# Fluctuation Theorem for Many-Body Pure Quantum States

###### Abstract

We prove the second law of thermodynamics and the nonequilibirum fluctuation theorem for pure quantum states. The entire system obeys reversible unitary dynamics, where the initial state of the heat bath is not the canonical distribution but is a single energy-eigenstate that satisfies the eigenstate-thermalization hypothesis (ETH). Our result is mathematically rigorous and based on the Lieb-Robinson bound, which gives the upper bound of the velocity of information propagation in many-body quantum systems. The entanglement entropy of a subsystem is shown connected to thermodynamic heat, highlighting the foundation of the information-thermodynamics link. We confirmed our theory by numerical simulation of hard-core bosons, and observed dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our result reveals a universal scenario that the second law emerges from quantum mechanics, and can experimentally be tested by artificial isolated quantum systems such as ultracold atoms.

Introduction. Although the microscopic laws of physics do not prefer a particular direction of time, the macroscopic world exhibits inevitable irreversibility represented by the second law of thermodynamics. Modern researches has revealed that even a pure quantum state, described by a single wave function without any genuine thermal fluctuation, can relax to macroscopic thermal equilibrium by a reversible unitary evolution [1, 10, 2, 3, 11, 4, 7, 5, 6, 8, 9]. Thermalization of isolated quantum systems, which is relevant to the zeroth law of thermodynamics, is now a very active area of researches in theory [1, 2, 3, 4, 5, 6], numerics [10, 13, 11, 12, 14, 15, 16], and experiments [17, 18, 19, 20, 21, 22, 23]. Especially, the concepts of typicality [24, 25, 26, 9] and the eigenstate thermalization hypothesis (ETH) [27, 28, 10, 29, 11, 30, 12, 31, 34, 32, 33, 36, 35] have played significant roles.

However, the second law of thermodynamics, which states that the thermodynamic entropy increases in isolated systems, has not been fully addressed in this context. We would emphasize that the informational entropy (i.e., the von Neumann entropy) of such a genuine quantum system never increases, but is always zero [37]. In this sense, a fundamental gap between the microscopic and macroscopic worlds has not yet been bridged: How does the second law emerge from pure quantum states?

In a rather different context, a general theory of the second law and its connection to information has recently been developed even out of equilibrium [38, 39, 40, 41], which has also been experimentally verified in laboratories [42, 43, 44, 45]. This has revealed that information contents and thermodynamic quantities can be treated on an equal footing, as originally illustrated by Szilard and Landauer in the context of Maxwell’s demon [46, 47]. This research direction invokes a crucial assumption that the heat bath is, at least in the initial time, in the canonical distribution [48]; this special initial condition effectively breaks the time-reversal symmetry and leads to the second law of thermodynamics. The same assumption has been employed in various modern researches on thermodynamics, such as the nonequilibrium fluctuation theorem [50, 49, 51, 52, 53, 54, 55, 48, 56, 57, 58] and the thermodynamic resource theory [59, 60].

Based on these streams of researches, in this Letter we rigorously derive the second law of thermodynamics for isolated quantum systems in pure states. We consider a small system and a large heat bath, where the bath is initially in a single energy-eigenstate. Such an eigenstate is a pure quantum state, and does not include any statistical mixture as is the case for the canonical distribution. The second law that we show here is formulated with the von Neumann entropy of the system, ensuring the information-thermodynamics link, which is a characteristic of our study in contrast to previous approaches [61, 62, 36]. Furthermore, we prove the integral fluctuation theorem [50, 53, 54, 63], a universal relation in nonequilibrium statistical mechanics, which expresses the second law as an equality rather than an inequality.

The key of our theory is combining the following two fundamental concepts. One is the Lieb-Robinson bound [64, 65], which characterizes the finite group velocity of information propagation in quantum many-body systems with local interaction. The other is the ETH, which states that even a single energy-eigenstate can behave as thermal [27, 28, 10, 29, 11, 30, 12, 31, 34, 32, 33, 36, 35]. In this Letter, we newly prove a variant of the ETH [12, 35], which is referred to as the weak ETH and states that most of the energy eigenstate satisfies the ETH, if an eigenstate is randomly sampled from the microcanonical energy shell.

Our theory provides a rigorous scenario of the emergence of the second law from quantum mechanics, which is relevant to experiments of artificial isolated quantum systems. Furthermore, our approach to the second law would be applicable to quite a broad class of modern researches of thermodynamics, from thermalization in ultracold atoms [20] to scrambling in black holes [66, 67, 68, 69].

Setup. We first formulate our setup with a heat bath in a pure state. Suppose that the entire system consists of system S and bath B. We assume that bath B is a quantum many-body system on a -dimensional hypercubic lattice with sites. The Hamiltonian is given by

(1) |

where and are respectively the Hamiltonians of system S and bath B, and represents their interaction. We assume that is translation invariant with local interaction, and system S is locally in contact with bath B (see Fig. 1(a)). We also assume that the correlation functions in the canonical distribution with respect to is exponentially decaying for any local observables, which implies that bath B is not on a critical point.

The initial state of the total system is given by

(2) |

where is the initial density operator of system S, and is the initial energy eigenstate of bath B. We sample from the set of the energy eigenstates in the microcanonical energy shell in a uniformly random way, as will be described in detail later. We can then define temperature of as the temperature of the corresponding energy shell. We note that the initial correlation between S and B is assumed to be zero.

The total system then obeys a unitary time evolution by the Hamiltonian: with . Such a situation can experimentally be realized with ultracold atoms by quenching an external potential at time . Let and be the density operators of system S and bath B at time , respectively. The change in the von Neumann entropy of S is given by with . We also define the heat emitted from bath B by .

If the initial state of system S is pure (i.e., ), the total system is also pure, whose von Neumann entropy vanishes. In such a case, the final state remains in a pure state because of the unitarity, but is entangled. Correspondingly, the final state of S is mixed and has non-zero von Neumann entropy, which is regarded as the entanglement entropy.

Second law. We now discuss our first main result. If is a typical energy eigenstate, that satisfies the ETH, the second law of thermodynamics is shown to hold within a small error:

(3) |

where is a positive error term. We can rigorously prove that can be arbitrarily small if bath B is sufficiently large. The left-hand side of inequality (3) is regarded as the average entropy production , where describes the ensemble average, and is the stochastic entropy production that will be introduced later. We note that, if the initial state of bath B is not pure but in the canonical distribution , inequality (3) exactly holds without any error [48].

The second law (3) implies that the information-thermodynamics link emerges in genuine quantum systems, if we look at the informational entropy of subsystem S, though that of the total system remains unchanged. A significant consequence of inequality (3) is the Landauer erasure principle [46]. Suppose that the initial state of S stores one bit of information such that , and it is erased in the final state: . We then have , and the heat emission from S, represented by , is bounded by within a small error. While the Landauer principle and its generalizations have been derived in various ways [38, 70, 71, 72, 73, 74, 75], we here showed that it emerges in the presence of a pure quantum bath.

We will prove inequality (3) in Supplemental Material in a mathematically rigorous way. Here We only discuss the essentials of the proof, where the key ingredients are the Lieb-Robinson bound [64, 65] and the weak ETH [12, 35].

Lieb-Robinson bound. The Lieb-Robinson bound gives an upper bound of the velocity of information propagation, and is applicable to any system on a generic lattice with local interaction. To apply the Lieb-Robinson bound, we divide bath B into and , such that is near system S and is far from S (see Fig. 1(b)). Then, the Lieb-Robinson bound [64, 65] sets the shortest time , at which information about reaches S across . We refer to as the Lieb-Robinson time.

The detailed formulation of the Lieb-Robinson bound is the following. Let be the union of S and the support of . Let and be arbitrary operators with the supports and , respectively, where is the boundary between and . Let be the spatial distance between and on the lattice, and let and be the numbers of the sites in and respectively. The Lieb-Robinson bound is formulated in terms of the operator norm as

(4) |

where represents the time evolution in the Heisenberg picture, and , , are positive constants. In particular, represents the velocity of information propagation. The Lieb-Robinson time is then given by .

Weak ETH. We next consider the concept of the weak ETH. Let be the dimension of the Hilbert space of the microcanonical energy shell of bath B, which is exponentially large with respect to , and let be an orthonormal set of the energy eigenstates of in the energy shell. Suppose that we choose from in a uniformly random way. As proved in Supplemental Material, if is sufficiently larger than , we typically have that

(5) |

which implies that is indistinguishable from if we only look at any operator on with . We refer to this theorem as the weak ETH, which is a variant of a theorem shown in Ref. [12, 35]. We note that the equivalence of the canonical and the microcanonical ensembles for reduced density operators [76, 77] plays an important role here.

We note that the weak ETH is true even if bath B is an integrable system [31, 34]. However, it has been shown that atypical states that do not satisfy Eq. (5) have large weights after quantum quench in the case of integrable systems [12], and the steady values of macroscopic observables are consistent with the generalized Gibbs ensemble (GGE) [78, 13, 79] rather than the microcanonical ensemble. In this sense, the weak ETH is physically significant only for nonintegrable systems, though it is mathematically true even for integrable systems. The reason why the weak ETH is called “weak” is that there is another concept called the “strong” ETH (or just ETH) that is believed to be true only for nonintegrable systems, where every energy eigenstate satisfies Eq. (5) without exception [31].

Outline of the proof of (3). We are now in the position to discuss the outline of the proof of the second law (3). In the short time regime , system S cannot feel the existence of , and the heat bath effectively reduces to . From the weak ETH, if is sufficiently larger, the initial state of bath B is typically indistinguishable from the canonical distribution, if we only look at any operator in . Thus, the reduced density operators of system S at time are almost the same for the initial energy eigenstate and the initial canonical distribution. Therefore, the conventional proof of the second law with the canonical bath approximately applies to the present situation, leading to inequality (3).

Integral fluctuation theorem. We next discuss the IFT for the case that bath B is initially in an energy eigenstate, which is our second main result. Let be the stochastic entropy production defined as follows. Suppose that one performs projection measurements of at the initial and the final times, where the first and the second terms on the right-hand side respectively represent the informational and the energetic terms, corresponding to the first and the second terms on the left-hand side of inequality (3). Then, is given by the difference between the obtained outcomes of these measurements. The average of the stochastic entropy production is equivalent to the aforementioned average entropy production: .

The conventional IFT states that, if the initial state of bath B is the canonical distribution,

(6) |

We note that the IFT holds even when system S is far from equilibrium. A crucial feature of the IFT is that it reproduces the second law from the Jensen inequality . Furthermore, the IFT can reproduce the fluctuation-dissipation theorem [54].

Our result is the IFT in the case that bath B is initially in a typical energy eigenstate (i.e., with initial condition (2)):

(7) |

where is a positive error term. We can rigorously prove that can be arbitrarily small if bath B is sufficiently large. We note that the detailed fluctuation theorem [54] also holds with an initial typical eigenstate, from which we can prove the IFT as a corollary (see Supplemental Material for details).

The central idea of the proof of the IFT (7) is almost the same as that of the second law, which is outlined above. On the other hand, we need to make an additional assumption to prove inequality (7) that

(8) |

which means that the sum of the energies of S and B is conserved at the level of fluctuations, and does not necessarily mean that itself is small. We note that assumption (8) is consistent with the concept of “thermal operation” in the thermodynamic resource theory [59, 60], where the left-hand side of Eq. (8) is assumed to be exactly zero. The rigorous meaning of “” is discussed in Supplemental Material. If the left-hand side of Eq. (8) is nonzero but small, a small positive error term should be added to the right-hand side of inequality (7), which cannot be arbitrarily small even in the large-bath limit. However, we will numerically show later that this additional term is negligible in practice.

Estimation of the error terms. We evaluate the error terms in inequalities (3) and (7) with respect to the size of bath B. We set with . The error from the weak ETH is bounded by , where is an unimportant constant that can be arbitrarily small, and is the fraction of atypical eigenstates in the weak ETH. The error from the Lieb-Robinson bound is bounded by , which is negligible compared with the error term from the weak ETH for sufficiently large , but increases in time with up to .

Numerical simulation. We performed numerical simulation of hard-core bosons with nearest-neighbor repulsion by exact diagonalization. System S is on a single site and bath B is on a two-dimensional lattice (see the inset of Fig. 2). The annihilation (creation) operator of a boson at site is written as (), which satisfies the commutation relations for , , and . The occupation number is defined as . Let be the index of the site of system S. The Hamiltonians in Eq. (1) are then given by

(9) | ||||

(10) |

where is the on-site potential, is the hopping rate in bath B, is the hopping rate between system S and bath B, and is the repulsion energy. The initial state of system S is given as . We set the size of bath B by , and the initial number of bosons in bath B by . To evaluate the Lieb-Robinson time, we set . We can then evaluate that and if , and therefore the Lieb-Robinson time is given by . We set the inverse temperature of the initial eigenstate as .

Figure 2 shows the time dependence of , which implies that the second law indeed holds. The average entropy production gradually increases up to , and then saturates with some oscillations around the time average. We note that the oscillation for is the Rabi oscillation between system S and a part of B. Remarkably, we observed that the second law holds even in a much longer time regime , though our theorem ensures the second law only up to . This implies that the second law is so robust against bare quantum fluctuations of pure quantum states.

We next confirmed the IFT (7). As shown in Fig. 3, is very close to unity up to , as predicted by our theorem. After , however, the deviation of from unity becomes significant, which is consistent with our theorem. This deviation comes from the effect of bare quantum fluctuations of the initial state, because if the initial state was the canonical distribution, such deviation would never be observed. As time increases, system S more and more feels the effect of bare quantum fluctuations, and the deviation becomes larger. This is regarded as a dynamical crossover from emergent thermal fluctuations to bare quantum fluctuations across the Lieb-Robinson time ; The IFT holds only for the former. Such a crossover is not clearly observed in the second law (Fig. 2), because the second law only concerns the average of the stochastic entropy production, while its fluctuations play a significant role in the IFT. Our numerical result also clarifies that our theory indeed accounts for the validity of IFT in the short time regime, because the numerically observed time scale of the breakdown of the IFT is consistent with our theory.

As shown in the inset of Fig. 3, the error term of the IFT is proportional to up to in our numerical simulation. In fact, our error evaluation based on the Lieb-Robinson bound predicts that an error term increases in time with -dependence as discussed before, if the additional error term , which could also increase in time, is zero (or equivalently, the left-hand side of (8) is zero). Therefore, our numerical result clarifies that the contribution from the left-hand side of (8) is negligible in our setup, though it is not exactly zero in our Hamiltonians (9) and (10).

Concluding remarks. In this Letter, we have established the second law (3) and the IFT (7) for unitary dynamics in the presence of a heat bath that is initially in a typical energy eigenstate. Our result implies that thermal fluctuations can emerge from quantum fluctuations in a short time regime, and the former crosses over to the latter in time. Our rigorous mathematical proof is based on the Lieb-Robinson bound (4) and the weak ETH (5). We also performed numerical simulation of two-dimensional hard-core bosons, and confirmed our theoretical results.

We remark that inequality (3) only ensures the entropy increase between the initial and final times, and does not imply the monotonic entropy increase in continuous time. The extension of our result to the monotonic second law in continuous time could be possible with controlled approximations such as the Born-Markov approximation [80], as is the case for the standard master equation approach [81, 82]. This is a future direction of our work, though it would be technically nontrivial.

Our results can experimentally be tested with artificial isolated quantum systems such as ultracold atoms on an optical lattice, by employing the technique of single-site addressing [83]. Another candidate of experimentally relevant systems is superconducting qubits, where fully-controlled dynamics of thermalization can be observed [84]. To examine the relevance of our theory to non-artificial complex materials in noisy open environment is an future issue.

Acknowledgement The authors are most grateful to Hal Tasaki for valuable discussions, especially on the equivalence of ensembles. The authors also thank Takashi Mori and Jae Dong Noh for valuable discussions. E.I. and T.S. are supported by JSPS KAKENHI Grant Number JP16H02211. E.I. is also supported by JSPS KAKENHI Grant Number 15K20944. T.S. is also supported by JSPS KAKENHI Grant Number JP25103003.

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Supplemental Material : Fluctuation Theorem for Many-Body Pure Quantum States

Eiki Iyoda, Kazuya Kaneko, Takahiro Sagawa

[1] Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

[2] Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan

###### Contents

- A The conventional second law and the fluctuation theorem
- B Norms of operators
- C The Lieb-Robinson bound
- D The canonical and the microcanonical ensembles
- E Weak eigenstate-thermalization hypothesis (ETH)
- F Our setup
- G Truncated dynamics around system S
- H Proof of the second law of thermodynamics
- I Proof of the fluctuation theorem
- J Typicality in the Hilbert space
- K Discussion
- L Details of the numerical simulation

In this Supplemental Material, we prove the second law of thermodynamics and the fluctuation theorem for a system in contact with a heat bath in a pure state [inequalities (3) and (7) in the main text] in a mathematically rigorous way. For this purpose, we need several technical assumptions that will be discussed in detail. However, the bare essentials of the proof are the same as illustrated in the main text. In addition, we show supplementary results of numerical simulation.

The organization of Supplemental Material is as follows. In Sec. A, we briefly review the established proof of the conventional second law and the fluctuation theorem with a heat bath that is initially in the canonical distribution. In Sec. B, we remark on the operator norm and the trace norm. In Sec. C, we formulate the Lieb-Robinson bound that is a key of our proof. In Sec. D, we define the canonical ensemble and the microcanonical ensemble. In Sec. E, we show our theorems on the weak eigenstate-thermalization hypothesis (ETH). In Sec. F, we describe our basic setup of the proof of the second law and the fluctuation theorem. In Sec. G, we prove two important lemmas. In particular, Lemma G.2 plays a key role in the proof of our main results. In Secs. H and I, we prove our main results: the second law (Theorem 1) and the fluctuation theorem (Theorem 2), respectively. In Sec. J, we discuss the typicality in the Hilbert space. In Sec. K, we remark on some assumptions in our setup. In Sec. L, we show the details of our numerical calculation and supplementary numerical results.

Throughout Supplemental Material, we set and for simplicity. We will only consider finite-dimensional Hilbert spaces.

## Appendix A The conventional second law and the fluctuation theorem

In this section, we review the conventional proof of the second law and the fluctuation theorem for a system in contact with a heat bath that is initially in the canonical distribution.

### a.1 Setup

Suppose that the total system consists of system S and bath B as shown in Fig. S1. The Hamiltonian of the composite system is given by

(S1) |

where and are the Hamiltonian of system S and bath B, respectively. The interaction Hamiltonian between system S and bath B is denoted by . We note that there is no additional assumption on (such as locality) in this section. The coupling between system S and bath B can either be weak or strong.

The initial state is represented by a product state without any correlation between S and B:

(S2) |

The initial state of system S is arbitrary, and the initial state of bath B is described by the canonical distribution , where is the partition function and is the inverse temperature.

The composite system obeys the unitary evolution described by , and the final state of the system is given by . The reduced density operators for the final state of system S and bath B are defined as and , respectively.

### a.2 The second law of thermodynamics

We first note that the von Neumann entropy of a density operator is given by

(S3) |

The change in the von Neumann entropy of system S is represented as . We also define the heat as . The second law of thermodynamics in this setup is then stated as follows:

Proposition 1 (Second law of thermodynamics with the canonical bath [S1])
(S4)

Proof. The von Neumann entropy of the composite system in the initial and the final states are given by

(S5) |

where we used the Klein inequality: . Since the von Neumann entropy is invariant under unitary dynamics , we obtain inequality (S4).

### a.3 The fluctuation theorem

We consider the concept of stochastic entropy production [S2]. We first define

(S6) |

in the Schrödinger picture, and consider the projection measurements of at time and . Let and be the measurement outcomes at time and , respectively. The joint probability distribution of and is given by

(S7) |

where and are the projection operators corresponding to eigenvalues and of and , respectively. We note that and are, and thus and are, commutable. The index in means the forward time evolution. The stochastic entropy production is then defined as the difference between the two measurement outcomes:

(S8) |

The probability distribution of the stochastic entropy production is written as , which is given by

(S9) |

where is the delta function. We note that in the proof of our main results in the subsequent sections, we will not use the concept of the delta function in order to avoid introducing advanced mathematical tools for the rigorous argument.

The characteristic function of the stochastic entropy production is given by the Fourier transformation of [S2]:

(S10) |

where is referred to as the counting field. The th differential of the characteristic function gives the th order moment of the stochastic entropy production:

(S11) |

We note that the ensemble average of the entropy production is given by

(S12) |

which is the left-hand side of the second law (S4). Therefore, we can rewrite the second law (S4) in terms of the entropy production by

(S13) |

By noting that , the characteristic function is rewritten as

(S14) |

which can be regarded as the definition of in the case that we do not use the delta function.

We next consider the reversed time evolution that obeys unitary evolution . The initial state of the reversed process is given by the product state with the final state of system S and the canonical distribution of bath : , where index represents the reversed processes. The density operator at time of the reversed process is given by .

We then define the stochastic entropy production in the reversed processes by the difference between the measurement outcomes of and at the initial and the final times of the reversed processes, respectively. Let be the probability distribution of the stochastic entropy production in the reversed processes, and be the corresponding characteristic function. By noting that , we have

(S15) |

We note that in general. Therefore, the entropy production defined above is not necessarily related to the change in the von Neumann entropy of system S in the reversed process. In this sense, the physical interpretation of the entropy production is not very clear in the reversed process. If happens to be true, the entropy production in the reversed process has a clear interpretation related to the von Neumann entropy of the reversed process. Such a situation is physically realizable if the initial and the final states of system S are in thermal equilibrium in both of the forward and the reversed processes.

Apart from its physical interpretation, the entropy production in the reversed process is always mathematically well-defined, and the following argument including the reversed process is true for any case. In addition, the entropy production in the reversed process is a useful concept to prove some theorems that only include quantities in the forward process, such as the integral fluctuation theorem. Therefore, we will not go into details of the physical interpretation of the reversed processes in the following argument.

We now discuss the fluctuation theorem. The detailed fluctuation theorem characterizes a universal relationship between the probabilities of the entropy changes in the forward and the reversed processes. In the following argument, it is convenient to formulate the detailed fluctuation theorem in terms of the characteristic functions:

Proposition 2 (Detailed fluctuation theorem with the canonical bath [S2])
(S16)

We next discuss the integral fluctuation theorem (IFT) [S3, S4]. We consider the following quantity:

(S18) |

or equivalently

(S19) |

We then have the following corollary.

Corollary 1 (Integral fluctuation theorem with the canonical bath)
(S20)

Proof. We note that and . By substituting to the detailed fluctuation theorem (S16), we obtain the IFT (S20).

We make some remarks on the fluctuation theorem. Applying the inverse Fourier transform to Eq. (S16), we obtain the detailed fluctuation theorem in terms of the probability distributions:

(S21) |

By integrating the both-hand sides of , which is equivalent to Eq. (S21), we again obtain the IFT:

(S22) |

By using the Jensen inequality to Eq. (S20), we obtain , which reproduces the second law (S13).

## Appendix B Norms of operators

As a preliminary, we briefly review the operator norm and the trace norm for finite-dimensional Hilbert spaces. Let be a vector of a Hilbert space, and be its norm.

We consider operators on the Hilbert space, which is not necessarily Hermitian. First, the operator norm of an operator is defined as

(S23) |

which is equal to the largest singular value of . Second, the trace norm of an operator is defined as

(S24) |

which is the sum of the singular values of . If is Hermitian, the above definition reduces to

(S25) |

We note some useful (and well-known) properties of the norms:

Proposition 3
For any operators,
(S26)
(S27)

## Appendix C The Lieb-Robinson bound

In this section, we review the Lieb-Robinson bound [S5, S6] that plays a key role in our study. The Lieb-Robinson bound gives an upper bound of the velocity of information propagation in a quantum many-body system on a general graph, which is formulated in terms of the operator norm of the commutator of two operators in spatially distinct regions on the graph.

We consider a general setup for the Lieb-Robinson bound as follows. The system is defined on a finite graph, written as , where the sets of all vertices (sites) and all edges (bonds) are denoted by and , respectively. A region X in the graph is defined as a set of sites: . We write the number of elements of a set as . For example, describes the number of all the sites in the graph.

We next define the spatial distance on the graph. The distance between two sites is defined as the number of bonds in the shortest path that connects x and y, which we denote as . The distance between two regions and is correspondingly defined as

(S28) |

The Hamiltonian in our setup is expressed as the sum of local Hamiltonians:

(S29) |

where is a local Hamiltonian on a bounded support . The sum on the right-hand side of Eq. (S29) is over a particular set of bounded supports. We then make the following assumption.

Assumption 1 (Conditions on the local Hamiltonians and the graph [S5, S6])
There exist , , and , such that for any ,
(S30)
(S31)

The constants and are determined by the Hamiltonian and the graph,
which represent the interaction strength and the interaction range, respectively.
For example, if we consider the case of the nearest-neighbor interaction on a hypercubic lattice,
and are determined to satisfy .
If the interaction is bounded as with being a constant,
is given by with arbitrary .
The constant is determined by the graph structure and .
For the case of nearest-neighbor interaction on the two-dimensional square lattice, is given by .

The Lieb-Robinson bound states that there exists an upper bound of the velocity of information propagation for a general Hamiltonian which satisfies Assumption C. In Sec. F, we will formulate a more specific setup for our study, where Assumption C is also assumed and the Lieb-Robinson bound is applicable.

We now state the Lieb-Robinson bound as follows.

Proposition 4 (Lieb-Robinson bound [S5, S6])
Let and be operators with supports A and , respectively.
We consider the Heisenberg picture of , defined as with .
Under Assumption C, defining and , the Lieb-Robinson bound states that
(S32)

The constants and depend on the interaction and the graph structure through and .
Here,
we refer to as the Lieb-Robinson velocity.
We then define a characteristic time as

(S33) |

to which we refer as the Lieb-Robinson time. If , the right-hand side of inequality (S32) becomes exponentially small with respect to the distance between and .

In the case of nearest-neighbor interaction on the two-dimensional square lattice, we have , , and . Since is arbitrary, we can choose it to maximize the Lieb-Robinson time, and is maximized when , where and .

## Appendix D The canonical and the microcanonical ensembles

In this section, we rigorously define the canonical and the microcanonical ensembles, and discuss their equivalence in line with Ref. [S7]. We consider a quantum many-body system on a lattice, to which we refer as “bath B”, as it will play a role of a heat bath in our main theorems in Secs. F-J. We assume that bath B is on a -dimensional hypercubic lattice with the periodic boundary condition. We denote the set of sites of bath B by the same notation, B. We denote the number of sites in bath B by , and the side length of by such that . Let be the local Hilbert space on site . The total Hilbert space of B is denoted by . The dimension of is denoted by .

We assume that the Hamiltonian of bath B is represented as the sum of local Hamiltonians:

(S34) |

where the sum is taken over a particular set of bounded region Z. We write the spectrum decomposition of as

(S35) |

where is an energy eigenvalue, and is the corresponding eigenstate. We assume the locality of the interaction on the lattice, which implies that the interaction range of the Hamiltonian is independent of the size of bath B:

Assumption 2 (Locality of the interaction)
There exists an integer that is independent of , such that for any with support Z, holds for .
We refer to as the interaction range inside bath B.
In addition, we assume that is independent of for any .

We also assume that bath is translation invariant:

Assumption 3 (Translation invariance)
The local Hilbert spaces with are identical,
and the local Hamiltonians are the same for any Z in Eq. (S34),
where ’s are defined over the lattice in a translation invariant way.
Let be the dimension of .

We now define the canonical ensemble.

Definition 1 (Canonical ensemble)
For a given inverse temperature ,
the density operator of the canonical ensemble is defined as
(S36)
where
is the partition function.

We also define the average energy density of the canonical ensemble at inverse temperature by

(S37) |

To define the microcanonical ensemble, we employ the framework of Ref. [S7], where the equivalence of the canonical and the microcanonical ensembles has rigorously been proved. Let be the width of the microcanonical energy shell, which can depend on but is bounded from below by -independent constant . This implies that can be in the order of .

Definition 2 (Microcanonical energy shell)
The energy shell with energy and width is defined as the following set of indices of energy eigenvalues of :
(S38)
We also define the Hilbert space
that is spanned by the energy eigenstates
in the energy shell.
We denote the dimension of the Hilbert space of the energy shell as .

To relate the microcanonical and the canonical ensembles, we define energy that is determined by a given inverse temperature . In line with Ref. [S7], we employ the following definition.

Definition 3
For a given inverse temperature ,
is defined as
(S39)

The above definition is motivated by the Legendre transformation from to , and is roughly rewritten as

(S40) |

where is the thermodynamic entropy.

We now define the microcanonical ensemble for a given inverse temperature .

Definition 4 (Microcanonical ensemble)
The density operator of the microcanonical ensemble corresponding to the energy shell is defined as
(S41)

In the following, we write for simplicity.

We next discuss two fundamental assumptions that are required to prove the equivalence of the ensembles. We first consider the correlation function.

Definition 5 (Correlation function)
Let be an arbitrary density operator.
The correlation function of arbitrary two operators and
is defined as
(S42)

We then assume the following property of the correlation functions for the canonical ensemble.

Assumption 4 (Exponential decay of correlation functions: Assumption I of [S7])
For a given inverse temperature ,
there exist positive constants and that satisfy the following.
Let and be arbitrary operators with supports .
For any , we have
(S43)
where
(S44)

Assumption D can be rigorously proved for any and sufficiently small ,
by using the cluster expansion method [S8, S9].
We note that a slightly different version of the exponential decaying of correlation functions have been proved in Ref. [S10] for sufficiently small .

We next consider the Massieu function , which is related to the free energy by .

Definition 6 (Massieu function)
We define the Massieu function of and as
(S45)
It is known rigorously in general [S11] that the following limit exists:
(S46)

We then assume the following properties of the Massieu function.

Assumption 5 (Properties of the Massieu function: Assumption II of [S7])
For a given inverse temperature ,
there exist and such that ,
and the following two properties are valid.
First, there exists such that
(S47)
for any and .
Second, the Massieu function is twice continuously differentiable,
and satisfies
with a constant in interval .

Assumption D can also be proved for any and sufficiently small ,
by using the cluster expansion method [S8, S9].

We now define the thermodynamic limit with the inverse temperature being fixed:

Definition 7 (Thermodynamic limit)
The thermodynamic limit is given by with being fixed.
In the thermodynamic limit, the interaction range in Assumption D is kept constant.

In the following argument, the phrase of “sufficiently large ” will be used in the sense of the above thermodynamic limit.

We are now in the position to state the equivalence of ensembles in terms of the reduced density operators of a subsystem.

Proposition 5 (Equivalence of ensembles: Main theorem in [S7])