \in \mathcal {X}\) can be connected by a geodesic \((\gamma _t)_{t \in [0,1]}\) in \((\mathcal {X}, d)\) along which \(\mathcal {F}\) satisfies the \(\lambda \)-convexity inequality$$\begin{aligned} \mathcal {F}(\gamma _t) \le (1-t) \mathcal {F}(\gamma _0) + t \mathcal {F}(\gamma _1) - \frac{\kappa }{2} t(1-t) d(x_0, x_1)^2\ . \end{aligned}$$(10.1)*strongly geodesically*\(\lambda \)*-convex*if (10.1) holds for any geodesic \((\gamma _t)_{t \in [0,1]}\) in \((\mathcal {X}, d)\).

### Theorem 10.2

- (1)
\({{\,\mathrm{Ent}\,}}_\sigma \) is weakly geodesically \(\lambda \)-convex on \(({{\mathfrak {P}}},\mathscr {W})\);

- (2)
\({{\,\mathrm{Ent}\,}}_\sigma \) is strongly geodesically \(\lambda \)-convex on \(({{\mathfrak {P}}},\mathscr {W})\);

- (3)For all \(\rho , \nu \in {{\mathfrak {P}}}\), the following ‘evolution variational inequality’ holds for all \(t \ge 0\):$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}^+}{\mathrm {d}t}\mathscr {W}^2(\mathscr {P}_t^\dagger \rho , \nu ) + \frac{\lambda }{2} \mathscr {W}^2(\mathscr {P}_t^\dagger \rho , \nu ) \le {{\,\mathrm{Ent}\,}}_\sigma (\nu ) - \mathcal {H}(\mathscr {P}_t^\dagger \rho )\;; \end{aligned}$$(10.2)
- (4)For all \(\rho \in {{\mathfrak {P}}}_+\) and \(A \in \mathcal {A}_0\) we have$$\begin{aligned} {{\,\mathrm{Hess}\,}}_\mathscr {K}{{\,\mathrm{Ent}\,}}_\sigma (\rho )[A, A] \ge \lambda \tau [A \mathscr {K}_\rho A] \ . \end{aligned}$$

### Proof

“\((4) \Rightarrow (3)\)” This can be proved by an argument from [14]; see [18, Theorem 4.5] for a proof in a similar setting.

“\((3) \Rightarrow (2)\)”: This follows from an application of [14, Theorem 3.2] to the metric space \(({{\mathfrak {P}}}, \mathscr {W})\).

“\((2) \Rightarrow (1)\)”: Since \(({{\mathfrak {P}}}, \mathscr {W})\) is a geodesic space, this implication is immediate.

“\((1) \Rightarrow (4)\)”: Obvious. \(\square \)

In the classical setting, the Ricci curvature on a Riemannian manifold \(\mathscr {M}\) is bounded from below by \(\lambda \in {{\mathbb {R}}}\) if and only if the entropy (with respect to the volume measure) is geodesically \(\lambda \)-convex in the space of probability measures \(\mathscr {P}(\mathscr {M})\) endowed with the Kantorovich metric \(W_2\). This characterisation is the starting point for the synthetic theory of metric measure spaces with lower Ricci curvature bounds, which has been pioneered by Lott, Sturm and Villani.

By analogy, we make the following definition in the non-commutative setting, which extends the corresponding definition in the discrete setting [18].

### Definition 10.3

(*Ricci curvature*) Let \(\lambda \in {{\mathbb {R}}}\). We say that a differential structure \((\mathcal {A}, \nabla , \sigma )\) has Ricci curvature bounded from below by \(\lambda \) if the equivalent conditions of Theorem 10.2 hold. In this case, we write \({{\,\mathrm{Ric}\,}}(\mathcal {A}, \nabla , \sigma ) \ge \lambda \).

It is possible to characterize Ricci curvature in terms of a gradient estimate in the spirit of Bakry–Émery; see [17] for the corresponding statement in the setting of finite Markov chains and [46] for an implementation in the Lindblad setting.

### Theorem 10.4

### Proof

We follow a standard semigroup interpolation argument. Clearly, (10.3) holds for any \(\rho \in {{\mathfrak {P}}}\) if and only if it holds for any \(\rho \in {{\mathfrak {P}}}_+\).

*s*. This implies that \(f(t) \ge f(0)\), which is (10.3).

An immediate consequence of a Ricci curvature bound is the following contractivity estimate for the associated semigroup, which was independently proved by Rouzé in [44].

### Proposition 10.5

### Proof

This is a well-known consequence of the evolution variational inequality (10.2); see [14, Proposition 3.1]. \(\square \)

Using the techniques developed in this paper, we can explicitly compute the Ricci curvature for the depolarizing channel defined in Sect. 5.6. The result has been obtained independently by Rouzé in [44].

### Theorem 10.6

(Ricci bound for the depolarizing channel) Let \(\gamma > 0\), and let \((\mathcal {A}, \nabla , \tau )\) be a differential structure for the generator of the depolarizing channel given by \(\mathscr {L}A = \gamma (\tau [A]{\mathbf{1}}- A)\). Then \({{\,\mathrm{Ric}\,}}(\mathcal {A}, \nabla , \tau ) \ge \gamma \).

### Proof

Since \(\mathscr {L}A = \gamma (\tau [A]{\mathbf{1}}- A)\) and \(\partial _j {\mathbf{1}}= 0\), we have \(\partial _j \mathscr {L}A = - \gamma \partial _j A\), independently of the choice of the operators \(\partial _j\). We will show that the result follows from this identity.

Since the spectral gap of \(\mathscr {L}\) equals \(\gamma \), it follows from the results in Sect. 11 that the obtained constant is optimal.

### 10.1 Geodesic Convexity Via Intertwining

In this subsection we provide a useful technique for proving Ricci curvature bounds, which has the advantage that it does not require an explicit computation of the Hessian of the entropy. Instead, it relies on the following intertwining property between the gradient and the quantum Markov semigroup.

### Definition 10.7

*Intertwining property*) For \(\lambda \in {{\mathbb {R}}}\), we say that a collection of linear operators \((\mathbf {\mathscr {P}_t})_{t \ge 0}\) on \(\mathcal {B}\) is \(\lambda \)-intertwining for the quantum Markov semigroup \((\mathscr {P}_t)_{t \ge 0}\), if the following conditions hold:

- (1)
For all \(A \in \mathcal {A}\) and \(t \ge 0\), we have \(\nabla \mathscr {P}_t A = \mathbf {\mathscr {P}_t} \nabla A\);

- (2)For all \(\rho \in {{\mathfrak {P}}}_+\), \(\mathbf {B}= (B_j) \in \mathcal {B}\) and \(t \ge 0\), we have$$\begin{aligned} \mathscr {A}\big (\rho , \mathbf {\mathscr {P}_t^\dagger } \mathbf {B}\big ) \le e^{-2\lambda t } \mathscr {A}\big (\rho , (\mathscr {P}_t^\dagger B_j)_j \big ) \ . \end{aligned}$$(10.6)

### Lemma 10.8

Let \(\lambda \in {{\mathbb {R}}}\), and suppose that \(\partial _j \mathscr {L}A = (\mathscr {L}- \lambda ) \partial _j A\) for all \(A \in \mathcal {A}\). Then the semigroup \((\mathbf {\mathscr {P}_t})_t\) defined by \((\mathbf {\mathscr {P}_t} \mathbf {B})_j = e^{-\lambda t} \mathscr {P}_t B_j\) is \(\lambda \)-intertwining for the quantum Markov semigroup \((\mathscr {P}_t)_{t \ge 0}\).

### Proof

By spectral theory, the stated condition on the generator is equivalent to the semigroup property \(\partial _j \mathscr {P}_t A = e^{-\lambda t} \mathscr {P}_t \partial _j A\) for all \(t \ge 0\). Thus, the semigroup \((\mathbf {\mathscr {P}_t})_t\) satisfies (1) in Definition 10.7. Since \((\mathbf {\mathscr {P}_t^\dagger } \mathbf {B})_j = e^{-\lambda t} \mathscr {P}_t^\dagger B_j\), condition (2) follows as well. \(\square \)

### Theorem 10.9

(Lower Ricci bound via intertwining) Let \((\mathcal {A}, \nabla , \sigma )\) be a differential structure, and let \(\lambda \in {{\mathbb {R}}}\). If there exists a collection of linear operators \((\mathbf {\mathscr {P}_t})_{t \ge 0}\) on \(\mathcal {B}\) that is \(\lambda \)-intertwining for the associated QMS \((\mathscr {P}_t)_{t \ge 0}\), then \({{\,\mathrm{Ric}\,}}(\mathcal {A}, \nabla , \sigma ) \ge \lambda \).

### Proof of Theorem 10.9

The proof is a variation on an argument by Dolbeault, Nazaret and Savaré [16].

### Remark 10.10

As pointed out by an anonymous referee, the condition from Lemma 10.8 is preserved under taking tensor products of quantum Markov semigroups. Therefore, Theorem 10.9 yields a lower Ricci curvature bound for tensor product semigroups of this type. It is an interesting open question whether such a tensorisation property holds for arbitrary quantum Markov semigroups, as is known to be true in the Markov chain setting [18].

We finish the section with the example of the Fermionic Ornstein–Uhlenbeck equation from Sect. 5.5, which was already discussed in [10]. For the convenience of the reader we provide the details.

### Proposition 10.11

(Intertwining for fermions) In the fermionic setting, we have the commutation relations \( [\partial _j, \mathscr {L}] = - \partial _j \) for \(j = 1, \ldots , n\). Consequently, the intertwining property holds with \(\lambda = 1\).

### Proof

We use the well-known fact that the differential operator \(\partial _j\) is the *annihilation operator*: it maps the *k*-particle space \(\mathcal {H}^k\) into the \((k-1)\)-particle space \(\mathcal {H}^{k-1}\) for any \(0 \le k \le n\) (with the convention that \(\mathcal {H}^{-1} = \{0\})\). On the other hand, \(-\mathscr {L}\) is the *number operator*, which satisfies \(\mathscr {L}A = - k A\) for all \(A \in \mathcal {H}^k\). Hence, for \(A \in \mathcal {H}^{k}\), we have \(\partial _j \mathscr {L}A = - k \partial _j A\), whereas \(\mathscr {L}\partial _j A = - (k-1)\partial _j A\). This yields the desired commutation relation \([\partial _j, \mathscr {L}] = - \partial _j\) on \(\mathcal {H}^{k}\), which extends to \(\mathfrak {C}^n\) by linearity. The result thus follows from Lemma 10.8. \(\square \)

We immediately obtain the following result.

### Corollary 10.12

The differential structure for the fermionic Ornstein–Uhlenbeck equation in Sect. 5.5 satisfies \({{\,\mathrm{Ric}\,}}(\mathfrak {C}^n, \nabla , \tau ) \ge 1\) in any dimension \(n \ge 1\).

It follows from the results in the following section that the constant 1 is optimal.

## 11 Functional Inequalities

One of the advantages of the framework of this paper is that it allows one to prove a sequence of implications between several useful functional inequalities. Throughout this section we assume that \((\mathscr {P}_t)_t\) is ergodic.

### Definition 11.1

- (1)a
*modified logarithmic Sobolev inequality*with constant \(\lambda >0\) if for all \(\rho \in \mathscr {P}(\mathcal {X})\), - (2)an \(H\mathscr {W}I\)
*inequality*with constant \(\kappa \in {{\mathbb {R}}}\) if for all \(\rho \in \mathscr {P}(\mathcal {X})\), - (3)a
*modified Talagrand inequality*with constant \(\lambda > 0\) if for all \(\rho \in {{\mathfrak {P}}}\), - (4)a \(T_1\)
*-transport inequality*with constant \(\lambda > 0\) if for all \(\rho \in {{\mathfrak {P}}}\), - (5)a
*Poincar*é*inequality*(or*spectral gap inequality*) with constant \(\lambda > 0\) if for all \(A \in \mathcal {A}_h\) with \(\tau [\int _0^1 \sigma ^{1-s} A \sigma ^s \; \mathrm {d}s] = 0\),

*metric derivative*

### Proposition 11.2

### Proof

The following result is a non-commutative analogue of a well-known result by Otto and Villani [39].

### Theorem 11.3

Assume that \({{\,\mathrm{Ric}\,}}(\mathcal {A},\nabla ,\sigma )\ge \kappa \) for some \(\kappa \in {{\mathbb {R}}}\). Then \({{\,\mathrm{H{\mathscr {W}}I}\,}}(\kappa )\) holds as well.

### Proof

The following result is now a simple consequence.

### Theorem 11.4

(Quantum Bakry–Émery Theorem) Suppose that \({{\,\mathrm{Ric}\,}}(\mathcal {A},\sigma ,\nabla )\ge \lambda \) for some \(\lambda >0\). Then the modified logarithmic Sobolev inequality \({{\,\mathrm{MLSI}\,}}(\lambda )\) holds.

### Proof

### Theorem 11.5

(Quantum Otto–Villani Theorem) Suppose that the differential structure \((\mathcal {A},\nabla ,\sigma )\) satisfies \({{\,\mathrm{MLSI}\,}}(\lambda )\) for some \(\lambda >0\). Then the Talagrand inequality \({{\,\mathrm{T_\mathscr {W}}\,}}(\lambda )\) holds as well.

### Proof

It suffices to prove \({{\,\mathrm{T_\mathscr {W}}\,}}(\lambda )\) for \(\rho \in {{\mathfrak {P}}}_+\), since the inequality for general \(\rho \in {{\mathfrak {P}}}\) can then be obtained by approximation.

It is known that the modified logarithmic Sobolev inequality implies a Poincaré inequality by a linearization argument. The following result shows that Poincaré inequality is in fact implied by the Talagrand inequality, which is weaker than the MLSI in view of the previous theorem. The BKM metric in the left-hand side of P(\(\lambda \)) appears since it also appears in the second order expansion of the relative entropy of \({{\,\mathrm{Ent}\,}}_\sigma (\rho )\) around \(\rho = \sigma \); see (6.12).

### Proposition 11.6

Assume that the triple \((\mathcal {A},\sigma ,\nabla )\) satisfies T\(_{\mathscr {W}}\)(\(\lambda \)) for some \(\lambda >0\). Then the Poincaré inequality P(\(\lambda \)) and the \(T_1\)-transport inequality T\(_{1}\)(\(\lambda \)) hold as well. Moreover, \({{\,\mathrm{Ric}\,}}(\mathcal {A},\sigma ,\nabla ) \ge \lambda \) implies \({{\,\mathrm{P}\,}}(\lambda )\).

### Proof

The fact that T\(_{\mathscr {W}}\)(\(\lambda \)) implies the \(T_1\)-inequality is an immediate consequence of Proposition 9.4.

Suppose that T\(_{\mathscr {W}}\)(\(\lambda \)) holds and let us show \({{\,\mathrm{P}\,}}(\lambda )\). Fix \(\nu \in \mathcal {A}_0\) and set \(\rho ^\varepsilon : = \sigma + \varepsilon \nu \). Then \(\rho ^\varepsilon \in {{\mathfrak {P}}}_+\) for sufficiently small \(\varepsilon >0\). For such \(\varepsilon > 0\), let \((\rho _t^\varepsilon , \mathbf {B}_t^\varepsilon )_t\) be an action minimizing curve connecting \(\rho _0^\varepsilon = \rho ^\varepsilon \) and \(\rho _1^\varepsilon = \sigma \). Thus we have \(\partial _t \rho _t^\varepsilon + {{\,\mathrm{div}\,}}({\widehat{\rho }}_t^\varepsilon \# \mathbf {B}_t^\varepsilon ) = 0\) and \(\int _0^1 \tau [(\mathbf {B}_t^\varepsilon )^*{\widehat{\rho }}_t^\varepsilon \# \mathbf {B}_t^\varepsilon ] \; \mathrm {d}t = \mathscr {W}(\rho ^\varepsilon , \sigma )^2\).

The final assertion of the proposition follows by combining this result with Theorem 11.4 and Theorem 11.5. \(\square \)

## Notes

### Acknowledgements

Open access funding provided by Institute of Science and Technology (IST Austria). Eric A. Carlen gratefully acknowledges support through NSF grant DMS-174625. Jan Maas gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No 716117), and by the Austrian Science Fund (FWF), Project SFB F65. We are grateful to the anonymous referees for carefully reading the original manuscript and making useful comments.

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