Let \(\mathcal {A}\subseteq B(\mathscr {H})\) be a \(C^*\)-algebra of operators acting on a finite-dimensional Hilbert space \(\mathscr {H}\). Let \(\tau \) be a tracial and faithful positive linear functional on \(\mathcal {A}\). A 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 \(n \in {{\mathbb {N}}}\).
Note that (2) implies that \(\mathscr {P}_{t}\) is 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.
It is well known [24, 31] that the generator \(\mathscr {L}\) of the semigroup \(\mathscr {P}_t = e^{t \mathscr {L}}\) can be written in Lindblad form
$$\begin{aligned} \mathscr {L}A&= i [{\widetilde{H}},A] + \sum _{j \in \mathcal {J}} V_j^* [A, V_j] + [V_j^* , A] V_j \ , \end{aligned}$$
(2.1)
$$\begin{aligned} \mathscr {L}^\dagger \rho&= - i [{\widetilde{H}},\rho ] + \sum _{j \in \mathcal {J}} [V_j, \rho V_j^*] + [V_j \rho , V_j^*] \ , \end{aligned}$$
(2.2)
where \(\mathcal {J}\) is a finite index set, \(V_j \in B(\mathscr {H})\) (not necessarily belonging to \(\mathcal {A}\)) for all \(j \in \mathcal {J}\), and the Hamiltonian \({\widetilde{H}} \in B(\mathscr {H})\) is self-adjoint.
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 \(\langle {f,g}\rangle _\sigma = \sum _{j=1}^n \sigma _j \overline{f_j}g_j\).
In the quantum setting, with a reference density matrix \(\sigma \) that is not a multiple of the identity, there are many candidates for such a weighted inner product. E.g., given \(\sigma \in {{\mathfrak {P}}}_+\), and \(s\in [0,1]\) one can define an inner product on \(\mathcal {A}\) by
$$\begin{aligned} \langle {X,Y}\rangle _s = \tau [X^* \sigma ^{s} Y\sigma ^{1-s}]\ . \end{aligned}$$
(2.3)
Note that by cyclicity of the trace, \(\langle {X,X}\rangle _s = \tau [|\sigma ^{s/2}X \sigma ^{(1-s)/2}|^2] \ge 0\), so that \(\langle {\cdot , \cdot }\rangle _s\) is indeed a positive definite sesquilinear form. The inner products for \(s=0\) and \(s=\frac{1}{2}\) will come up frequently in what follows, and they have their own names: \(\langle {\cdot ,\cdot }\rangle _0\) is the Gelfand–Naimark–Segal inner product, denoted \(\langle {\cdot , \cdot }\rangle _{L^{2}_\mathrm{GNS}(\sigma )}\), and \(\langle {\cdot , \cdot }\rangle _{1/2}\) is the Kubo–Martin–Schwinger inner product, denoted \(\langle {\cdot , \cdot }\rangle _{L^2_\mathrm{KMS}(\sigma )}\). We shall write \(\mathcal {A}= L^2_\mathrm{GNS}(\mathcal {A}, \sigma )\) (resp. \(\mathcal {A}= L^2_\mathrm{KMS}(\mathcal {A}, \sigma )\)) if we want to stress this Hilbert space structure.
Suppose, for some \(s\in [0,1]\), that \(\mathscr {P}_t\) is self-adjoint with respect to the \(\langle \cdot ,\cdot \rangle _s\) inner product. Then, for all \(A \in \mathcal {A}\),
$$\begin{aligned} \tau [(\mathscr {P}_t^\dagger \sigma ) A]= & {} \tau [\sigma \mathscr {P}_t A ] = \tau [\sigma ^{1-s} {\mathbf{1}}\sigma ^s \mathscr {P}_t A ] =\langle {\mathbf{1}}, \mathscr {P}_t A\rangle _s\\= & {} \langle \mathscr {P}_t{\mathbf{1}}, A\rangle _s = \langle {\mathbf{1}}, A\rangle _s = \tau [\sigma A ] \ . \end{aligned}$$
Hence for each of these inner products, self-adjointness of \(\mathscr {P}_t\) implies that \(\sigma \) is invariant under \(\mathscr {P}_t^\dagger \).
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
$$\begin{aligned} \langle {A, B}\rangle _{L^2_\mathrm{BKM}(\sigma )} = \int _0^1 \langle {A,B}\rangle _s \; \mathrm {d}s\ . \end{aligned}$$
(2.4)
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.,
$$\begin{aligned} \tau [\sigma A^* \mathscr {P}_t B ] = \tau [\sigma (\mathscr {P}_t A)^* B ] \quad \text { for all } A, B \in \mathcal {A}\ . \end{aligned}$$
We shall write that \((\mathscr {P}_t)_t\) satisfies \(\sigma \)-DBC for brevity.
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
(Structure of Lindblad operators with detailed balance) Let \(\mathscr {P}_t = e^{t \mathscr {L}}\) be a quantum Markov semigroup on \(B(\mathscr {H})\) satisfying detailed balance with respect to \(\sigma \in {{\mathfrak {P}}}_{+}\). Then the generator \(\mathscr {L}\) and its adjoint \(\mathscr {L}^\dagger \) have the form
$$\begin{aligned} \mathscr {L}&= \sum _{j \in \mathcal {J}} e^{-\omega _j/2} \mathscr {L}_j \ , \qquad \mathscr {L}_{j}(A) = V_j^* [A, V_j] + [V_j^* , A] V_j \ , \end{aligned}$$
(2.5)
$$\begin{aligned} \mathscr {L}^\dagger&= \sum _{j \in \mathcal {J}} e^{-\omega _j/2} \mathscr {L}_{j}^\dagger \ , \qquad \mathscr {L}^\dagger _{j}(\rho ) = [V_j, \rho V_j^*] + [V_j \rho , V_j^* ] \ , \end{aligned}$$
(2.6)
where \(\mathcal {J}\) is a finite index set, the operators \(V_j \in B(\mathscr {H})\) satisfy \(\{ V_{j} \}_{j \in \mathcal {J}} = \{ V_{j}^{*} \}_{j \in \mathcal {J}}\), and \(\omega _{j} \in {{\mathbb {R}}}\) satisfies
$$\begin{aligned} \Delta _\sigma V_j&= e^{-\omega _j} V_j \quad \text {for all } j \in \mathcal {J}\ . \end{aligned}$$
(2.7)
For \(j \in \mathcal {J}\), let \(j^{*} \in \mathcal {J}\) be an index such that \(V_{j^{*}} = V_{j}^{*}\). It follows from (2.7) that
$$\begin{aligned} \omega _{j^{*}} = - \omega _{j} \ . \end{aligned}$$
Moreover, if we define \(H = -\log \sigma \), (2.7) is equivalent to the commutator identity \( [V_j,H] = - \omega _{j}V_j \). Furthermore, in our finite-dimensional context, the identity
$$\begin{aligned} \Delta _\sigma ^t V_j = e^{- \omega _j t} V_j \end{aligned}$$
(2.8)
is valid for some \(t\ne 0\) in \({{\mathbb {R}}}\) if and only if it is valid for all \(t \in {{\mathbb {C}}}\).
Gradient Flow Structure for the Non-commutative Dirichlet Energy
Let \((\mathscr {P}_t)_{t\ge 0}\) be a quantum Markov semigroup satisfying detailed balance with respect to \(\sigma \in {{\mathfrak {P}}}_+(\mathcal {A})\). Let \(\mathscr {L}\) be the generator, so that for each \(t>0\), \(\mathscr {P}_t = e^{t\mathscr {L}}\). As explained in the discussion leading up to Definition 2.3, for each 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
$$\begin{aligned} \mathscr {E}(A,A) = \lim _{t\downarrow 0}\frac{1}{t} \langle A, (I - \mathscr {P}_t )A\rangle \end{aligned}$$
(2.9)
where the inner product is either the GNS or the KMS inner product. Then, either way, the 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.
Consider a Lindblad generator \(\mathscr {L}\) given as in Theorem 2.4. To bring out the analogy with classical Kolmogorov backward diffusion equations of the form
$$\begin{aligned} \frac{\partial }{\partial t}f(x,t) = \Delta f(x,t) + (\nabla \log \sigma (x))\cdot \nabla f(x,t)\ , \end{aligned}$$
(2.10)
where \(\sigma \) is a smooth, strictly positive probability density on \({{\mathbb {R}}}^n\), we define the following partial derivative operators on \(\mathcal {A}\):
$$\begin{aligned} \partial _j A = [V_j, A] \ , \end{aligned}$$
(2.11)
\(j\in \mathcal {J}\). Note that \(\partial _{j}^\dagger = \partial _{j^{*}}\), where we recall that \(j^{*}\) denotes an index such that \(V_{j^*} = V_j^*\). An easy computation shows that the adjoint of \(\partial _{j}\) with respect to \(\langle {\cdot ,\cdot }\rangle _{L_\mathrm{KMS}^{2}(\sigma )}\) is given by
$$\begin{aligned} \partial _{j,\sigma }^\dagger A = \sigma ^{-1/2} \partial _{j}^\dagger \big ( \sigma ^{1/2} A \sigma ^{1/2} \big ) \sigma ^{-1/2} \ . \end{aligned}$$
(2.12)
Proposition 2.5
(Divergence form representation of \(\mathscr {L}\)) For all \(A \in \mathcal {A}\) we have
$$\begin{aligned} \mathscr {L}A = - \sum _{j \in \mathcal {J}} \partial _{j,\sigma }^\dagger \partial _{j} A \ . \end{aligned}$$
Proof
Using (2.12) and (2.8) we obtain
$$\begin{aligned} \sum _{j \in \mathcal {J}} \partial _{j,\sigma }^\dagger \partial _{j} A&= \sum _{j \in \mathcal {J}} \partial _{j,\sigma }^\dagger (V_j A - A V_j )\\&= \sum _{j \in \mathcal {J}} \sigma ^{-1/2} \partial _{j}^\dagger \big ( \sigma ^{1/2} (V_j A - A V_j ) \sigma ^{1/2} \big ) \sigma ^{-1/2} \\&= \sum _{j \in \mathcal {J}} \sigma ^{-1/2} \Big ( V_j^* \sigma ^{1/2} (V_j A - A V_j ) \sigma ^{1/2} - \sigma ^{1/2} (V_j A - A V_j ) \sigma ^{1/2} V_j^* \Big ) \sigma ^{-1/2} \\&= \sum _{j \in \mathcal {J}} \Big ( e^{-\omega _j/2} V_j^* (V_j A - A V_j ) - e^{\omega _j/2} (V_j A - A V_j ) V_j^* \Big ) \\&= \sum _{j \in \mathcal {J}} \Big ( e^{-\omega _j/2} V_j^* (V_j A - A V_j ) - e^{-\omega _j/2} (V_j^* A - A V_j^* ) V_j \Big )\\&= -\sum _{j \in \mathcal {J}} e^{-\omega _j/2} \mathscr {L}_j(A) = -\mathscr {L}A\ , \end{aligned}$$
as desired. \(\square \)
Proposition 2.5 can be stated equivalently as an integration by parts identity
$$\begin{aligned} \sum _{j \in \mathcal {J}} \langle {\partial _j A, \partial _j B}\rangle _{L_\mathrm{KMS}^{2}(\sigma )} = - \langle {A,\mathscr {L}B}\rangle _{L_\mathrm{KMS}^{2}(\sigma )} \quad \text { for } A, B \in \mathcal {A}\ . \end{aligned}$$
(2.13)
It is now immediate that the backward equation \(\partial _t A = \mathscr {L}A\) with \(\mathscr {L}\) given by (2.1), is the gradient flow equation for the energy \(\mathscr {E}(A,A)\) with respect to the KMS inner product induced by \(\sigma \). What makes this particular gradient flow representation especially useful is that the Dirichlet form \(\mathscr {E}\) is written, in (2.13), as the expectation of a squared gradient. That is, the gradient flow structure given here is analogous to the gradient flow formulation for the Kolmogorov backward equation (2.10) for the Dirichlet energy \(\mathcal {D}_{class}(f) = \frac{1}{2}\int _{{{\mathbb {R}}}^n} |\nabla f(x)|^2 \sigma (x)\; \mathrm {d}x\). This would not be the case if we had considered the Dirichlet form based on the GNS inner product: We would have a gradient flow structure, but the Dirichlet form would not be the expectation of a squared gradient in any meaningful sense; see however, Proposition 4.12 below for a related representation.
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.
A Gradient Flow Structure for the Quantum Relative Entropy
Consider the quantum relative entropy functionals \({{\,\mathrm{Ent}\,}}_{\sigma }: {{\mathfrak {P}}}_{+} \rightarrow {{\mathbb {R}}}\) given by
$$\begin{aligned} {{\,\mathrm{Ent}\,}}_{\sigma }(\rho ) := \tau [\rho ( \log \rho - \log \sigma ) ] \ . \end{aligned}$$
Our goal is to sketch a proof of one of the results of [10, 36], namely that the quantum master equation \(\partial _t \rho = \mathscr {L}^\dagger \rho \), which is a Kolmogorov 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 \(A, B \in \mathcal {A}_{0}\).
Let \((g_{\rho })_{\rho \in {{\mathfrak {P}}}_{+}}\) be a Riemannian metric on \({{\mathfrak {P}}}_{+}\), i.e., a collection of positive definite bilinear forms \(g_{\rho } : T_{\rho }{{\mathfrak {P}}}_{+} \times T_{\rho }{{\mathfrak {P}}}_{+} \rightarrow {{\mathbb {R}}}\) depending smoothly on \(\rho \in {{\mathfrak {P}}}_{+}\). Consider the associated operator \(\mathscr {G}_{\rho } : T_{\rho }{{\mathfrak {P}}}_{+} \rightarrow T_{\rho }^{\dagger }{{\mathfrak {P}}}_{+}\) defined by \(\langle {A, \mathscr {G}_{\rho } B}\rangle = g_{\rho }(A, B)\) for \(A, B \in T_{\rho } {{\mathfrak {P}}}_{+}\). Clearly, \(\mathscr {G}_{\rho }\) is invertible and self-adjoint with respect to the Hilbert–Schmidt inner product on \(\mathcal {A}_{0}\). Define \(\mathscr {K}_{\rho } : T_{\rho }^{\dagger }{{\mathfrak {P}}}_{+} \rightarrow T_{\rho } {{\mathfrak {P}}}_{+}\) by \(\mathscr {K}_{\rho } = (\mathscr {G}_{\rho })^{-1}\), so that
$$\begin{aligned} g_{\rho }(A,B) = \langle { A, \mathscr {K}_{\rho }^{-1} B}\rangle \ . \end{aligned}$$
(2.14)
In many situations of interest it is convenient to define the metric \(g_{\rho }\) by specifying the operator \(\mathscr {K}_{\rho }\). In such cases, there is often no explicit formula available for \(\mathscr {G}_{\rho }\) and \(g_{\rho }\).
For a smooth functional \(\mathcal {F}: {{\mathfrak {P}}}_{+} \rightarrow {{\mathbb {R}}}\) and \(\rho \in {{\mathfrak {P}}}_{+}\), its 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
$$\begin{aligned} \nabla _g \mathcal {F}(\rho ) = \mathscr {K}_{\rho } \mathrm {D}\mathcal {F}(\rho ) \ . \end{aligned}$$
The gradient flow equation \(\partial _{t}\rho = - \nabla _g \mathcal {F}(\rho )\) takes the form
$$\begin{aligned} \partial _{t}\rho = - \mathscr {K}_{\rho } \mathrm {D}\mathcal {F}(\rho ) \ . \end{aligned}$$
Let us now focus on the relative entropy functional \({{\,\mathrm{Ent}\,}}_\sigma \) for some \(\sigma \in {{\mathfrak {P}}}_+\), and note that its differential is given by
$$\begin{aligned} \mathrm {D}{{\,\mathrm{Ent}\,}}_{\sigma }(\rho ) = \log \rho - \log \sigma \ . \end{aligned}$$
(2.15)
Consider a generator \(\mathscr {L}^\dagger \) written in the form (2.6), i.e.,
$$\begin{aligned} \mathscr {L}^\dagger&= \sum _{j \in \mathcal {J}} e^{-\omega _j/2} \mathscr {L}_{j}^\dagger \ , \qquad \mathscr {L}^\dagger _{j}(\rho ) = [V_j, \rho V_j^*] + [V_j \rho , V_j^* ] \ , \end{aligned}$$
where \(\{V_j\}_{j\in \mathcal {J}}\) is a finite set of eigenvectors of \(\Delta _\sigma \) such that \(\{V_j^*\}_{j\in \mathcal {J}} = \{V_j\}_{j\in \mathcal {J}}\), and where \(\Delta _\sigma V_j = e^{-\omega _j}V_j\) for some \(\omega _j \in {{\mathbb {R}}}\). As before, we use the notation \(\partial _j A := [V_j, A]\).
For \(\rho \in {{\mathfrak {P}}}\) we define \({\widehat{\rho }}_j\in \mathcal {A}\otimes \mathcal {A}\) by
$$\begin{aligned} \widehat{\rho _{j}}= \int _0^1 \big (e^{\omega _j /2} \rho \big )^{1-s} \otimes \big (e^{-\omega _j /2} \rho \big )^{s}\; \mathrm {d}s \ . \end{aligned}$$
We shall frequently make use of the contraction operator \(\# : (\mathcal {A}\otimes \mathcal {A}) \times \mathcal {A}\rightarrow \mathcal {A}\) defined by
$$\begin{aligned} (A \otimes B) \# C&:= ACB \end{aligned}$$
(2.16)
and linear extension. A crucial step towards obtaining the gradient flow structure is the following chain rule for the commutators \(\partial _j\), which involves the differential of the entropy.
Lemma 2.6
(Chain rule for the logarithm) For all \(\rho \in {{\mathfrak {P}}}_+\) and \(j \in \mathcal {J}\) we have
$$\begin{aligned} e^{-\omega _j/2}V_j \rho - e^{\omega _j/2}\rho V_j&= \widehat{\rho _{j}}\# \partial _{j}(\log \rho - \log \sigma ) \ . \end{aligned}$$
(2.17)
Proof
Using (2.7) we infer that
$$\begin{aligned} \partial _{j}( \log \rho - \log \sigma ) = V_j \log (e^{-\omega _j/2} \rho ) - \log (e^{\omega _j/2} \rho ) V_j \ . \end{aligned}$$
Consider the spectral decomposition \(\rho = \sum _\ell \lambda _\ell E_\ell \), where \(\lambda _\ell > 0\) for all 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
$$\begin{aligned} {\widehat{\rho }}_{j} = \sum _{\ell ,m} \Lambda (e^{\omega _j/2} \lambda _\ell , e^{-\omega _j/2} \lambda _m) E_\ell \otimes E_m \ , \end{aligned}$$
where \(\Lambda (\xi ,\eta ) = \int _0^1 \xi ^{1-s} \eta ^s \; \mathrm {d}s = \frac{\xi - \eta }{\log \xi - \log \eta }\) denotes the logarithmic mean of \(\xi \) and \(\eta \). Thus,
$$\begin{aligned}&\widehat{\rho _{j}}\# \big ( \partial _{j}( \log \rho - \log \sigma ) \big ) \\&\quad = \sum _{\ell , m, p} \Lambda (e^{\omega _j/2} \lambda _\ell , e^{-\omega _j/2} \lambda _m) E_\ell \Big ( \log (e^{-\omega _j/2} \lambda _p) V_j E_p - \log (e^{\omega _j/2} \lambda _p) E_p V_j \Big ) E_m\\&\quad = \sum _{\ell , m} \Lambda (e^{\omega _j/2} \lambda _\ell , e^{-\omega _j/2} \lambda _m) \Big ( \log (e^{-\omega _j/2} \lambda _m) - \log (e^{\omega _j/2} \lambda _\ell ) \Big ) E_\ell V_j E_m\\&\quad = \sum _{\ell , m} \big ( e^{-\omega _j/2} \lambda _m - e^{\omega _j/2} \lambda _\ell \big ) E_\ell V_j E_m\\&\quad = e^{-\omega _j/2} V_j \rho - e^{\omega _j/2} \rho V_j\ , \end{aligned}$$
which proves (2.17). \(\square \)
For \(\rho \in {{\mathfrak {P}}}_+\) we define the operator \(\mathscr {K}_\rho : \mathcal {A}\rightarrow \mathcal {A}\) by
$$\begin{aligned} \mathscr {K}_\rho A := \sum _{j \in \mathcal {J}} \partial _{j}^\dagger \big ( \widehat{\rho _{j}}\# \partial _{j} A \big ) \ . \end{aligned}$$
(2.18)
Since \({{\,\mathrm{Tr}\,}}(A^* \mathscr {K}_\rho B) = \overline{{{\,\mathrm{Tr}\,}}(B^* \mathscr {K}_\rho A)}\) for \(A, B \in \mathcal {A}\), it follows that \(\mathscr {K}_\rho \) is a non-negative self-adjoint operator on \(L^2(\mathcal {A}, \tau )\) for each \(\rho \in {{\mathfrak {P}}}_+\). Assuming that \(\mathscr {P}_t\) is ergodic, the operator \(\mathscr {K}_\rho : \mathcal {A}_0 \rightarrow \mathcal {A}_0\) is invertible for each \(\rho \in {{\mathfrak {P}}}_+\) (see Corollary 7.4 below for a proof of this statement). Since \(\mathscr {K}_\rho \) depends smoothly on \(\rho \), it follows that \(\mathscr {K}_\rho \) induces a Riemannian metric on \({{\mathfrak {P}}}_+\) defined by (2.14).
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
For \(\rho \in {{\mathfrak {P}}}_+\) we have the identity
$$\begin{aligned} \mathscr {L}^\dagger \rho = - \mathscr {K}_\rho \mathrm {D}{{\,\mathrm{Ent}\,}}_{\sigma }(\rho )\ , \end{aligned}$$
hence the gradient flow equation of \({{\,\mathrm{Ent}\,}}_\sigma \) with respect to the Riemannian metric induced by \((\mathscr {K}_\rho )_\rho \) is the master equation \(\partial _t \rho = \mathscr {L}^\dagger \rho \).
Proof
Using the identity (2.15), the chain rule from Lemma 2.6, and the fact that \(\{V_j\} = \{V_j^*\}\) and \(\omega _{j^*} = -\omega _j\), we obtain
$$\begin{aligned} \mathscr {K}_\rho \mathrm {D}{{\,\mathrm{Ent}\,}}_{\sigma }(\rho )&= \sum _{j \in \mathcal {J}} \partial _{j}^\dagger \big ( \widehat{\rho _{j}}\# \partial _{j} (\log \rho - \log \sigma ) \big )\\&= \sum _{j \in \mathcal {J}} \partial _{j}^\dagger \big ( e^{-\omega _j/2}V_j \rho - e^{\omega _j/2}\rho V_j \big )\\&= \frac{1}{2} \sum _{j \in \mathcal {J}} \Big ( \partial _{j}^\dagger \big ( e^{-\omega _j/2}V_j \rho - e^{\omega _j/2}\rho V_j \big ) + \partial _{j} \big ( e^{\omega _{j}/2}V_j^* \rho - e^{-\omega _j/2}\rho V_j^* \big ) \Big ) \\&= - \frac{1}{2}\sum _{j \in \mathcal {J}} e^{-\omega _j/2} \Big ( [V_j, \rho V_j^*] + [V_j \rho , V_j^* ] \Big ) + e^{\omega _j/2} \Big ( [V_j^*, \rho V_j] + [V_j^* \rho , V_j ]\Big ) \\&= - \sum _{j \in \mathcal {J}} e^{-\omega _j/2} \Big ( [V_j, \rho V_j^*] + [V_j \rho , V_j^* ] \Big ) \\&= - \mathscr {L}^\dagger \rho \ , \end{aligned}$$
which is the desired identity. \(\square \)
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
The gradient flow structure given in Proposition 2.7 can be viewed as a non-commutative analogue of the Kantorovich gradient flow structure obtained by Jordan, Kinderlehrer and Otto [29] for the Kolmogorov backward equation
$$\begin{aligned} \frac{\partial }{\partial t}\rho (x,t) = \Delta \rho (x,t) - \nabla \cdot ( \rho (x,t) \nabla \log \sigma (x) ) \ . \end{aligned}$$
This structure is formally given in terms of the operator \(K_\rho \) defined by
$$\begin{aligned} K_{\rho } \psi = - \nabla \cdot (\rho \nabla \psi ) \ , \end{aligned}$$
for probability densities \(\rho \) on \({{\mathbb {R}}}^n\) and suitable functions \(\psi : {{\mathbb {R}}}^n \rightarrow {{\mathbb {R}}}\) in analogy with (2.18). As the differential of the relative entropy \({{\,\mathrm{Ent}\,}}_\sigma (\rho ) = \int _{{{\mathbb {R}}}^n} \rho (x)\log \frac{\rho (x)}{\sigma (x)} \; \mathrm {d}x\) is given by \(\mathrm {D}{{\,\mathrm{Ent}\,}}_\sigma (\rho ) = 1 + \log \frac{\rho }{\sigma }\), we have
$$\begin{aligned} K_\rho \mathrm {D}{{\,\mathrm{Ent}\,}}_\sigma (\rho ) = - \Delta \rho + \nabla \cdot (\rho \nabla \log \sigma )\ , \end{aligned}$$
which is the commutative counterpart of Proposition 2.7.
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 \(\sigma \).
Before beginning the proof, we recall some relevant facts, and introduce some notation. Regarding \(\sigma \) as an element of \({{\mathbb {M}}}_n({{\mathbb {C}}})\), we define the operator \(\mathscr {M}\) on \({{\mathbb {M}}}_n({{\mathbb {C}}})\) by
$$\begin{aligned} \mathscr {M}A = \int _0^1 \sigma ^{1-s}A \sigma ^s \; \mathrm {d}s \ . \end{aligned}$$
A simple calculation shows that \({\mathscr {M}}\) is the derivative of the matrix exponential function. Its inverse is the derivative of the matrix logarithm function:
$$\begin{aligned} \mathscr {M}^{-1} A = \int _0^\infty \frac{1}{t+\sigma } A \frac{1}{t+\sigma } \; \mathrm {d}t \ , \end{aligned}$$
(see Example 6.5 below for more details). While the matrix logarithm function is monotone, the matrix exponential is not. Thus \(\mathscr {M}^{-1}\) preserves positivity, but \(\mathscr {M}\) does not. In fact \(A \mapsto \mathscr {M}^{-1} A\) is evidently completely positive. The BKM inner product can now be written as
$$\begin{aligned} \langle {A,B}\rangle _{L^2_\mathrm{BKM}(\sigma )} = \tau [A^* \mathscr {M}B ] = \tau [\mathscr {M}(A^*) B ] \ . \end{aligned}$$
Proof of Theorem 2.9
As before, it will be convenient to consider the operators \((\mathscr {K}_\rho )\) defined by (2.14). Since \(\mathrm {D}{{\,\mathrm{Ent}\,}}_{\sigma }(\rho ) = \log \rho - \log \sigma \), the gradient flow equation \(\partial _t \rho = - \mathscr {K}_{\rho } \mathrm {D}{{\,\mathrm{Ent}\,}}_{\sigma }(\rho )\) becomes
$$\begin{aligned} \mathscr {L}^\dagger \rho = - \mathscr {K}_\rho (\log \rho - \log \sigma ) \ . \end{aligned}$$
(2.19)
Applying this identity to \(\rho _\varepsilon = \sigma + \varepsilon A\) for \(A \in \mathcal {A}_0\), and differentiating at \(\varepsilon = 0\), we obtain using the identity \(\partial _\varepsilon |_{\varepsilon = 0} \log \rho _\varepsilon = \mathscr {M}^{-1} A\) that
$$\begin{aligned} \mathscr {L}^\dagger A = - \mathscr {K}_\sigma \mathscr {M}^{-1} A \ , \end{aligned}$$
(2.20)
Consequently, for \(A, B \in \mathcal {A}\),
$$\begin{aligned} \langle {\mathscr {L}A, B}\rangle _{L^2_\mathrm{BKM}(\sigma )} = \tau [ (\mathscr {L}A)^* \mathscr {M}B] = \tau [ A^* \mathscr {L}^\dagger \mathscr {M}B] = - \tau [ A^* \mathscr {K}_\sigma B] \ . \end{aligned}$$
As \(g_\sigma \) is a symmetric bilinear form, the operator \(\mathscr {K}_\sigma \) is self-adjoint with respect to the Hilbert-Schmidt scalar product. This implies the result. \(\square \)
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 \(\sigma \).
Let \(\mathscr {P}\) be a unital completely positive map such that \(\mathscr {P}^\dagger \sigma = \sigma \), and define
$$\begin{aligned} {{\widetilde{\mathscr {P}}}}(A) = {\mathscr {M}}^{-1}(\sigma ^{1/2} \mathscr {P}(A) \sigma ^{1/2})\ . \end{aligned}$$
Note that
$$\begin{aligned} {\mathscr {M}}^{-1}(\sigma ^{1/2} A \sigma ^{1/2}) = \int _0^\infty \frac{\sigma ^{1/2}}{t + \sigma } A \frac{\sigma ^{1/2}}{t + \sigma } \; \mathrm {d}t \end{aligned}$$
defines a completely positive and unital operator, and hence \({\widetilde{\mathscr {P}}}\) is completely positive and unital. Moreover,
$$\begin{aligned} {\widetilde{\mathscr {P}}}^\dagger (A) = \mathscr {P}^\dagger ( {\mathscr {M}}^{-1}(\sigma ^{1/2} A \sigma ^{1/2}))\ , \end{aligned}$$
and hence \({\widetilde{\mathscr {P}}}^\dagger \sigma = \sigma \). Now observe that \({\widetilde{\mathscr {P}}}\) is self-adjoint with respect to the BKM inner product if and only if \(\mathscr {P}\) is self-adjoint for the KMS inner product. In fact, for all \(A, B \in \mathcal {A}\),
$$\begin{aligned} \langle {{\widetilde{\mathscr {P}}} A , B}\rangle _{L^2_\mathrm{BKM}(\sigma )} = \tau [ \mathscr {M}( \mathscr {M}^{-1} (\sigma ^{1/2} \mathscr {P}(A^*) \sigma ^{1/2})) B] = \langle {\mathscr {P}A, B}\rangle _{L^2_\mathrm{KMS}(\sigma )}\ . \end{aligned}$$
Next, it is clear that \({\widetilde{\mathscr {P}}}\) commutes with \(\Delta _\sigma \) if and only if \(\mathscr {P}\) commutes with \(\Delta _\sigma \). Since there exist completely positive unital maps \(\mathscr {P}\) satisfying \(\mathscr {P}^\dagger \sigma = \sigma \) that are KMS symmetric but do not commute with \(\Delta _\sigma \), there exists completely positive unital maps \({\widetilde{\mathscr {P}}}\) satisfying \({\widetilde{\mathscr {P}}}^\dagger \sigma = \sigma \) that are BKM symmetric but do not commute with \(\Delta _\sigma \).
Moreover, the class of completely positive unital maps \({\widetilde{\mathscr {P}}}\) satisfying \({\widetilde{\mathscr {P}}}^\dagger \sigma = \sigma \) that are BKM symmetric is in some sense larger than the class of completely positive unital maps \(\mathscr {P}\) satisfying \(\mathscr {P}^\dagger \sigma = \sigma \) that are KMS symmetric: The map \(\mathscr {P}\mapsto {\widetilde{\mathscr {P}}}\) is invertible, but \({\mathscr {M}}\) is not even positivity preserving, let alone completely positive, so that
$$\begin{aligned} \mathscr {P}(A) = \sigma ^{-1/2} {\mathscr {M}}({\widetilde{\mathscr {P}}}(A) )\sigma ^{-1/2} \end{aligned}$$
need not be completely positive. It is therefore an interesting problem to characterize the QMS generators that are self-adjoint with respect to the BKM inner product.