**Finite Gibbs variational principle.** Set $P(\beta) = \log \text{tr}(e^{-\beta H})$ and denote by $S(v) = -\text{tr}(v \log v)$ the von Neumann entropy of $v \in \mathcal{S}_{\mathcal{K}}$. Then, for any $v \in \mathcal{S}_{\mathcal{K}}$,

$$S(v) - \beta v(H) \leq P(\beta),$$

with equality iff $v = \omega_{\beta}$.

The relative entropy of two states $v, \rho \in \mathcal{S}_{\mathcal{K}}$ is defined by⁷

$$\mathrm{Ent}(v|\rho) = \mathrm{tr}(v(\log \rho - \log v)). \qquad (19.2)$$

Its basic property is that $\mathrm{Ent}(v|\rho) \leq 0$ with equality iff $v = \rho$. One computes

$$\mathrm{Ent}(v|\omega_{\beta}) = S(v) - \beta v(H) - P(\beta),$$

and so the finite Gibbs variational principle is an immediate consequence of the above property of relative entropy.

In the full generality of algebraic quantum statistical mechanics, observables of a quantum system are described by elements of a $C^*$-algebra $\mathcal{O}$ with unit $1$. For a large part of the general theory, no other structure is imposed on $\mathcal{O}$. The set of selfadjoint elements in $\mathcal{O}$ is denoted by $\mathcal{O}_{\mathrm{sa}}$. Let $\mathcal{O}^*$ be the dual of $\mathcal{O}$ and $\mathcal{S}_{\mathcal{O}}$ the set of positive normalized elements in $\mathcal{O}^*$. We equip $\mathcal{O}^*$ with the weak$^*$-topology, with respect to which $\mathcal{S}_{\mathcal{O}}$ is a compact, convex subset of $\mathcal{O}^*$. The physical states are described by elements of $\mathcal{S}_{\mathcal{O}}$. If the system is in a state $v$, then the number $v(A)$ is interpreted as the expectation value of the observable $A$.

The notion of Hamiltonian is lost, and the dynamics in the Heisenberg picture is, from the outset, described by a strongly continuous $^8$group $\tau = \{\tau^t | t \in \mathbb{R}\}$ of $*-\mathrm{automorphisms}$ of $\mathcal{O}$. The group $\tau$ is called $C^*$-dynamics and the pair $(\mathcal{O}, \tau)$ a $C^*$-dynamical system. Its dual action $\tau^*$ preserves $\mathcal{S}_{\mathcal{O}}$ and describes the dynamics in the Schrödinger picture. We write $A_t$ for $\tau^t(A)$ and $v_t$ for $\tau^{t^*}(v) = v \circ \tau^t$. The function $F_{v,A,B}(t)$ is defined in the same way as in the finite case.

A state $v$ is called $\tau$-invariant (or stationary) if $v_t = v$ for all $t$. The set of all $\tau$-invariant states is denoted by $\mathcal{S}_{\tau}$ and is always non-empty. A triple $(\mathcal{O}, \tau, \omega)$, where $\omega$ is the initial (reference) state of the system, is called a $C^*$-quantum dynamical system.

A state $\omega \in \mathcal{S}_{\tau}$ is called ergodic if

$$\lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \omega(B^* A_t B) dt = \omega(A)\omega(B^* B)$$

holds for all $A, B \in \mathcal{O}$. Upon passing to the GNS representation, it becomes clear that this definition is a special case of the one presented in Section 9.1.

Time-reversal plays an important role in statistical mechanics. An anti-linear involutive $*-\mathrm{automorphism}$ $\Theta$ of $\mathcal{O}$ is a time-reversal of the $C^*$-dynamical system $(\mathcal{O}, \tau)$ if

$$\Theta \circ \tau^t = \tau^{-t} \circ \Theta$$

for all $t \in \mathbb{R}$. A state $\omega$ is called time-reversal invariant if $\omega(\Theta(A)) = \omega(A^*)$ for all $A \in \mathcal{O}$. In the sequel we abbreviate time-reversal invariance by TRI.

Since the notion of Hamiltonian is lost, so is the notion of the Gibbs canonical ensemble (19.1). The KMS-condition, however, survives and extends to the general setting as:

<footnote>$^7$This relative entropy is finite iff $\mathrm{Ker} \rho \subseteq \mathrm{Ker} v$, otherwise it takes the value $-\infty$.</footnote>

<footnote>$^8\lim_{t \to -0} \|\tau^t(A) - A\| = 0$ for all $A \in \mathcal{O}$</footnote>
**KMS-condition.** $v \in \mathcal{S}_{\mathcal{O}}$ is a $(\tau, \beta)$-KMS state if, for all $A, B \in \mathcal{O}$, the function $\mathbb{R} \ni \tau \mapsto F_{v,A,B}(\tau)$ has an analytic extension to the strip $0 < \text{sign}(\beta) \text{Im} z < |\beta|$ that is bounded and continuous on its closure and satisfies the KMS-boundary condition

$$F_{v,A,B}(t + i\beta) = F_{v,B,A}(t)$$

for all $t \in \mathbb{R}$.

We denote by $\mathcal{S}_{(\tau, \beta)}$ the set of all $(\tau, \beta)$-KMS states. At the current level of generality this set might be empty. One always has $\mathcal{S}_{(\tau, \beta)} \subset \mathcal{S}_{\tau}$. A quantum dynamical system $(\mathcal{O}, \tau, \omega)$ is said to be in thermal equilibrium at inverse temperature $\beta \in \mathbb{R}$ (or just thermal) if $\omega$ is a $(\tau, \beta)$-KMS state.

A state $v$ is called modular if there exists $C^*$-dynamics $\zeta_v$ on $\mathcal{O}$ such that $v$ is $(\zeta_v, -1)$-KMS state. $\zeta_v$ is called modular dynamics of $v$ and is unique if it exists. We denote by $\delta_v$ the generator of $\zeta_v$ with the convention $\zeta_v^t = e^{t\delta_v}$. If $v$ is a $(\tau, \beta)$-KMS state and $\beta \neq 0$, then $\zeta_v^t = \tau^{-\beta t}$, or equivalently, $\delta_v = -\beta\delta$ where $\delta$ is the generator of $\tau$. The modular dynamics plays an important role in non-equilibrium statistical mechanics. In the equilibrium setting of quantum spin systems, its a priori existence allows for the formulation of the Gibbs condition as characterization of thermal equilibrium; we will briefly discuss this topic in the next section.

Given $(\mathcal{O}, \tau)$, the two basic questions of equilibrium statistical mechanics are:

(QI) Describe the properties of KMS-states and the structure of the sets $\mathcal{S}_{(\tau, \beta)}$.

(QII) Elucidate the dynamical, and in particular the ergodic, properties of thermal quantum dynamical systems.

Note that (QI) and (QII) have trivial answers in the finite setting: $\mathcal{S}_{(\tau, \beta)}$ is always a singleton and no thermal system is ergodic if $\dim(\mathcal{K}) > 1$. It is precisely this triviality that forces consideration of infinitely extended systems from the outset in the study of (QI) and (QII).

From a general perspective, substantial progress has been made on (QI) and (QII) in the 1970s. The volumes [BR81; BR87] are devoted to presenting these developments. The link between KMS-condition and Tomita-Takesaki's modular theory has played a central role in this progress. However, from the standpoint of concrete physically relevant models, advancement has been much slower, and comparatively little is known at a mathematically rigorous level.

The original definition of relative entropy of a pair $(v, \rho)$ of states on $\mathcal{O}$ goes back to Araki and is based on modular theory. This definition was discussed in Section 12.1 and we will review it again in Section 25. The relative entropy is also characterized by the Pusz-Woronowicz-Kosaki variational formula

$$\mathrm{Ent}(v|\rho) = \inf \int_{0}^{\infty} \left[ \frac{1}{t} v(x^{*}(t)x(t)) + \rho(y^{*}(t)y(t)) - \frac{1}{1+t} \right] \frac{dt}{t}, \qquad (19.3)$$

where $y(t) = 1 - x(t)$ and the infimum is taken over all countably valued step functions $[0, \infty] \ni t \mapsto x(t) \in \mathcal{O}$ vanishing in a neighborhood of zero; see [OP93, Section 5]. Its basic property is $\mathrm{Ent}(v|\rho) \le 0$ with equality iff $v = \rho$. In the finite case (19.3) reduces to (19.2). The relative entropy plays a very important role in both equilibrium and non-equilibrium quantum statistical mechanics. In the equilibrium case and in the context of quantum lattice spin systems, relative entropy – through its relation with the specific entropy – plays a central role in the derivation of the Gibbs variational principle. This topic is discussed in detail in Section 6.2.3 of [BR81].

We conclude this section with three general remarks.

Modular theory and the closely related Araki's perturbation theory of KMS-structure play a central role in both equilibrium and non-equilibrium quantum statistical mechanics. A basic introduction to this subject can be
found in [BR81; BR87]; see also [DJC03] and references therein for modern expositions. A pedagogical introduction to the modular theory in the context of finite quantum systems can be found in [Jak+12a]

$W^*$-dynamical systems play a distinguished role in modular theory. A $W^*$-dynamical system is a pair $(\mathfrak{M}, \tau)$ where $\mathfrak{M}$ is a $W^*$-algebra and $\tau = \{\tau^t | t \in \mathbb{R}\}$ is a pointwise $\sigma$-weakly continuous group of $*$-automorphisms on $\mathfrak{M}$. We shall refer to such $\tau$ as $W^*$-dynamics. A triple $(\mathfrak{M}, \tau, \omega)$, where $\omega$ is a normal state on $\mathfrak{M}$, is called a $W^*$-dynamical system. In the general development of non-equilibrium quantum statistical mechanics, the $C^*$-quantum dynamical systems are preferred starting point since the central notion of non-equilibrium steady states cannot be naturally defined in the $W^*$-setting. In this context, the $W^*$-systems and modular structure emerge through the GNS-representation of $\mathcal{O}$ associated to the reference state $\omega$. This being said, there are important physical models that can be only described by $W^*$-dynamical system; a well-known example are Pauli-Fierz systems with bosonic reservoirs. It is most natural to develop the non-equilibrium statistical mechanics of such systems on a case-by-case basis.

The thermodynamic (often abbreviated TD) limit plays a distinguished role in statistical mechanics. It realizes infinitely extended systems through a limiting procedure involving only finite quantum system and is central for the identification of physically relevant objects in the extended setting. The precise way the TD limit is taken depends on the structure of the specific physical model under consideration, and often a number of different approximation routes are possible. This topic is well-understood and discussed on many places in the literature; see [BR81; BR87; Rue69] and [Jak+12a] for a pedagogical introduction to the topic. Since the early days it is well-known that the modular structure is stable under TD limit [AI74]; this fact plays an important role in the foundations of quantum statistical mechanics. The customary route in discussions of the structural theory is the following:

**Step 1.** A physical notion, introduced in the context of finite quantum systems, is expressed in a modular form, and through this form is directly extended, by definition, to a general $C^*$ or $W^*$-dynamical system. One basic example of such procedure is the introduction of the KMS-condition as characterization of thermal equilibrium states.

**Step 2.** In concrete physical models the definitions of Step 1 are justified by the TD limit.

Step 2 has been extensively studied in the early days of quantum statistical mechanics, and the wealth of obtained results make its implementation in modern literature most often a routine exercise. For this reason, this step is often skipped. There are rare exceptions to this rule, one of which emerged in the study of entropy production of open quantum systems [Ben+24a; Ben+25a].

## 20 Lattice quantum spin systems

We follow [BR81]; see also [Isr79; Sim93; Rue69].

**$C^*$-algebra.** The lattice $G$ is taken to be a countably infinite set. At this stage, no additional structure on $G$ is assumed. The collection of all finite subsets of $G$ is denoted by $\mathfrak{G}_{\text{fin}}$. If $(\Lambda_\alpha)$ is a net in $\mathfrak{G}_{\text{fin}}$, $\Lambda_\alpha \to \infty$ means that $\Lambda_\alpha$ eventually contains any finite subset of $G$.

The single spin Hilbert space is finite dimensional and is denoted by $\mathfrak{h}$. To each $x \in G$ we associate a copy $\mathfrak{h}_x$ of $\mathfrak{h}$, and to each $\Lambda \in \mathfrak{G}_{\text{fin}}$ the Hilbert space

$$ \mathcal{K}_{\Lambda} = \bigotimes_{x \in \Lambda} \mathfrak{h}_x. $$

In the sequel, $\mathfrak{O}_\Lambda$ stands for $\mathfrak{O}_{\mathcal{K}_\Lambda}$. The $C^*$-algebra $\mathfrak{O}_\Lambda$ describes observables of spins located at the points in $\Lambda$. For $\Lambda \subset \Lambda'$ one naturally identifies $\mathfrak{O}_\Lambda$ with a $C^*$-subalgebra of $\mathfrak{O}_{\Lambda'}$. The algebra of local observables is

$$ \mathfrak{O}_{\text{loc}} = \bigcup_{\Lambda \in \mathfrak{G}_{\text{fin}}} \mathfrak{O}_{\Lambda}. $$
Finally, the G-lattice spin system $C^*$-algebra $\mathcal{O}_G$ is the completion of the $*$-algebra of local observables. The algebra $\mathcal{O}_G$ is unital, simple, and separable. For any $G_0 \subset G$ one has the natural identification $\mathcal{O}_G = \mathcal{O}_{G_0} \otimes \mathcal{O}_{G_0^c}$. Whenever G is understood, we write $\mathcal{O}$ for $\mathcal{O}_G$.

**Dynamics.** An interaction is a map $\Phi: \mathfrak{G}_{\text{fin}} \to \mathfrak{O}_{\text{sa}}$ such that $\Phi(X) \in \mathfrak{O}_X$. For $\Lambda \in \mathfrak{G}_{\text{fin}}$, the local Hamiltonians are defined by

$$H_{\Lambda}(\Phi) = \sum_{X \subset \Lambda} \Phi(X).$$

They generate the local $C^*$-dynamics $\tau_{\Phi,\Lambda}$ on $\mathcal{O}$, given by

$$\tau_{\Phi, \Lambda}^{t}(A) = e^{itH_{\Lambda}(\Phi)} A e^{-itH_{\Lambda}(\Phi)}.$$

To extend $\tau_{\Phi,\Lambda}$ to a $C^*$-dynamics on $\mathcal{O}$ in the limit $\Lambda \to \infty$, one needs a suitable regularity assumption. We choose the following one<fnref n="9" />

(SR) For some $\lambda > 0$,

$$\|\Phi\|_{\lambda} := \sum_{n \ge 0} e^{\lambda n} \left( \sup_{x \in G} \sum_{\substack{x \in X \\ |X|=n+1}} \|\Phi(X)\| \right) < \infty.$$

**Theorem 20.1.** Suppose that (SR) holds. Then:

1. For all $A \in \mathcal{O}$, the limit

$$\tau_{\Phi}^{t}(A) := \lim_{\Lambda \to \infty} \tau_{\Phi, \Lambda}^{t}(A)$$

exists and is uniform for $t$ in compact sets.

2. $\tau_{\Phi} = \{\tau_{\Phi}^t | t \in \mathbb{R}\}$ is a $C^*$-dynamics on $\mathcal{O}$. We denote by $\delta_{\Phi}$ its generator.

3. $\mathcal{O}_{\text{loc}} \subset \text{Dom}(\delta_{\Phi})$, and for $A \in \mathcal{O}_{\Lambda}$,

$$\delta_{\Phi}(A) = i \sum_{X \cap \Lambda \neq \emptyset} [\Phi(X), A].$$

Moreover, $\mathcal{O}_{\text{loc}}$ is a core for $\delta_{\Phi}$.

4. For $A \in \mathcal{O}_{\text{loc}}$ and $n \ge 1$,

$$\|\delta_{\Phi}^{n}(A)\| \le \frac{2^{n} n!}{\lambda^{n}} e^{\lambda |\Lambda|} \|\Phi\|_{\lambda}^{n} \|A\|,$$

where $|\Lambda|$ denotes the cardinality of $\Lambda$. In particular, for all $A \in \mathcal{O}_{\text{loc}}$, the map

$$\mathbb{R} \ni t \mapsto \tau_{\Phi}^{t}(A) \in \mathcal{O}$$

has an analytic extension to the strip $|\operatorname{Im} z| < \frac{\lambda}{2} \|\Phi\|_{\lambda}^{-1}$.

For the proof, we refer the reader to [Rue69, Lemma 7.6.1 and Theorem 7.6.2] and [BR81, Theorem 6.2.4].

Until the end of this section we assume that (SR) holds.

**KMS-states.** Let $\beta > 0$ be the inverse temperature. For $\Lambda \in \mathfrak{G}_{\text{fin}}$, the local Gibbs state is defined by

$$\omega_{\Lambda, \beta} = e^{-\beta H_{\Lambda}(\Phi)} / \text{tr}(e^{-\beta H_{\Lambda}(\Phi)}). \qquad (20.1)$$

Using the identification $\mathcal{O} = \mathcal{O}_{\Lambda} \otimes \mathcal{O}_{\Lambda^c}$, one extends $\omega_{\Lambda,\beta}$ (in an arbitrary way) to a state on $\mathcal{O}$. Denoting this extension by the same letter, it is a basic result that any weak\*-limit point of the net $(\omega_{\beta}(\Lambda))_{\Lambda \in \mathfrak{G}_{\text{fin}}}$ as $\Lambda \to \infty$ is

<footnote n="9">⁹See Theorem 6.2.4 in [BR81].</footnote>
a $(\tau_\Phi, \beta)$-KMS state on $\mathcal{O}$. The $(\tau_\Phi, \beta)$-KMS states that arise in this way are called thermodynamic-limit point $\beta$-KMS states.<fnref n="10" /> This construction, in particular, shows that the set $\mathcal{S}_{(\tau_\Phi, \beta)}$ is non-empty.

If it happens that $\mathcal{S}_{(\tau_\Phi, \beta)}$ is a singleton, then all the nets $(\omega_\beta(\Lambda))_{\Lambda \in \mathcal{G}_{\text{fin}}}$ converge, as $\Lambda \to \infty$, to the unique $(\tau_\Phi, \beta)$-KMS state. This is known to be the case in the high-temperature regime, i.e., for $\beta \|\Phi\|_\lambda$ small enough. For concrete estimates, see [BR81, Proposition 6.2.45], or [FU15] for more recent results.

If $\mathcal{S}_{(\tau_\Phi, \beta)}$ is not a singleton, one needs to take into account boundary conditions to reach all KMS-states by thermodynamic limit. That is our next topic.

**Araki-Gibbs condition.** For any $\Lambda \in \mathcal{G}_{\text{fin}}$, the so called surface energies

$$W_{\Lambda}(\Phi) := \sum_{\substack{X \cap \Lambda \neq \emptyset \\ X \cap \Lambda^{c} \neq \emptyset}} \Phi(X)$$

are in $\mathcal{O}_{\text{sa}}$. Let $\beta > 0$ and $V_\Lambda = \beta W_\Lambda(\Phi)$. Suppose that $\omega$ is a modular state on $\mathcal{O}$ and let $\delta_\omega$ be the generator of its modular $C^*$-dynamics $\zeta_\omega$, $\zeta_\omega^t = e^{t\delta_\omega}$. Consider the perturbed dynamics $\zeta_{\omega, V_\Lambda}$ generated by $\delta_\omega + i[V_\Lambda, \cdot]$, and let $\omega_{V_\Lambda}$ be the $(\zeta_{\omega, V_\Lambda}, -1)$-KMS state associated to $\omega$ by Araki's perturbation theory. We say that $\omega$ satisfies $(\beta, \Phi)$ Gibbs condition if, for all $\Lambda \in \mathcal{G}_{\text{fin}}$, the restriction of $\omega_{V_\Lambda}$ to $\mathcal{O}_\Lambda$ is given by (20.1). The Gibbs condition is the quantum counterpart of the DLR equation in the equilibrium theory of classical spin systems. For the proofs, see [BR81, Proposition 6.2.17].

**Theorem 20.2.** Suppose that $\omega$ is a modular state on $\mathcal{O}$ and $\beta > 0$. The following statements are equivalent.

1. $\omega$ is a $(\tau_\Phi, \beta)$-KMS state.

2. $\omega$ satisfies the $(\beta, \Phi)$ Gibbs condition.

**Gibbs variational principle.** We will discuss the Gibbs variational principle only in the translation invariant setting; for a general lattice theory see [BR81, Section 6.2.3]. Suppose that $G = \mathbb{Z}^d$, and let $\varphi = \{\varphi^x | x \in \mathbb{Z}^d\} \subset \text{Aut}(\mathcal{O})$ denote the natural action of the translation group $\mathbb{Z}^d$ on $\mathcal{O}$. We denote by $\mathcal{S}_1$ the set of all translation-invariant (that is, $\varphi$-invariant) states on $\mathcal{O}$. For any $v \in \mathcal{S}_1$, the limit

$$s(v) = \lim_{\Lambda \to \infty} \frac{S(v_\Lambda)}{|\Lambda|}$$

exists, where $v_\Lambda$ denotes the restriction of $v$ to $\mathcal{O}_\Lambda$ and $S(v_\Lambda)$ is its von Neumann entropy. $s(v)$ is the specific entropy of the state $v$. The entropy map $\mathcal{S}_1 \ni v \mapsto s(v)$ is affine, upper-semicontinuous, and takes values in $[0, \log \dim h]$. An interaction $\Phi$ is translation invariant if

$$\varphi^x(\Phi(X)) = \Phi(X + x)$$

holds for all finite subsets $X \subset \mathbb{Z}^d$ and all $x \in \mathbb{Z}^d$. For translation invariant interactions,

$$\|\Phi\|_{\lambda} = \sum_{X \ni 0} e^{\lambda(|X|-1)} \|\Phi(X)\|.$$

In what follows we fix a translation invariant $\Phi$ satisfying $\|\Phi\|_\lambda < \infty$ for some $\lambda > 0$. The specific energy observable of $\Phi$ is

$$E_{\Phi} = \sum_{X \ni 0} \frac{\Phi(X)}{|X|},$$

and for any $\omega \in \mathcal{S}_1$,

$$\lim_{\Lambda \to \infty} \frac{1}{|\Lambda|} \omega(H_{\Lambda}(\Phi)) = \omega(E_{\Phi}).$$

<footnote n="10">¹⁰We follow the terminology introduced in [BR81, Section 6.2.2.].</footnote>
Finally, the finite limit

$$P(\Phi) = \lim_{\Lambda \to \infty} \frac{1}{|\Lambda|} \log \mathrm{tr}(e^{-H_{\Lambda}(\Phi)})$$

exists and is called the pressure of $\Phi$. The next theorem is known as the Gibbs variational principle.

**Theorem 20.3.** For any $\beta > 0$,

$$P(\beta\Phi) = \sup_{v \in \mathcal{S}_1} (s(v) - \beta v(E_\Phi)).$$

Moreover,

$$\mathcal{S}_{\mathrm{eq}}(\beta\Phi) = \{v \in \mathcal{S}_1 | P(\beta\Phi) = s(v) - \beta v(E_\Phi)\}$$

is a non-empty convex compact subset of $\mathcal{S}_1$. The elements of $\mathcal{S}_{\mathrm{eq}}(\beta\Phi)$ are called equilibrium states for interaction $\Phi$ at inverse temperature $\beta$.

Finally, equilibrium states and KMS states are related by

**Theorem 20.4.** For any $\beta > 0$ the following statements are equivalent.

1. $\omega \in \mathcal{S}_{\mathrm{eq}}(\beta\Phi)$.

2. $\omega$ is a translation invariant $(\tau_{\Phi}, \beta)$-KMS state.

For further discussion of Theorems 20.3 and 2, we refer the reader to [BR81, Proposition 6.2.39 and Theorem 6.2.40], and [BR81, Theorem 6.2.42].

# Part VI

# Non-equilibrium quantum statistical mechanics

The works [JP01; Rue00b; Rue01; Pil01]$^{11}$ initiated modern developments in non-equilibrium quantum statistical mechanics by introducing the notions of entropy production observable and non-equilibrium steady states (NESS). The concept of quantum entropy production observable goes back at least to [PW78]$^{12}$ and was reintroduced independently in the literature several times since; see [OHI88; Oji89; Oji91]. The concept of NESS in the quantum setting was novel and motivated by developments in classical non-equilibrium statistical mechanics; see [Rue99]. The introduction of the quantum entropy production observable in conjunction with NESS has spurred rapid developments in non-equilibrium quantum statistical mechanics—both in structural theory and in the study of concrete models—that continue to this day. For a non-exhaustive list of references, we refer the reader to [Ben+24a].

## 21 General setting

We follow [JP01]. Let $(\mathcal{O}, \tau, \omega)$ be a $C^*$-quantum dynamical system whose reference state $\omega$ is not $\tau$-invariant. The NESS of $(\mathcal{O}, \tau, \omega)$ are the limit points of the net

$$\left\{ \frac{1}{T} \int_{0}^{T} \omega \circ \tau^{t} dt \right\}_{T>0}$$

<footnote>$^{11}$The work [HA00], although focused on the concrete exactly solvable non-equilibrium XY-spin chain, foreshadowed these developments.</footnote>

<footnote>$^{12}$See the Remark on page 281 of this paper. Another pioneering work on the subject is [SL78].</footnote>
as $T \uparrow \infty$. The set $\mathcal{S}_{\tau+}(\omega)$ of NESS is non-empty and its elements are $\tau$-invariant.

To introduce the entropy production observable, we assume that the state $\omega$ is modular. Recall that $\delta_{\omega}$ denotes the generator of the modular dynamics $\zeta_{\omega}$. We further assume that the generator $\delta$ of $\tau$ has the form

$$ \delta = \delta_{\mathrm{fr}} + i[V, \cdot], $$

where $V \in \mathcal{O}_{sa}$ and $\delta_{\mathrm{fr}}$ generates a "free" $C^*$-dynamics $\tau_{\mathrm{fr}}$ for which $\omega \in \mathcal{S}_{\tau_{\mathrm{fr}}}$. Lacking a better terminology, we shall say that such a system $(\mathcal{O}, \tau, \omega)$ is locally perturbed (abbreviated LP). If $V \in \mathrm{Dom}(\delta_{\omega})$, the entropy production observable of a LP system is defined by

$$ \sigma := \delta_{\omega}(V). $$

When we wish to indicate the dependence of $\tau$ and $\sigma$ on $V$ we will write $\tau_V, \sigma_V$.

The starting point of the theory is the entropy balance equation

$$ \mathrm{Ent}(\omega_t|\omega) = - \int_0^t \omega_s(\sigma) \mathrm{d}s. \qquad (21.1) $$

Since the relative entropy is non-positive, it follows that

$$ \omega_+(\sigma) \geq 0 $$

for any NESS $\omega_+ \in \mathcal{S}_{\tau+}(\omega)$.

The two basic questions of non-equilibrium quantum statistical mechanics are:

(QIII) Describe the structure of the set $\mathcal{S}_{\tau+}$ and the properties of its elements.

(QIV) Elucidate dynamical mechanisms that ensure $\omega_+(\sigma) > 0$ for all $\omega_+ \in \mathcal{S}_{\tau+}$.

There are two general approaches to (QIII). The first is the $C^*$-scattering method that goes back to Robinson [Rob73] and in the context of NESS was further developed by Ruelle [Rue00b]; see also [Asc+06]. In this method the goal is to prove that for $A \in \mathcal{O}$ the limit

$$ \gamma^+(A) := \lim_{t \to \infty} \tau_{\mathrm{fr}}^{-t} \circ \tau^t(A) $$

exists. In this case $\gamma^+ \in \mathrm{Aut}(\mathcal{O})$, $\lim_{t \to \infty} \omega_t = \omega_+ = \omega \circ \gamma^+$, and $\mathcal{S}_{\tau+} = \{\omega_+\}$. A sufficient condition that ensures the existence of the Møller morphism $\gamma^+$ is that there exists a dense subset $A \subset \mathcal{O}$ such that for all $A \in A$,

$$ \int_0^\infty \| [\tau^t(A), V] \| dt < \infty. $$

Conditions of this type are called $L^1$-asymptotic Abelianness and have been extensively used in the literature; see [BR81]. For some physically important models where this condition has been verified, see [BM83; FMU03; JOP07].

The second general approach to (QIII) was developed in [JP02b] based on the spectral theory of quantum transfer operators. It is a quantum extension of Ruelle transfer operator method; see [Bal00]. For a pedagogical introduction to quantum transfer operators in finite quantum system setting, see [Jak+12a]. For modern developments, see [Ben+24b].

Regarding (QIV), we refer the reader to [Asc+06; JP02a] and references therein. Both questions (QIII) and (QIV) remain poorly understood and much work remains to be done.
A special class of LP systems, the so-called open quantum systems, play a privileged role in the study of non-equilibrium quantum statistical mechanics and we proceed to describe them.

Consider finitely many, say $M$, thermal reservoirs $\mathcal{R}_j$ described by $C^*$-quantum dynamical systems $(\mathcal{O}_j, \tau_j, \omega_j)$. We denote by $\delta_j$ the generator of $\tau_j$. The reservoir $\mathcal{R}_j$ is assumed to be in thermal equilibrium at inverse temperature $\beta_j > 0$, that is, we assume that $\omega_j$ is a $(\tau_j, \beta_j)$-KMS state on $\mathcal{O}_j$. In the absence of interaction, the combined reservoir system $\mathcal{R} = \mathcal{R}_1 + \cdots + \mathcal{R}_M$ is described by the quantum dynamical system $(\mathcal{O}_{\mathcal{R}}, \tau_{\mathcal{R}}, \omega_{\mathcal{R}})$, where<fnref n="13" />

$$ \mathcal{O}_{\mathcal{R}} := \mathcal{O}_1 \otimes \cdots \otimes \mathcal{O}_M, $$

$$ \tau_{\text{fr}} := \tau_1 \otimes \cdots \otimes \tau_M, $$

$$ \omega_{\mathcal{R}} := \omega_1 \otimes \cdots \otimes \omega_M. $$

There are two ways to couple the reservoirs. In the first case of directly coupled reservoirs, $\tau_{\text{fr}} = \tau_{\mathcal{R}}$, the interaction is described by $V \in \mathcal{O}_{\mathcal{R},\text{sa}}$, and the interacting dynamics $\tau$ is generated by $\delta = \delta_{\mathcal{R}} + i[V, \cdot]$.

The reservoirs can be also coupled through a finite quantum system $\mathcal{S}$ with Hilbert space $\mathcal{K}_{\mathcal{S}}$. Let $(\mathcal{O}_{\mathcal{S}}, \tau_{\mathcal{S}}, \omega_{\mathcal{S}})$ be a finite-dimensional quantum system dynamical describing $\mathcal{S}^{14}$, where we assume that $\omega_{\mathcal{S}} > 0$. The generator of $\tau_{\mathcal{S}}$ is $\delta_{\mathcal{S}} = i[H_{\mathcal{S}}, \cdot]$, where $H_{\mathcal{S}}$ is the Hamiltonian of $\mathcal{S}$. In the absence of interaction, the joint system $\mathcal{S} + \mathcal{R}$ is described by the $C^*$-quantum dynamical system $(\mathcal{O}, \tau_{\text{fr}}, \omega)$ where

$$ \mathcal{O} = \mathcal{O}_{\mathcal{S}} \otimes \mathcal{O}_{\mathcal{R}}, \quad \tau_{\text{fr}} = \tau_{\mathcal{S}} \otimes \tau_{\mathcal{R}}, \quad \omega = \omega_{\mathcal{S}} \otimes \omega_{\mathcal{R}}. $$

The interaction of $\mathcal{S}$ with $\mathcal{R}_j$ is described by a selfadjoint $V_j \in \mathcal{O}_{\mathcal{S}} \otimes \mathcal{O}_j$, and the full interaction by $V := \sum_j V_j$. The interacting dynamics $\tau$ is generated by $\delta := \delta_{\text{fr}} + i[V, \cdot]$. We assume that $V_j \in \text{Dom}(\delta_j)$ for all $j$, which ensures that the entropy production observable is well-defined.

In what follows, we will always take

$$ \omega_{\mathcal{S}} = 1 / \dim \mathcal{K}_{\mathcal{S}} \qquad (21.2) $$

for the reference state of $\mathcal{S}$. This choice is made for convenience. It is easy to show that none of the asymptotic results of this paper depend on the choice of $\omega_{\mathcal{S}}$ as long as $\omega_{\mathcal{S}}$ is faithful. With the choice (21.2),

$$ \sigma = - \sum \beta_j \mathcal{J}_j, $$

where the observable $\mathcal{J}_j := \delta_j(V_j)$ describes the energy flux out of the j-th reservoir.

The above description of open quantum system needs to be adapted in the case of fermionic systems. The modifications are straightforward; see [Asc+06; JOP07].

## 22 Open lattice quantum spin systems

We follow [Rue01], and consider the lattice quantum spin system of Section 20, the interaction $\Phi$ satisfying Assumption (SR). We further assume:

(a) For some finite $M$,

$$ G = S \cup \left( \bigcup_{j=1}^{M} R_j \right) $$

where $S$ is a finite set and each $R_j$ is an infinite set. The sets $S, R_1, \dots, R_M$ are assumed to be disjoint.

<footnote n="13">¹³Whenever the meaning is clear within the context, we write $A$ for $A \otimes 1$ and $1 \otimes A$, $\delta_j$ for $\delta_j \otimes \text{Id}$, $\text{Id} \otimes \delta_j$, etc.</footnote>

<footnote n="14">¹⁴We abbreviate $\mathcal{O}_{\mathcal{K}_{\mathcal{S}}}$ by $\mathcal{O}_{\mathcal{S}}$</footnote>
We adapt the notation of Section 20 to that of Section 21 by setting $\mathcal{O}_{\mathcal{R}_j} = \mathcal{O}_{R_j}$ for $j \in \{1, \dots, M\}$, $\mathcal{O}_{\mathcal{S}} = \mathcal{O}_{S}$, and $\mathcal{K}_{\mathcal{S}} := \otimes_{x \in S} \mathfrak{h}_x$. One then has the decomposition

$$ \mathcal{O} = \mathcal{O}_{\mathcal{S}} \otimes \mathcal{O}_{\mathcal{R}_j} = \mathcal{O}_{\mathcal{S}} \otimes \mathcal{O}_{\mathcal{R}_1} \otimes \cdots \otimes \mathcal{O}_{\mathcal{R}_M}. $$

In the framework of Section 21, the subsystems described by the subalgebras $\mathcal{O}_{\mathcal{R}_j}$ act as reservoirs, and the next assumption forbids direct coupling between these reservoirs.

(b) For any $X \in \mathcal{G}_{\text{fin}}$, if $X \cap R_i \neq \emptyset$, $X \cap R_j \neq \emptyset$ for some $i \neq j$, then $\Phi(X) = 0$.

We denote by $\mathcal{G}_{j,\text{fin}}$ the collection of all finite subsets of $R_j$, and by $\Phi_j$ the restriction of $\Phi$ to $\mathcal{G}_{j,\text{fin}}$. $\tau_{\mathcal{R}_j}$ denotes the $C^*$-dynamics on $\mathcal{O}_{\mathcal{R}_j}$ generated by $\Phi_j$ and $\delta_{\mathcal{R}_j}$ its generator. The coupling of the finite subsystem $\mathcal{S}$ with the $j$-th reservoir is given by

$$ V_j := \sum_{X \cap S \neq \emptyset, X \cap R_j \neq \emptyset} \Phi(X), $$

and we set $V := \sum_j V_j$. The "free" $C^*$-dynamics $\tau_{\text{fr}}$ on $\mathcal{O}$ is generated by

$$ \delta_{\text{fr}} = \delta_{\mathcal{S}} + \sum_j \delta_{\mathcal{R}_j}, $$

where $\delta_{\mathcal{S}} = i[H_S(\Phi), \cdot]$. The fully coupled dynamics $\tau = \tau_{\Phi}$ on $\mathcal{O}$ is generated by

$$ \delta_{\Phi} = \delta_{\text{fr}} + i[V, \cdot]. $$

The model is completed by a choice of $(\tau_{\mathcal{R}_j}, \beta_j)$-KMS state $\omega_{\beta_j}$ on $\mathcal{O}_{\mathcal{R}_j}$ for $j = 1, \dots, M$, and by taking<fnref n="15" />

$$ \omega = \omega_{\mathcal{S}} \otimes \omega_{\beta_1} \otimes \cdots \otimes \omega_{\beta_M} $$

for the reference state of $(\mathcal{O}, \tau)$. $(\mathcal{O}, \tau, \omega)$ is an example of open quantum system discussed in Section 21.

Assumption (SR) ensures that

$$ \sum_j \sum_Y \sum_{X \cap S \neq \emptyset, X \cap R_j \neq \emptyset} \| [\Phi(Y), \Phi(X)] \| < \infty. $$

Since $\delta_{\omega} = -\sum_j \beta_j \delta_{\mathcal{R}_j}$, we deduce that $V \in \text{Dom}(\delta_{\omega})$, the entropy production observable being given by

$$ \sigma = \delta_{\omega}(V) = \sum_j \sum_{Y \subset R_j} \sum_{X \cap S \neq \emptyset, X \cap R_j \neq \emptyset} -i\beta_j [\Phi(Y), \Phi(X)]. $$

We now turn to the discussion of the thermodynamic limit of open lattice quantum spin systems. One starts with $\Lambda \in \mathcal{G}_{\text{fin}}$ of the form

$$ \Lambda = S \cup \left( \bigcup_{j=1}^{M} \Lambda_j \right), $$

where $\Lambda_j \in \mathcal{G}_{j,\text{fin}}$, and

$$ \mathcal{O}_{\Lambda} = \mathcal{O}_{\mathcal{S}} \otimes \mathcal{O}_{\Lambda_1} \otimes \cdots \otimes \mathcal{O}_{\Lambda_M}. $$

The finite volume dynamics $\tau_{\Lambda}$ is generated by the Hamiltonian

$$ H_{\Lambda}(\Phi) = H_{\text{fr},\Lambda} + V_{\Lambda}, $$

where

$$ H_{\text{fr},\Lambda} := H_S(\Phi) + \sum_j H_{\Lambda_j}(\Phi_j), \quad V_{\Lambda} := \sum_j V_{j,\Lambda}, $$

<footnote n="15">¹⁵$\omega_{\mathcal{S}}$ is given by (21.2).</footnote>
with

$$V_{j,\Lambda} := \sum_{\substack{X \subseteq S \cup \Lambda_j \\ X \cap S \neq \emptyset, X \cap \Lambda_j \neq \emptyset}} \Phi(X).$$

The finite volume reference state has a product structure

$$\omega_{\Lambda} = \omega_{\mathcal{S}} \otimes \omega_{\Lambda_1} \otimes \cdots \otimes \omega_{\Lambda_M}, \qquad (22.1)$$

where

$$\omega_{\Lambda_j} = e^{-\beta_j H_{\Lambda_j}(\Phi_j)} / \text{tr}(e^{-\beta_j H_{\Lambda_j}(\Phi_j)}). \qquad (22.2)$$

The first basic question regarding the thermodynamic limit in the present setting concerns the entropy production observable. In a finite volume $\Lambda$, the entropy production observable is given by

$$\sigma_{\Lambda} = -i \sum_{j} \beta_{j} [H_{\Lambda_{j}}(\Phi_{j}), V_{j,\Lambda}] = i \sum_{j} \beta_{j} [H_{\Lambda}(\Phi), H_{\Lambda_{j}}(\Phi_{j})]. \qquad (22.3)$$

The finite volume entropy balance equation takes the form

$$\text{Ent}(\omega_{\Lambda} \circ \tau_{\Lambda}^{t} | \omega_{\Lambda}) = - \int_{0}^{t} \omega_{\Lambda}(\tau_{\Lambda}^{s}(\sigma_{\Lambda}))ds,$$

and of course has an elementary proof.

**Theorem 22.1.** 1. $\lim_{\Lambda \to \infty} \sigma_{\Lambda} = \sigma$.

2. Suppose that each $\omega_{\beta_j}$ is a thermodynamic-limit point $\beta_j$-KMS state on $\mathcal{R}_{\beta_j}$. Let

$$\Lambda_{\alpha} = S \cup \left( \bigcup_{j=1}^{M} \Lambda_{j,\alpha} \right),$$

be a net in $\mathcal{G}_{\text{fin}}$ such that for each $j$,

$$\lim_{\Lambda_{j,\alpha} \to \infty} \omega_{\Lambda_{j,\alpha}} = \omega_{\beta_j}.$$

Then

$$\begin{align*}
\text{Ent}(\omega \circ \tau^t | \omega) &= \lim_{\Lambda_\alpha \to \infty} \text{Ent}(\omega_{\Lambda_\alpha} \circ \tau^t_{\Lambda_\alpha} | \omega_{\Lambda_\alpha}) \\
&= - \lim_{\Lambda_\alpha \to \infty} \int_0^t \omega_{\Lambda_\alpha} (\tau^s_{\Lambda_\alpha}(\sigma_{\Lambda_\alpha})) ds \\
&= - \int_0^t \omega(\tau^s(\sigma)) ds.
\end{align*}$$

In [Rue01], Ruelle considered a somewhat more general setting by allowing for suitable "boundary terms" in (22.2). The reason for introducing these "boundary terms" is to reach a broader class of reservoir KMS-states than the "free boundary condition" (22.2) allows. To address this issue more generally, one can start with an arbitrary $(\tau_{\mathcal{R}_j}, \beta_j)$-KMS state $\omega_{\beta_j}$ on $\mathcal{R}_{\beta_j}$, set $\omega_{\Lambda_j} = \omega_{\beta_j} \upharpoonright_{\mathcal{R}_{\beta_j}}$, and take (22.1) defined with these new $\omega_{\Lambda_j}$. The entropy production observable $\sigma_{\Lambda}$ remains defined by (22.3). In this case, the total entropy production over the time-interval $[0, t]$,

$$- \int_{0}^{t} \omega_{\Lambda}(\tau_{\Lambda}^{s}(\sigma_{\Lambda}))ds,$$
cannot be directly linked to the relative entropy and does not need to be non-negative. However, since

$$-\lim_{\Lambda} \int_{0}^{s} \omega_{\Lambda}(\tau_{\Lambda}^{t}(\sigma_{\Lambda})) dt = -\int_{0}^{s} \omega_{t}(\sigma) dt = \text{Ent}(\omega_{s}|\omega),$$

the basic properties of entropy production are restored in the thermodynamic limit. For further discussion of this topic, we refer the reader to [Ben+25a, Section 3.4].

## 23 Equilibrium versus non-equilibrium

There are good reasons why our understanding of equilibrium quantum statistical mechanics vastly surpasses its non-equilibrium counterpart. Perhaps the most important reasons concern the role of dynamics, the variational principle, and the thermodynamic limit. Although thermal equilibrium states have dynamical characterization in terms of the KMS-condition, in the concrete setting of lattice spin or fermionic systems the KMS condition is equivalent to the Gibbs variational principle and Gibbs condition. The latter two, in conjunction with thermodynamic limit, provide powerful tools in the study of the question (QI) and sharply separate it from the much less understood question (QII). There is no such separation in the non-equilibrium case, and a detailed understanding of the dynamics is central to both (QIII) and (QIV).

Another important reason concerns the relation of (QIV) with the singularity of NESS. If $\omega_+ = \omega_n + \omega_s$ is the decomposition of a NESS into its $\omega$-normal part $\omega_n$ and its $\omega$-singular part $\omega_s$, then $\omega_n(\sigma) = 0$, and so $\omega_+(\sigma) > 0$ iff $\omega_s(\sigma) > 0$. Moreover, if the following boundedness condition holds,

$$\sup_{T>0} \left| \int_{0}^{T} (\omega(\tau^{t}(\sigma)) - \omega_{+}(\sigma)) dt \right| < \infty,$$

then $\omega_+(\sigma) > 0$ iff $\omega_s \neq 0$; see [JP02b, Theorem 1.1]. Although physically natural, the fact that the entropy production of NESS is carried by its singular part makes its study particularly delicate. This emergence of singularity is specific to non-equilibrium: if $\omega$ is a $(\tau_{fr}, \beta)$-KMS state, then the NESS is unique and $\omega$-normal. These points are discussed in detail in [Asc+06; JP02a].

All the reasons discussed above are common to the classical and quantum case, underscoring that the conceptual difficulty of non-equilibrium statistical mechanics is not a specifically quantum phenomenon.

# Part VII

## Quantum phase space construction

In open (classical or quantum) systems entropy production is directly related to the heat generation. The phase-space contraction reflects the loss of microscopic information as the system evolves, in the large-time limit, to a singular NESS. These two notions are closely related. Much of this section follows [Ben+24b].

## 24 Classical setting

We consider a classical dynamical system $(\mathcal{X}, \phi)$ where $\mathcal{X}$ is a compact metric space and $\phi = \{\phi^t | t \in \mathbb{R}\}$ is a group of homeomorphisms of $\mathcal{X}$ such that the map

$$\mathbb{R} \times \mathcal{X} \ni (t, x) \mapsto \phi^t(x) \in \mathcal{X}$$
is continuous. We denote by $C(\mathcal{X})$ the vector space of all continuous complex-valued functions on $\mathcal{X}$ and equip it with the sup norm $\|f\|_{\infty} := \sup_{x \in \mathcal{X}} |f(x)|$. The observables are functions $f \in C(\mathcal{X})$ and they evolve in time as $f \mapsto f_t := f \circ \phi^t$. The states are Borel probability measures on $\mathcal{X}$ and we write $v(f) = \int_{\mathcal{X}} f \, dv$. They evolve in time as $v \mapsto v_t := v \circ \phi^{-t}$. A state $v$ is $\phi$-invariant if $v_t = v$ for all $t$. The classical relative entropy of two states $v$ and $\rho$ is defined by

$$ \mathrm{Ent}(v|\rho) = \int_{\mathcal{X}} \log \frac{d\rho}{dv} dv $$

if $\rho \ll v$, and is set to $-\infty$ otherwise. Its basic property is $\mathrm{Ent}(v|\rho) \le 0$ with equality iff $v = \rho$.

A time-reversal of $(\mathcal{X}, \phi)$ is an involutive homeomorphism $\vartheta: \mathcal{X} \to \mathcal{X}$ such that

$$ \vartheta \circ \phi^t = \phi^{-t} \circ \vartheta $$

for all $t \in \mathbb{R}$. A state $v$ is called time-reversal invariant (TRI) if $v \circ \vartheta = v$.

The starting point is a classical dynamical system $(\mathcal{X}, \phi, \omega)$, assuming that the initial (reference) state $\omega$ is not $\phi$-invariant. The system is TRI if $\omega$ is TRI for some time-reversal of $(\mathcal{X}, \phi)$.

We set the regularity assumptions. First, throughout this section we assume

(C1) For all $t \in \mathbb{R}$ the measures $\omega$ and $\omega_t$ are equivalent, i.e., have the same sets of measure zero.

Let

$$ \Delta_{\omega_t|\omega} := \frac{d\omega_t}{d\omega}, \quad \ell_{\omega_t|\omega} := \log \Delta_{\omega_t|\omega}. $$

(C2) $\ell_{\omega_t|\omega} \in C(\mathcal{X})$ for all $t \in \mathbb{R}$.

We set

$$ c^t := \ell_{\omega_t|\omega} \circ \phi^t, $$

and denote by $Q_t$ the law of $c^t$ w.r.t. $\omega$. For $\alpha \in \mathbb{C}$, let

$$ \mathcal{F}_t(\alpha) := \int_{\mathbb{R}} e^{-\alpha s} dQ_t(s). $$

(C3) The map

$$ \mathbb{R} \ni t \mapsto \ell_{\omega_t|\omega} \in C(\mathcal{X}) $$

is differentiable at $t = 0$.

The entropy production observable, or the phase space contraction rate, is defined by

$$ \sigma = \frac{d}{dt} \ell_{\omega_t|\omega}|_{t=0} = \frac{d}{dt} c^t|_{t=0}. $$

**Proposition 24.1.** Suppose that (C1)–(C2) hold.

1. For all $t, s \in \mathbb{R}$,

$$ \Delta_{\omega_{t+s}|\omega} = \Delta_{\omega_s|\omega} \circ \phi^{-t} \Delta_{\omega_t|\omega} $$

$$ \ell_{\omega_{t+s}|\omega} = \ell_{\omega_s|\omega} \circ \phi^{-t} + \ell_{\omega_t|\omega}, $$

$$ c^{t+s} = c^s + c^t \circ \phi^s. $$
2. $\mathrm{Ent}(\omega_t|\omega) = -\omega(c^t)$.

In the remaining statements we assume that (C3) also holds.

3.

and

$$ \ell_{\omega_t|\omega} = \int_0^t \sigma_{-s} ds, \quad c^t = \int_0^t \sigma_s ds, $$

$$ \mathrm{Ent}(\omega_t|\omega) = -\int_0^t \omega_s(\sigma)ds. $$

4. $\omega(\sigma) = 0$.

In the remaining statements we assume that, in addition, $(\mathcal{X}, \phi, \omega)$ is TRI with time-reversal $\vartheta$.

5. $c^t \circ \vartheta = c^{-t}$ and $\sigma \circ \vartheta = -\sigma$.

6. For all $t \in \mathbb{R}$ and $\alpha \in \mathbb{C}$,

$$ \mathfrak{F}_t(\alpha) = \mathfrak{F}_t(1 - \bar{\alpha}). $$

7. Let $\nu: \mathbb{R} \to \mathbb{R}$ be the reflection $\nu(s) = -s$ and $\bar{Q}_t = Q_t \circ \nu$. Then, for any $t \in \mathbb{R}$, the measures $Q_t$ and $\bar{Q}_t$ are equivalent, and

$$ \frac{d\bar{Q}_t}{dQ_t}(s) = e^{-s}. \qquad (24.1) $$

The NESS of $(\mathcal{X}, \phi, \omega)$ are defined as the weak-limit points of the net

$$ \left\{ \frac{1}{T} \int_{0}^{T} \omega_{t} dt \mid T > 0 \right\}_{T>0}. $$

The set of NESS is non-empty and any NESS is $\phi$-invariant. Moreover, for any NESS $\omega_+$ one has

$$ \omega_+(\sigma) \geq 0. $$

The two basic questions (QIII) and (QIV) of Section 21 have the same formulation in the classical case. In fact, the classical case can be directly embedded in the quantum case by considering commutative $C^*$-dynamical systems, see [Ben+24b].

## 25 GNS-representation and modular structure

Some of the material below has been already discussed; see in particular Sections 5 and 12.

We denote by $(\mathcal{H}_\omega, \pi_\omega, \Omega_\omega)$ the GNS-representation of $\mathcal{O}$ associated to $\omega$ and by $\mathfrak{M}_\omega = \pi_\omega(\mathcal{O})''$ the enveloping von Neumann algebra of bounded operators on $\mathcal{H}_\omega$. We drop subscript $\omega$ when no confusion is possible. Since the state $\omega$ is assumed to be modular, the vector $\Omega$ is separating for $\mathfrak{M}$,<fnref n="16" /> and in particular $\|\pi(A)\| = \|A\|$ for all $A \in \mathcal{O}$. Whenever the meaning is clear from the context, we will denote $\pi(A)$ by $A$.

$\mathcal{N}$ denotes the set of all normal states on $\mathfrak{M}$, i.e., the states described by density matrices on $\mathcal{H}$. Obviously elements of $\mathcal{N}$ also define states on $\mathcal{O}$. Any state on $\mathcal{O}$ that arises in this way is called $\omega$-normal. Again, whenever the meaning is clear from the context, we will denote such states by the same letter. In particular, the vector state $\mathfrak{M} \ni A \mapsto \langle\Omega, A\Omega\rangle$ is denoted by $\omega$.

<footnote n="16">¹⁶See [BR81, Corollary 5.3.9]</footnote>
We will assume that the reader is familiar with the basic notions of Tomita-Takesaki modular theory; see Section 5 and any of the references [BR87; BR81; DJ03; Haa96; OP93; SZ79]. We will use the same notation and terminology as in Section 5.3. $\mathcal{H}_+$ and $J$ denote the natural cone and modular conjugation associated to the pair $(\mathfrak{M}, \Omega)$. The unique vector representative of $v \in \mathcal{N}$ in the natural cone is denoted by $\Omega_v$. The modular operator of $v \in \mathcal{N}$ is denoted by $\Delta_v$. The relative modular operator of a pair $(v, \rho)$ of normal states is denoted by $\Delta_{v|\rho}$. The following perturbative result [JP03] is behind the entropy balance equation (21.1).

**Theorem 25.1.** Let $v$ be a modular state on $\mathcal{O}$ and $U \in \text{Dom}(\delta_v)$ a unitary. Consider the state $v_{U}(\cdot) = v(U^{*} \cdot U)$ and let $P = -iU\delta_v(U^{*})$. Then

$$ \log \Delta_{vU|v} = \log \Delta_v + P. $$

The relative entropy of a pair $(v, \rho)$ of normal faithful states is

$$ \text{Ent}(v|\rho) = \langle \Omega_v, \log \Delta_{\rho|v} \Omega_v \rangle. $$

This is the original definition of Araki [Ara75], with the sign and ordering convention of [BR81]; see also Section 12.

Since $\omega$ is $\zeta_{\omega}$ invariant, the family $\{\pi \circ \zeta_{\omega}^{t} | t \in \mathbb{R}\}$ extends to a $W^{*}$-dynamics on $\mathfrak{M}$ which we again denote by $\zeta_{\omega}$.<fnref n="17" /> For $A \in \mathfrak{M}$,

$$ \zeta_{\omega}^{t}(A) = \Delta_{\omega}^{it} A \Delta_{\omega}^{-it}. $$

More generally, to any faithful $v \in \mathcal{N}$ one associates a $W^{*}$-dynamics $\zeta_v$ by

$$ \zeta_{v}^{t}(A) = \Delta_{v}^{it} A \Delta_{v}^{-it}. $$

$\zeta_v$ is called the modular dynamics of $v$, and $v$ is a $(\zeta_v, -1)$-KMS state on $\mathfrak{M}$.

We make the assumption:

(St) The family $\{\pi \circ \tau^t | t \in \mathbb{R}\}$ extends to a $W^*$-dynamics on $\mathfrak{M}$ which we again denote by $\tau$.

This assumption is automatically satisfied for LP systems discussed in Section 21. It ensures that $\omega_t$ is a faithful $\omega$-normal state on $\mathfrak{M}$, and that there exists a unique selfadjoint operator $\mathcal{L}$ on $\mathcal{H}$, called the standard Liouvillean of $\tau$, such that for all $A \in \mathfrak{M}$ and $t \in \mathbb{R}$,

$$ \tau^t(A) = e^{it\mathcal{L}} A e^{-it\mathcal{L}}, \quad e^{-it\mathcal{L}} \mathcal{H}_+ = \mathcal{H}_+. $$

The vector representative of $\omega_t$ in $\mathcal{H}_+$ is $e^{-it\mathcal{L}}\Omega$. The standard Liouvillean is a basic example of a quantum transfer operator. Note that the standard Liouvillean of $\zeta_{\omega}$ is $\log \Delta_{\omega}$.

The basic properties of the standard Liouvillean are summarized in:

**Theorem 25.2.** 1. $e^{it\mathcal{L}} J = Je^{it\mathcal{L}}$.

2. The standard Liouvillean of an LP system is

$$ \mathcal{L} = \mathcal{L}_{\text{fr}} + V - JVI, $$

where $\mathcal{L}_{\text{fr}}$ is the standard Liouvillean of $\tau_{\text{fr}}$.

3. $\omega \in \mathcal{S}_{\tau} \Leftrightarrow \mathcal{L}\Omega = 0$.

4. Suppose that $\omega \in \mathcal{S}_{\tau}$. Then the quantum dynamical system $(\mathcal{O}, \tau, \omega)$ is ergodic iff 0 is a simple eigenvalue of $\mathcal{L}$.

<footnote n="17">¹⁷Pointwise $\sigma$-weakly continuous groups of $*$-automorphisms on $\mathfrak{M}$.</footnote>
The following well-known results identifies ergodicity with the so-called property of return to equilibrium.

**Theorem 25.3.** Suppose that $\omega$ is $\tau$-invariant. Then the quantum dynamical system $(\mathcal{O}, \tau, \omega)$ is ergodic iff, for any $\omega$-normal state $v$ and $A \in \mathcal{O}$, one has

$$ \lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} v(\tau^{t}(A))dt = \omega(A). $$

To any two faithful normal states $v$ and $\rho$ one associates a family of unitary operators called Connes' cocycle

$$ [Dv : D\rho]_{it} := \Delta_{v|\rho}^{it} \Delta_{\rho}^{-it}, \quad t \in \mathbb{R}. $$

Its basic properties are summarized in:

**Theorem 25.4.** 1. For all $t \in \mathbb{R}$, $[Dv : D\rho]_{it}$ is a unitary element of $\mathfrak{M}$.

$$ [Dv : D\rho]_{it}^* = [D\rho : Dv]_{it} \text{ for all } t \in \mathbb{R}. $$

3. The $\zeta_{\rho}$-cocycle relation

$$ [Dv : D\rho]_{is} \zeta_{\rho}^{s} ([Dv : D\rho]_{it}) = [Dv : D\rho]_{i(s+t)}, $$

holds for all $s, t \in \mathbb{R}$.

4. The intertwining relation

$$ [Dv : D\rho]_{it} \zeta_{\rho}^{t}(A) [Dv : D\rho]_{it}^* = \zeta_{v}^{t}(A), $$

holds for any $A \in \mathfrak{M}$ and $t \in \mathbb{R}$.

5. If $\mu$ is another faithful normal state, then the chain rule

$$ [Dv : D\rho]_{it} [D\rho : D\mu]_{it} = [Dv : D\mu]_{it}, $$

holds for any $t \in \mathbb{R}$.

For the proof we refer the reader to [AM82].

The family of Connes cocycles $([D\omega_t : D\omega]_\alpha)_{\alpha \in i\mathbb{R}}$ will be crucial for the identification of quantum phase space contraction in the next section. The following proposition describes its basic property, which is the quantum counterpart of the first relation in Theorem 24.1(i).

**Proposition 25.5.** For all $t, s \in \mathbb{R}$ and $\alpha \in i\mathbb{R}$,

$$ [D\omega_{t+s} : D\omega]_{\alpha} = \tau^{-t} ([D\omega_s : D\omega]_{\alpha}) [D\omega_t : D\omega]_{\alpha}. $$

In the development of non-equilibrium quantum statistical mechanics, the following regularity assumption plays an important role:

(Reg1) For all $t \in \mathbb{R}$ and $\alpha \in i\mathbb{R}$,

$$ [D\omega_t : D\omega]_{\alpha} \in \pi(\mathcal{O}). $$

When (Reg1) holds, we will denote $\pi^{-1}([D\omega_t : D\omega]_\alpha)$ by $[D\omega_t : D\omega]_\alpha$. For LP systems, (Reg1) holds if $V \in \text{Dom}(\delta_\omega)$.
## 26 Quantum phase space contraction

As in the classical case, our starting point is Proposition 25.5. We make two additional regularity assumptions:

(Reg2) For all $t \in \mathbb{R}$ the map
$$i\mathbb{R} \ni \alpha \mapsto [D\omega_t : D\omega]_\alpha \in \mathcal{O}$$
is differentiable at $\alpha = 0$.

Let

$$\ell_{\omega_t|\omega} := \frac{d}{d\alpha} [D\omega_t : D\omega]_\alpha|_{\alpha=0},$$

and

$$c^t := \tau^t(\ell_{\omega_t|\omega}).$$

Note that $\ell_{\omega_t|\omega}$, and consequently $c^t$, are selfadjoint elements of $\mathcal{O}$.

(Reg3) The map
$$\mathbb{R} \ni t \mapsto \ell_{\omega_t|\omega} \in \mathcal{O},$$
is differentiable at $t = 0$.

Let

$$\sigma = \left. \frac{d}{dt} \ell_{\omega_t|\omega} \right|_{t=0} = \left. \frac{d}{dt} c^t \right|_{t=0}. \qquad (26.1)$$

**Proposition 26.1.** 1. For all $t, s \in \mathbb{R}$, one has

$$\ell_{\omega_{t+s}|\omega} = \tau^{-t}(\ell_{\omega_s|\omega}) + \ell_{\omega_t|\omega}, \\ c^{t+s} = c^s + \tau^s(c^t).$$

2. For all $t \in \mathbb{R}$, $\log \Delta_{\omega_t|\omega} = \log \Delta_{\omega} + \ell_{\omega_t|\omega}$, and for $\alpha \in i\mathbb{R}$,
$$[D\omega_t : D\omega]_\alpha = e^{\alpha(\log \Delta_{\omega} + \ell_{\omega_t|\omega})} e^{-\alpha \log \Delta_{\omega}}.$$

3. For all $t \in \mathbb{R}$, $\mathrm{Ent}(\omega_t|\omega) = -\omega(c^t)$.

4. Setting $\sigma_t = \tau^t(\sigma)$ for $t \in \mathbb{R}$, one has

$$\ell_{\omega_t|\omega} = \int_0^t \sigma_{-s} \mathrm{d}s, \quad c^t = \int_0^t \sigma_s \mathrm{d}s, \\ \text{and} \\ \mathrm{Ent}(\omega_t|\omega) = - \int_0^t \omega_s(\sigma) \mathrm{d}s. \qquad (26.2)$$

5. $\omega(\sigma) = 0$.

6. If $(\mathcal{O}, \tau, \omega)$ is TRI with time-reversal $\Theta$, then $\Theta(c^t) = c^{-t}$ and $\Theta(\sigma) = -\sigma$.

This proposition is the direct quantum analogue of Proposition 24.1. $\sigma$ is the entropy production observable of $(\mathcal{O}, \tau, \omega)$ and (26.2) is the entropy balance equation.

Theorem 26.1 sheds light on the introduction to non-equilibrium quantum statistical mechanics presented in Section 21 in the context of the LP systems.
**Proposition 26.2.** Suppose that $(\mathcal{O}, \tau, \omega)$ is an LP system and that $V \in \text{Dom}(\delta_\omega)$. Then (Reg1), (Reg2) and (Reg3) hold, and $\sigma$ in (26.1) is equal to $\delta_\omega(V)$.

The two basic questions of non-equilibrium quantum statistical mechanics, (QIII) and (QIV) formulated in Section 21, extend in the obvious way to the more general setting considered here.

Motivated by the analogies between Theorem 24.1 and Theorem 26.1, we consider the map

$$i\mathbb{R} \ni \alpha \mapsto [D\omega_{-t} : D\omega]_\alpha \in \mathcal{O}$$

as the quantum analogue characterization of phase space contraction for $(\mathcal{O}, \tau, \omega)$ at time $t$.

# Part VIII

## How should one define quantum entropy production?

Entropy production quantifies irreversibility and information loss and is directly linked to the Second Law of Thermodynamics. While entropy production is a well-defined concept in classical physics, its interpretation becomes multifaceted and subtler in the quantum domain. In particular, multiple non-equivalent notions of entropy production arise.

## 27 Quantum phase space contraction and entropy production observable

The most direct approach to defining quantum entropy production proceeds by quantizing the corresponding classical entropy production observable, and has been described in Sections 21 and 26. This route has been adopted in [JP01; Rue01] and played an important role in the development of the subject.

The main criticism of this approach concerns the following two points:

1. The experimental status of the selfadjoint observable $c^t = \int_0^t \sigma_s ds$ is questionable.

2. The finite time fluctuation relation, the quantum analog of (24.1), fails for $c^t$ as an operator-valued observable. More precisely, if $Q_t$ is the spectral measure for $\omega$ and $c^t$, the relation

$$\frac{d\bar{Q}_t}{dQ_t}(s) = e^{-s}.$$

does not hold except in special cases; see [Jak+12a, Exercise 3.3].

Point (a) is connected to several perennial aspects of the so-called standard (or orthodox) view of the conceptual foundations of quantum mechanics. There are many excellent discussions of this view in the literature, beginning with the classical book of von Neumann [Neu55]. Here we limit ourselves to two further references: [Haa96, Section 7] and [Wig63]. Point (b), on the other hand, was emphasized forcefully in [TLH07].

We do not think that the validity of the finite time fluctuation relation can be raised to a fundamental principle of what is observable in quantum statistical mechanics, and hence we do not find criticism (b) tenable. Although (a) obviously must be taken more seriously, it is reasonable to accept that any proposal for the average entropy production over a time interval $[0, t]$ of $(\mathcal{O}, \tau, \omega)$ gives the same value $\omega(c^t)$, and this is certainly true for
all the known proposals. Thus, the questions (QIII) and (QIV) of Section 21 are unaffected by (a). The entropy production observable is also a natural starting point in the study of the linear response theory, Onsager reciprocity relations, and near-equilibrium fluctuation-dissipation mechanism [JOP06b; JOP06a; JOP06c; JOP07]; see also Section 12. It is the analysis of far from equilibrium entropy production fluctuations that brings (a) to focus.

We however emphasize that some foundational issues about the choice of entropy production observable σ remain, even if one focuses only on the averages ω(c^t). One of them is our next topic.

## 28 Ruelle's decomposition of entropy production

Shortly after the introduction of NESS and entropy production observable in [JP01; Rue01], David Ruelle wrote an important but little known paper [Rue02] concerning the interpretation of the quantum entropy production. We start with a brief summary of [Rue02] adjusting certain points to allow for a clearer comparison with subsequent developments.

Consider a finite open quantum system on the Hilbert space $\mathcal{H} = \bigotimes_{j=1}^{M} \mathcal{H}_j$, in the directly coupled reservoir setting. With a more involved notation the arguments easily extend to a general open quantum system. For $j = 1, \dots, M$ we set $\mathcal{K}_j := \bigotimes_{i \neq j} \mathcal{K}_i$. Let<fnref n="18" />

$$ \omega_{tj} := \text{tr}_{\mathcal{K}_j} \omega_t $$

be the state of the j-th reservoir at the time t and

$$ \omega_t^{\text{dec}} := \bigotimes_{j=1}^{M} \omega_{tj}. $$

Note that $\omega_0^{\text{dec}} = \omega$. Let<fnref n="19" />

$$ \Delta S(t) := S(\omega_t^{\text{dec}}) - S(\omega), $$

and

$$ \Delta \mathcal{G}(t) := - \text{Ent}(\omega_t^{\text{dec}}|\omega). $$

Obviously, $\Delta \mathcal{G}(t) \ge 0$. Moreover, since $S(\omega_t^{\text{dec}}) = -\text{tr}(\omega_t^{\text{dec}} \log \omega_t^{\text{dec}}) = -\text{tr}(\omega_t \log \omega_t^{\text{dec}})$ and $S(\omega) = S(\omega_t)$, one has

$$ \Delta S(t) = - \text{tr}(\omega_t(\log \omega_t^{\text{dec}} - \log \omega_t)) = - \text{Ent}(\omega_t|\omega_t^{\text{dec}}) \ge 0. $$

The starting point of [Rue02] is the identity

$$ \begin{align} \int_0^t \omega_s(\sigma) ds &= - \text{Ent}(\omega_t|\omega) = \text{tr}(\omega_t(\log \omega_t - \log \omega)) \nonumber \\ &= \text{tr}(\omega_t(\log \omega_t - \log \omega_t^{\text{dec}})) + \text{tr}(\omega_t(\log \omega_t^{\text{dec}} - \log \omega)) \tag{28.1} \\ &= - \text{Ent}(\omega_t|\omega_t^{\text{dec}}) - \text{Ent}(\omega_t^{\text{dec}}|\omega) = \Delta S(t) + \Delta \mathcal{G}(t), \nonumber \end{align} $$

expressing the mean entropy production over the time interval $[0, t]$ as the sum of mutual information $\Delta S(t)$ and a relative entropy-cost $\Delta \mathcal{G}(t)$ associated with the reduced reservoir dynamics.

The decomposition (28.1) remains valid for infinitely extended reservoirs and can be justified either by a thermodynamic limit argument or directly by an application of Corollary 5.20 and identity (5.22) of [OP93]

<footnote n="18">¹⁸tr$_{\mathcal{K}_j}$ denotes the partial trace over $\mathcal{K}_j$</footnote>

<footnote n="19">¹⁹We recall that $S(v) = -\text{tr}(v \log v)$ denotes the von Neumann entropy of v.</footnote>
In [Rue02], Ruelle proposes a heuristic framework in which $\Delta S(t)$ is expected to dominate (28.1). This perspective touches on foundational issues in non-equilibrium statistical mechanics, particularly concerning the role and structure of thermal reservoirs in open quantum systems. Ruelle's program remains only partially understood, and much work remains to be done.

Finally, we note that the Ruelle program is not specific to the quantum setting; the same analysis applies, and the same questions arise, in open classical systems, where equally little is known.

## 29 Two-time measurement entropy production (TTMEP)

In the finite dimensional setting, the notion of two-times entropy production goes back to [Kur00; Tas00]. Our presentations follows [Jak+12a].

Consider a finite quantum dynamical system ($\mathcal{O}_{\mathcal{K}}, \tau, \omega$). The observable to be measured is described by a partition of unity $(P_a)_{a \in \mathcal{A}}$ on $\mathcal{K}^{20}$ indexed by a finite alphabet $\mathcal{A}$ labeling the possible outcomes of the measurement. The two-times measurement protocol goes as follows. At time $t = 0$, when the system was in the state $\omega$, the first measurement is performed and the outcome $a \in \mathcal{A}$ is observed with probability

$$p(a) = \mathrm{tr}(\omega P_a).$$

After this first measurement, the system is in the reduced state

$$\frac{1}{p(a)} P_a \omega P_a,$$

which evolves over the time interval $[0, t]^{21}$ to

$$\frac{1}{p(a)} e^{-i t H} P_a \omega P_a e^{i t H}.$$

The second measurement, at the time $t$, yields the outcome $a' \in \mathcal{A}$ with probability

$$p_t(a'|a) = \frac{1}{p(a)} \mathrm{tr}(e^{-i t H} P_a \omega P_a e^{i t H} P_{a'}).$$

Finally, the probability of observing the pair $(a, a')$ in the two-times measurement protocol is

$$p_t(a, a') = p_t(a'|a) p(a) = \mathrm{tr}(e^{-i t H} P_a \omega P_a e^{i t H} P_{a'}).$$

Starting with the above protocol, there are two distinct routes to define measurement based entropy production. We describe here the one which is the main topic of this section. The second route involving repeated measurements is reviewed in Section 34.

Assume that $\omega > 0$ and consider the partition of unity $(P_a)_{a \in \mathcal{A}}$ that arises through the spectral decomposition of $\omega$,

$$\omega = \sum_{a \in \mathcal{A}} \lambda_a P_a,$$

where the eigenvalues $\lambda_a$ are assumed to be distinct. The entropy production random variable $\mathcal{E}: \mathcal{A} \times \mathcal{A} \to \mathbb{R}$ is defined by

$$\mathcal{E}(a, a') = \log \lambda_{a'} - \log \lambda_a,$$

<footnote>20The $P_a$ are orthogonal projections such that $\sum_{a \in \mathcal{A}} P_a = 1$.</footnote>

<footnote>21We allow for $t < 0$.</footnote>
and its probability distribution with respect to $p_t$ is denoted by $Q_t$,

$$Q_t(s) = \sum_{\mathcal{E}(a,a')=s} p_t(a, a').$$

The statistics of TTMEP over the time interval $[0, t]$ is described by $Q_t$. For $\alpha \in \mathbb{C}$, we set

$$\mathcal{F}_t^{\text{ttm}}(\alpha) := \int_{\mathbb{R}} e^{-\alpha s} dQ_t(s).$$

We then have:

**Proposition 29.1.** 1.

$$\mathcal{F}_t^{\text{ttm}}(\alpha) = \omega([D\omega_{-t} : D\omega]_\alpha) = \text{tr}(\omega_{-t}^\alpha \omega^{1-\alpha}). \qquad (29.1)$$

$$2. \qquad \int_{\mathbb{R}} s dQ_t(s) = -\text{Ent}(\omega_t|\omega).$$

In particular $\int_{\mathbb{R}} s dQ_t(s) \ge 0$ with equality iff $\omega = \omega_t$.

In the remaining statements we assume that the system is TRI.

$$3. \qquad \mathcal{F}_t^{\text{ttm}}(\alpha) = \mathcal{F}_t^{\text{ttm}}(1 - \bar{\alpha}).$$

$$4. \qquad Q_t(-s) = e^{-s} Q_t(s). \qquad (29.2)$$

The identity (29.2) is sometimes called finite time fluctuation relation.

We now turn to a general modular quantum dynamical system $(\mathcal{O}, \tau, \omega)$ satisfying (Reg1), (Reg2) and (Reg3). For $t \in \mathbb{R}$ and $\alpha \in i\mathbb{R}$, we set

$$\mathcal{F}_t^{\text{ttm}}(\alpha) := \omega([D\omega_{-t} : D\omega]_\alpha).$$

When $(\mathcal{O}, \tau, \omega)$ is finite, this reduces to (29.1).

**Proposition 29.2.** 1. There exists unique probability measure $Q_t$ on $\mathbb{R}$ such that

$$\mathcal{F}_t^{\text{ttm}}(\alpha) = \int_{\mathbb{R}} e^{-\alpha s} dQ_t(s).$$

$$2. \int_{\mathbb{R}} s dQ_t(s) = -\text{Ent}(\omega_t|\omega). \text{ In particular, } \int_{\mathbb{R}} s dQ_t(s) \ge 0 \text{ with equality iff } \omega = \omega_t.$$

$$3. \text{ For } \alpha \in i\mathbb{R},$$

$$\mathcal{F}_t^{\text{ttm}}(\alpha) = \langle \Omega, \Delta_{\omega_{-t}|\omega}^{\alpha} \Omega \rangle.$$

In particular, $Q_t$ is the spectral measure of the selfadjoint operator $-\log \Delta_{\omega_{-t}|\omega}$ for the vector $\Omega$.

In the remaining statements we assume that $(\mathcal{O}, \tau, \omega)$ is TRI.

$$4. \text{ The map } i\mathbb{R} \ni \alpha \mapsto \mathcal{F}_t^{\text{ttm}}(\alpha) \text{ has an analytic extension to the vertical strip } 0 < \text{Re } z < 1 \text{ that is bounded and continuous on its closure.}$$

$$5. \text{ For any } \alpha \text{ satisfying } 0 \le \text{Re } \alpha \le 1,$$

$$\mathcal{F}_t^{\text{ttm}}(\alpha) = \mathcal{F}_t^{\text{ttm}}(1 - \bar{\alpha}).$$

$$6. \text{ Let } \mathfrak{r}: \mathbb{R} \to \mathbb{R} \text{ be the reflection } \mathfrak{r}(s) = -s \text{ and } \overline{Q}_t = Q_t \circ \mathfrak{r}. \text{ Then the measures } Q_t \text{ and } \overline{Q}_t \text{ are equivalent and}$$

$$\frac{d\overline{Q}_t}{dQ_t}(s) = e^{-s}.$$

The measure $Q_t$ describes the statistics of the two-time measurement entropy production of $(\mathcal{O}, \tau, \omega)$ over the time interval $[0, t]$.
## 30 Thermodynamic limit of TTMEP in open quantum spin systems

The general modular theoretic approach to $Q_t$ described in the previous section requires justification of the thermodynamic limit for concrete models. In this section we address this question in the context of open quantum lattice spin systems introduced in Section 22. We denote by $Q_{t,\Lambda}$ the law of the TTMEP for $(\mathcal{O}_{\Lambda}, \tau_{\Lambda}, \omega_{\Lambda})$ over the time interval $[0, t]$ with $\omega_{\Lambda}$ defined by (22.1) and (22.2). Convergence of probability measures is understood in the weak sense.

**Theorem 30.1.** Suppose that each $\omega_{\beta_j}$ is a thermodynamic limit point $\beta_j$-KMS state on $\mathcal{O}_{R_j}$. Let

$$ \Lambda_a = S \cup \left( \bigcup_{j=1}^{M} \Lambda_{j,a} \right) $$

be a net in $\mathcal{G}_{\text{fin}}$ such that for each $j$,

$$ \lim_{\Lambda_{j,a} \to \infty} \bar{\omega}_{\Lambda_{j,a}} = \omega_{\beta_j}. $$

Then, for all $t \in \mathbb{R}$,

$$ \lim_{\Lambda_a \to \infty} Q_{t, \Lambda_a} = Q_t. $$

For the proof and additional information, see [Ben+25a].

## 31 Ancilla state tomography

Ancilla state tomography is a technique which extracts information on the state of a quantum system from the outcome of measurements performed on an auxiliary system, the ancilla, coupled to the system of interest. In this section, following [Ben+24b], we show how this technique allows for an indirect measurement of the function $\tilde{\mathfrak{F}}_t^{\text{ttm}}$.

We start with the finite system of Section 29. The state of the ancilla is described by a spin 1/2 on $\mathbb{C}^2$. We denote by $v_{\pm}$ the eigenvectors of the Pauli matrix $\sigma_z$ associated to the eigenvalues $\pm 1$. The state of ancilla is described by a density matrix $\rho_a$ such that $\langle v_+, \rho_a v_- \rangle \neq 0$. The Hilbert space of the coupled system is $\mathcal{H} = \mathcal{K} \otimes \mathbb{C}^2$, and its initial state is $\hat{\omega} = \omega \otimes \rho_a$. We introduce the following family of Hamiltonians, indexed by $\alpha \in i\mathbb{R}$,

$$ \hat{H}_{\alpha} = e^{\frac{\alpha}{2} \log \omega \otimes \sigma_z} (H \otimes 1) e^{-\frac{\alpha}{2} \log \omega \otimes \sigma_z}. $$

Setting $N := |v_-\rangle\langle v_+|$, a computation gives

$$ \text{tr} \left( e^{-i t \hat{H}_{\alpha}} \hat{\omega} e^{i t \hat{H}_{\alpha}} (1 \otimes N) \right) = \langle v_+, \rho_a v_- \rangle \tilde{\mathfrak{F}}_t^{\text{ttm}}(\alpha). \quad (31.1) $$

The left-hand side in (31.1) is interpreted as entropic ancilla state tomography of the finite quantum system over the interval $[0, t]$.

In the particular case of a finite LP system where $H = H_{\text{fr}} + V$ with $[\log \omega, H_{\text{fr}}] = 0$, one has

$$ \hat{H}_{\alpha} = H \otimes 1 + \hat{W}_{\alpha}, $$

where

$$ \hat{W}_{\alpha} = \frac{1}{2} W_{\alpha} \otimes (1 + \sigma_z) + \frac{1}{2} W_{-\alpha} \otimes (1 - \sigma_z), \quad (31.2) $$

and

$$ W_{\alpha} = \zeta_{\omega}^{-i\alpha/2}(V) - V. \quad (31.3) $$
Turning to the general modular quantum dynamical system of Section 29, we now consider a LP system satisfying $V \in \text{Dom}(\delta_\omega)$. Let $\hat{\mathcal{O}} := \mathcal{O} \otimes \mathcal{O}_{\mathcal{O}^2}$ and $\hat{\tau}^t = \tau^t \otimes \text{Id}$. For $\alpha \in i\mathbb{R}$, $\hat{W}_\alpha$ and $W_\alpha$ are given by (31.2) and (31.3). Note that $W_\alpha \in \mathcal{O}_{\text{sa}}$ and $\hat{W}_\alpha \in \hat{\mathcal{O}}_{\text{sa}}$. Let $\hat{\tau}_\alpha$ be the perturbation of $\hat{\tau}$ by $\hat{W}_\alpha$. Let $N$ as above and $\hat{\omega} = \omega \otimes \rho_a$. Then, again, for all $t \in \mathbb{R}$,

$$ \hat{\omega}(\hat{\tau}_{\alpha}^{t}(1 \otimes N)) = \langle v_{+}, \rho_{a}v_{-} \rangle \hat{\mathcal{F}}_{t}^{\text{ttm}}(\alpha). $$

We refer the reader to [Dor+13; Cam+13; Joh+16; RCP14; CRP15; GPM14; MDP13] for related theoretical studies in the physics literature and to [An+14; Bat+14; Bat+15; Pet+19] for experimental implementations.

## 32 Limitations of TTMEP

The statistics of the two-time measurement entropy production was initially introduced in the context of finite quantum systems (or, slightly more generally, confined quantum systems with a possibly infinite discrete energy spectra). Experimental studies have been carried out in this same setting, often relying on ancilla state tomography. The thermodynamic limit of these statistics is stable and preserves the connection with modular theory and ancilla state tomography, although – to the best of our knowledge – the statistics of the two-time measurement entropy production of infinitely extended system has not been experimentally explored. Obviously, the ancilla state tomography link is central to the physical relevance of the two-time measurement protocol for large quantum systems. It tells us that if the protocol could have been implemented, then the resulting statistics would coincide with those predicted by the ancilla-based tomography. This is certainly satisfactory.

On the other hand, the TTMEP exhibits a surprising degree stability (or rigidity) with respect to the instant of the first measurement – an effect with no classical counterpart – which has been recently investigated in [Ben+24a; Ben+25a; Ben+24b; Ben+26]. Suppose that the first measurement of the entropy production observable is not performed at $t = 0$, when the system was in the reference state $\omega$ from which the observable is defined, but instead at the later time $T$, when the system is in the state $\omega \circ \tau_T$. A second measurement is then performed at time $T + t$. After passing to the thermodynamic limit, and under a general and natural dynamical ergodicity assumption, the statistics of the TTMEP turn out to be independent of the instant $T$ of the first measurement. This effect arises from the invasive decoherence induced by the first measurement, which dominates the resulting statistics in the thermodynamic limit. This stability should be further examined both theoretically and experimentally before drawing any final conclusions regarding the foundational role of TTMEP in large quantum systems.

## 33 Bessis-Moussa-Villani entropy production

To motivate the introduction of this notion of entropy production, we return to the classical setting of Section 24. The variational formula for the classical relative entropy<fnref n="22" /> gives that for $\alpha \in \mathbb{R}$,

$$ \log \left[ \int_{\mathcal{X}} e^{-\alpha \int_{0}^{t} \sigma_s ds} d\omega \right] = \sup_{v} \left[ S(v|\omega) - \alpha \int_{0}^{t} \sigma_s ds dv \right], $$

where the supremum is taken over all Borel probability measures $v$ on $\mathcal{X}$. For finite quantum systems, one has

$$ \log \operatorname{tr} \left[ e^{\log \omega - \alpha \int_{0}^{t} \sigma_{s} ds} \right] = \sup_{v} \left[ S(v|\omega) - \alpha \int_{0}^{t} v(\sigma_{s}) ds \right], $$

where the supremum is taken over all density matrices $v$ on the system Hilbert space $\mathcal{X}$. We set

$$ \mathcal{F}_t^{\text{BMV}}(\alpha) = \operatorname{tr} \left[ e^{\log \omega - \alpha \int_0^t \sigma_s ds} \right]. \qquad (33.1) $$

<footnote n="22">²²See [OP93] or Theorem 2.1 in [JPR11].</footnote>
This function has been introduced and studied in [Jak+12a]. The acronym BMV in (33.1) reflects the connection of the right-hand side with the celebrated Bessis–Moussa–Villani conjecture [BMV75]$^{23}$, which was proved by Herbert Stahl [Sta13] in 2013.

**Theorem 33.1.** Let A and B be $n \times n$ Hermitian matrices, and assume that B is positive semidefinite. Define

$$[0, \infty) \ni \alpha \mapsto f(\alpha) = \operatorname{tr}(e^{A-\alpha B}).$$

Then there exists a non-negative Borel measure $\mu$ on $[0, \infty)$ such that

$$f(\alpha) = \int_{0}^{\infty} e^{-\alpha s} d\mu(s)$$

for all $\alpha \ge 0$.

Stahl's theorem gives that there exists a Borel probability measure $Q_t^{BMV}$ on $\mathbb{R}$ such that, for $\alpha \in \mathbb{C}$,

$$\mathcal{F}_t^{BMV}(\alpha) = \int_{\mathbb{R}} e^{-\alpha s} dQ_t^{BMV}(s). \qquad (33.2)$$

We will refer to $Q_t^{BMV}$ as the statistics of the BMV entropy production over the time interval $[0, t]$. Except in trivial cases, the measure $Q_t^{BMV}$ is not atomic. We have

**Proposition 33.2.** 1. $\int_{\mathbb{R}} s dQ_t^{BMV}(s) = -\operatorname{Ent}(\omega_t|\omega)$.

In the remaining statements we assume that the finite quantum system $(\mathcal{O}_\mathcal{X}, \tau, \omega)$ is TRI.

2. For $\alpha \in \mathbb{C}$,

$$\mathcal{F}_t^{BMV}(\alpha) = \mathcal{F}_t^{BMV}(1 - \bar{\alpha}).$$

3. Let $\tau: \mathbb{R} \to \mathbb{R}$ be the reflection $\tau(s) = -s$ and $\bar{Q}_t^{BMV} = Q_t^{BMV} \circ \tau$. Then the measures $Q_t^{BMV}$ and $\bar{Q}_t^{BMV}$ are equivalent and

$$\frac{d\bar{Q}_t^{BMV}}{dQ_t^{BMV}}(s) = e^{-s}.$$

In the general setting of a modular $C^*$-quantum dynamical systems satisfying (Reg1), (Reg2), and (Reg3), one has, for $\alpha \in \mathbb{R}$,

$$\log \|e^{\frac{1}{2}(\log \Delta_{\omega} - \alpha \int_{0}^{t} \sigma_{s} ds)} \Omega\|^2 = \sup_{v} \left[ S(v|\omega) - \alpha \int_{0}^{t} v(\sigma_{s}) ds \right], \qquad (33.3)$$

where the supremum is taken over all states $v$ on $\mathcal{O}$. For finite quantum systems the left-hand side of (33.3) is a rewriting of (33.3) in modular terms. For $\alpha \in \mathbb{R}$, we define

$$\mathcal{F}_t^{BMV}(\alpha) = \|e^{\frac{1}{2}(\log \Delta_{\omega} - \alpha \int_0^t \sigma_s ds)} \Omega\|^2.$$

It follows from Araki's perturbation theory of the KMS structure that $\mathcal{F}_t^{BMV}(\alpha)$ is finite for all $\alpha \in \mathbb{R}$, and that the map $\mathbb{R} \ni \alpha \mapsto \mathcal{F}_t^{BMV}(\alpha)$ extends to an entire analytic function. We also have

$$\left. \frac{\partial}{\partial \alpha} \mathcal{F}_t^{BMV}(\alpha) \right|_{\alpha=0} = \operatorname{Ent}(\omega_t|\omega).$$

In addition, if $(\mathcal{O}, \tau, \omega)$ is TRI, then for all $\alpha \in \mathbb{C}$,

$$\mathcal{F}_t^{BMV}(\alpha) = \mathcal{F}_t^{BMV}(1 - \bar{\alpha}). \qquad (33.4)$$

The existence of the BMV measure satisfying (33.2) is not known except for finite quantum systems and in the thermodynamic limit setting; see the conjecture at the end of this section.

Returning to the thermodynamic limit of non-equilibrium open quantum spin systems, we have

<footnote>$^{23}$See also [LS04].</footnote>
**Theorem 33.3.** Suppose that each $\omega_{\beta_j}$ is a thermodynamic limit point $\beta_j$-KMS state on $\mathcal{O}_{R_j}$. Let

$$ \Lambda_a = S \cup \left( \bigcup_{j=1}^{M} \Lambda_{j,a} \right) $$

be a net in $\mathfrak{G}_{\text{fin}}$ such that for each $j$,

$$ \lim_{\Lambda_{j,a} \to \infty} \omega_{\Lambda_{j,a}} = \omega_{\beta_j}. $$

where $\omega_{\Lambda_{j,a}}$ is the same as in Theorem 22.1. Then, for all $t \in \mathbb{R}$, the limit

$$ Q_t^{\text{BMV}} := \lim_{\Lambda_a \to \infty} Q_{t, \Lambda_a}^{\text{BMV}} $$

exists and for all $\alpha \in \mathbb{R}$,

$$ \mathfrak{F}_t^{\text{BMV}}(\alpha) = \int_{\mathbb{R}} e^{-\alpha s} dQ_t^{\text{BMV}}. $$

The proof of this result is implicit in [Ben+25a].

The BMV entropy possesses several appealing features. By Proposition 33.2(i), it correctly predicts the entropy production of NESS, and the symmetry (33.4), with its suitable generalization, leads to the linear response theory of NESS; see [Jak+12a]. The function $\mathfrak{F}_t^{\text{BMV}}$ is entire analytic without invoking any additional regularity assumptions.

Many interesting questions remain open; we mention here perhaps the two most important ones.

**Question 1.** For finite quantum systems, the physical status of the statistics $Q_t^{\text{BMV}}$ is unclear. Are these statistics experimentally accessible?

**Question 2.** At present, for infinitely extended systems the existence of the probability measures $Q_t^{\text{BMV}}$ can only be justified by thermodynamic limit arguments. These measures would exist for any modular $C^*$-dynamical system $(\mathcal{O}, \tau, \omega)$ provided that the following KMS extension of the BMV conjecture holds.

**Conjecture.** Let $(\mathfrak{M}, \tau)$ be a $W^*$-dynamical system on a Hilbert space $\mathcal{H}$ and $\omega(A) = (\Omega, A\Omega)$ a faithful $(\tau, \beta)$-KMS vector state on $\mathfrak{M}$. Let $\mathcal{L}$ be the standard Liouvillean of $\tau$, $V \in \mathfrak{M}$ selfadjoint, and consider the perturbed $W^*$-dynamics on $\mathfrak{M}$ given by

$$ \tau_V^t(A) = e^{it(\mathcal{L}+V)} A e^{-it(\mathcal{L}+V)}. $$

Let

$$ \omega_V(A) = \frac{(\Omega_V, A\Omega_V)}{\|\Omega_V\|^2}, \quad \Omega_V = e^{-\frac{\beta}{2}(\mathcal{L}+V)}\Omega, $$

be the perturbed $(\tau_V, \beta)$-KMS vector state on $\mathfrak{M}$. Then there exists a Borel probability measure $P$ on $\mathbb{R}$ such that for all $\alpha \in \mathbb{R}$,

$$ \|\Omega_{\alpha V}\|^2 = \int_{\mathbb{R}} e^{-\alpha s} dP(s). $$

In finite dimension, the above conjecture is equivalent to the BMV conjecture positively resolved by Stahl [Sta13]. The Stahl method, however, is restricted to finite-dimensional matrices, and novel ideas are needed to address the above general conjecture.

The material discussed in this section has not previously appeared in print.
## 34 Entropy production of repeated quantum measurements

Besides the dynamical system approach to classical entropy production, in which the reference state plays an important role, an altogether different random path approach has been developed in [Kur98; LS99; Mae99], that does not make use of the reference state and is applicable to stochastic processes. Its quantum formulation in the setting of repeated quantum measurement processes goes back to [Cro08], and was elaborated in [Ben+18; Ben+21]. The advent of experimental methods in cavity and circuit QED, and in particular the experimental breakthroughs of the Haroche-Raimond and Wineland groups [Har13; HR06; Win13], make this complementary approach particularly relevant.

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{B}(\mathcal{H})$ denote the algebra of bounded operators on $\mathcal{H}$. A linear completely positive map

$$ \Phi: \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H}) $$

is called a *quantum channel*. The channel $\Phi$ is said to be *unitary* if it preserves the identity operator, i.e., satisfies $\Phi(1) = 1$. The predual of a quantum channel $\Phi$ is the map $\Phi^*$ on the trace class operators on $\mathcal{H}$ such that

$$ \mathrm{tr}(\rho\Phi(A)) = \mathrm{tr}(\Phi^*(\rho)A) $$

for all $A \in \mathcal{B}(\mathcal{H})$. If $\Phi$ is unitary, then $\Phi^*$ is trace preserving and maps density matrices to density matrices.

**Definition 34.1.** Given a unitary quantum channel $\Phi$ on a separable Hilbert space $\mathcal{H}$ and a Polish space $\mathcal{A}$, a $(\mathcal{F}, \mathcal{A})$-instrument is a $\sigma$-additive map $\mathcal{J}$ from the Borel $\sigma$-algebra $\mathcal{A}$ of $\mathcal{A}$ to the set of completely positive maps on $\mathcal{B}(\mathcal{H})$ satisfying

$$ \mathcal{J}(\mathcal{A}) = \Phi. $$

An instrument models a repeatable quantum measurement as follows. Let the system be in the state (density matrix) $\rho$ at time $t=1$. A measurement is performed, and a random outcome $\omega_1 \in \mathcal{A}$ is observed with the law $\mathrm{tr}(\mathcal{J}(d\omega_1)^*(\rho))$. After the measurement, the system state conditioned on $\omega_1 \in A_1$ is

$$ \rho_{A_1} = \frac{\mathcal{J}(A_1)^*(\rho)}{\mathrm{tr}(\mathcal{J}(A_1)^*(\rho))}. $$

The law of the outcome $\omega_2$ of the next measurement at time $t=2$ is $\mathrm{tr}(\mathcal{J}(d\omega_2)^*(\rho_{A_1}))$, and the state of the system after the second measurement conditioned on $(\omega_1, \omega_2) \in A_1 \times A_2$ is

$$ \rho_{A_1 A_2} = \frac{\mathcal{J}(A_2)^*(\rho_{A_1})}{\mathrm{tr}(\mathcal{J}(A_2)^*(\rho_{A_1}))}. $$

The probability that the observed $(\omega_1, \omega_2)$ is in $A_1 \times A_2$ is

$$ \mathrm{tr}(\mathcal{J}(A_1)^*(\rho)) \mathrm{tr}(\mathcal{J}(A_2)^*(\rho_{A_1})) = \mathrm{tr}(\mathcal{J}(A_2)^* \circ \mathcal{J}(A_1)^*(\rho)) = \mathrm{tr}(\rho \mathcal{J}(A_1) \circ \mathcal{J}(A_2)(1)). $$

Continuing in this way, one derives that, after $n$ repeated measurements, the probability of observing a sequence of outcomes $(\omega_1, \dots, \omega_n) \in A_1 \times \dots \times A_n$ is given by

$$ \mathbb{P}_n(A_1 \times \dots \times A_n) = \mathrm{tr}(\rho \mathcal{J}(A_1) \circ \dots \circ \mathcal{J}(A_n)(1)). $$

This defines a probability measure $\mathbb{P}_n$ on $\Omega_n = \mathcal{A}^n$, and the unitality of $\mathcal{J}(\mathcal{A}) = \Phi$ implies that the family $(\mathbb{P}_n)_{n \in \mathbb{N}}$ is consistent. Let $\mathbb{P}$ be the unique probability measure induced on $\Omega = \mathcal{A}^N$, equipped with the usual product $\sigma$-algebra $\mathcal{F}$, and the filtration $(\mathcal{F}_n)_{n \in \mathbb{N}}$ generated by the cylinders

$$ [A_1, \dots, A_n] = \{\omega \in \Omega | (\omega_1, \dots, \omega_n) \in A_1 \times \dots \times A_n\}, \quad A_1, \dots, A_n \in \mathcal{A}. $$
We assume that the state $\rho$ is $\Phi$-invariant. Then $\mathbb{P} \circ \phi^{-1} = \mathbb{P}$ where $\phi$ is the left shift on $\Omega$. The dynamical system $(\Omega, \phi, \mathbb{P})$ thus describes the outcomes of the repeated measurement process. The measure $\mathbb{P}$ is sometimes called the $\rho$-statistics of the instrument $\mathcal{J}$.

A local reversal on $\mathcal{A}$ is a measurable involution $\theta: \mathcal{A} \to \mathcal{A}$. The associated $\theta$-time reversal on $\Omega_n$ is the involution

$$ \theta_n(\omega_1, \dots, \omega_n) = (\theta(\omega_n), \dots, \theta(\omega_1)). $$

The family of probability measures $(\hat{\mathbb{P}}_n)_{n \in \mathbb{N}}$ defined by

$$ \hat{\mathbb{P}}_n = \mathbb{P}_n \circ \theta_n $$

is also consistent. By Kolmogorov's extension theorem, there exists a unique $\phi$-invariant $\hat{\mathbb{P}} \in \mathcal{P}_{\phi}(\Omega)$ which describes the statistics of the $\theta$-time reversal of the dynamical system $(\Omega, \phi, \mathbb{P})$.

We shall say that the pair $(\Omega, \mathbb{P})$ is $\theta$-time-reversal invariant if $\mathbb{P} = \hat{\mathbb{P}}$, i.e., if $\mathbb{P}_n = \hat{\mathbb{P}}_n$ for all $n \in \mathbb{N}$.

The relative entropy of two probability measures $\mathbb{P}$ and $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, defined by

$$ \mathrm{Ent}(\mathbb{P}|\mathcal{Q}) = \begin{cases} \int_{\Omega} \log \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathcal{Q}}(\omega) \mathrm{d}\mathbb{P}(\omega) & \text{if } \mathbb{P} \ll \mathcal{Q}; \\ \infty & \text{otherwise,} \end{cases} $$

is non-negative and vanishes iff $\mathbb{P} = \mathcal{Q}$. The $\theta$-entropy production of $(\Omega, \phi, \mathbb{P})$ in the discrete-time interval $[1, n]$ is defined by

$$ \mathrm{Ep}(\mathbb{P}_n, \theta) = \mathrm{Ent}(\mathbb{P}_n|\hat{\mathbb{P}}_n). $$

The (possibly infinite) non-negative number

$$ \mathrm{ep}(\mathbb{P}, \theta) = \limsup_{n \to \infty} \frac{1}{n} \mathrm{Ep}(\mathbb{P}_n, \theta) $$

is called the $\theta$-entropy production rate of $(\Omega, \phi, \mathbb{P})$.

At the current level of generality, the $\theta$-entropy production rate can exhibit pathological behavior; see [And16]. The following upper-decoupling property excludes these pathologies, ensuring that $\theta$-time-reversal invariance is equivalent to the vanishing of entropy production, and in this sense is characteristic of equilibrium.

(UD) There is a constant $C > 0$ such that for any $n \in \mathbb{N}$, $A \in \mathcal{F}_n$ and $B \in \mathcal{F}$,

$$ \mathbb{P}(A \cap \phi^{-n}(B)) \leq C \mathbb{P}(A) \mathbb{P}(B). $$

Note that if (UD) holds for $\mathbb{P}$, then it also holds for $\hat{\mathbb{P}}$. (UD) automatically holds if $\dim \mathcal{H} < \infty$, $\rho > 0$, and the set $\mathcal{A}$ is finite.<fnref n="24" />

The following result is an immediate extension of Theorem 2.1 and Proposition 2.2 in [Ben+18] using the Donsker-Varadhan variational formula for the relative entropy. For details see [Ben+25b, Proposition 1.3]

**Proposition 34.2.** Suppose that (UD) holds. Then:

1. The following (possibly infinite) limit exists:

$$ \mathrm{ep}(\mathbb{P}, \theta) = \lim_{n \to \infty} \frac{1}{n} \mathrm{Ep}(\mathbb{P}_n, \theta). $$

2. Assume in addition that $\mathbb{P}$ is $\phi$-ergodic. Then $\mathrm{ep}(\mathbb{P}, \theta) = 0$ iff $\mathbb{P} = \hat{\mathbb{P}}$.

This result is a starting point of [Ben+18; Ben+21] which focus on the non-equilibrium case $\mathrm{ep}(\mathbb{P}, \theta) > 0$. In [Ben+25b] the vanishing of $\mathrm{ep}(\mathbb{P}, \theta)$ for a suitable class of instruments is linked to the detailed balance condition induced by the inner product (16.2).

<footnote n="24">²⁴These assumptions cover many cases of physical interest.</footnote>
## 35 Conclusion

The study of entropy production in the setting of C\*-quantum dynamical systems has revealed a somewhat unexpected structural richness in non-equilibrium quantum statistical mechanics. There exist several distinct notions that correctly predict the mean entropy production of NESS and its linear response theory, while at the same time capturing different facets of the subject. The richness of the subject is further enhanced by the study of entropic fluctuations in the large-deviations regime and their links to quantum fluctuation theorems [Ben+24b] and to hypothesis testing of the arrow of time [Jak+12b].

In Parts V–VIII, we have touched on the complexities and subtleties that arise in the study of quantum entropy production; much theoretical and experimental work remains to be done to achieve a deeper understanding of this notion.

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