Saturation of entropy production in quantum many-body systems

Saturation of entropy production in quantum many-body systems

Kazuya Kaneko Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan    Eiki Iyoda Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan    Takahiro Sagawa Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
July 12, 2019
Abstract

Bridging the second law of thermodynamics and microscopic reversible dynamics has been a longstanding problem in statistical physics. Here, we address this problem on the basis of quantum many-body physics, and discuss how the entropy production saturates in isolated quantum systems under unitary dynamics. First, we rigorously prove that the entropy production does indeed saturate in the long time regime, even when the total system is in a pure state. Second, we discuss the non-negativity of the entropy production at saturation, implying the second law of thermodynamics. This is based on the eigenstate thermalization hypothesis (ETH), which states that even a single energy eigenstate is thermal. We also numerically demonstrate that the entropy production saturates at a non-negative value even when the initial state of a heat bath is a single energy eigenstate. Our results reveal fundamental properties of the entropy production in isolated quantum systems at late times.

pacs:
05.30.d,03.65.w,05.70.a,05.70.Ln

I Introduction

In the mid-nineteenth century, Rudolf Clausius introduced entropy as a measure of thermodynamic irreversibility, by arguing that, while the energy of the universe is constant, the entropy of the universe tends to a maximum Clausius (1867). Ever since, understanding inevitable irreversibility in the macroscopic world is one of the most fundamental problems in statistical physics. Especially, in light of the fact that isolated quantum systems have reversible dynamics, it has not been clear whether such a behavior of the entropy production is consistent with quantum theory.

From the modern point of view, thermalization of isolated quantum systems has been of much interest in light of state-of-the-art technologies such as ultracold atoms Kinoshita et al. (2006); Hofferberth et al. (2007); Trotzky et al. (2012); Gring et al. (2012); Langen et al. (2013); Kaufman et al. (2016), trapped ions Clos et al. (2016), and superconducting qubits Neill et al. (2016). For example, it has been established that even a pure quantum state in an isolated system exhibits relaxation toward thermal equilibrium by unitary dynamics von Neumann (1929); Jensen and Shankar (1985); Tasaki (1998); Rigol et al. (2008); Reimann (2008); Linden et al. (2009); Goldstein et al. (2010); Reimann (2010); Riera et al. (2012); Mueller et al. (2013); Goldstein et al. (2015); Reinmann (2015); Tasaki (2016a); Farrelly et al. (2017); Gogolin and Eisert (2016). Several key theoretical ideas have been developed to understand such relaxation processes. It has been rigorously proved that an overwhelming majority of pure states in the Hilbert space describe thermal equilibrium, which is referred to as typicality Popescu et al. (2006); Goldstein et al. (2006); Sugita (2007); Reimann (2007). The eigenstate thermalization hypothesis (ETH) is another cornerstone to understand thermalization Jensen and Shankar (1985); Rigol et al. (2008); Berry (1977); Peres (1984); Deutsch (1991); Srednicki (1994); Biroli et al. (2010); Kim et al. (2014); Beugeling et al. (2014); Garrison and Grover (2015), which states that all the energy eigenstates in an energy shell are thermal. The ETH (or the strong ETH) is believed to be true for non-integrable systems without disorder. We note that there is a weaker version of the ETH (the weak ETH) Biroli et al. (2010), which states that most of the energy eigenstate are thermal. The weak ETH is rigorously proved for both integrable and non-integrable systems without disorder Mori (2016); Iyoda et al. (2017).

On the other hand, research into the second law, such as the fluctuation theorem Jarzynski (1997); Crooks (1999); Jarzynski (1999); Kurchan (2000); Tasaki (2000a); Esposito et al. (2009); Campisi et al. (2011); Sagawa (2012); Alhambra et al. (2016); Morikuni et al. (2017) and thermodynamic resource theories Horodecki and Oppenheim (2013); Brandão et al. (2015), is based on a special assumption that the initial state of a heat bath is the canonical ensemble. This assumption is very strong, given that a thermal equilibrium state can even be a pure quantum state. If we do not assume the initial canonical distribution, the non-negativity of the entropy production cannot be proved by the conventional arguments, i.e., the fluctuation theorem or the positivity of relative entropy Sagawa (2012). In this sense, except for only a few seminal works Tasaki (2000b); Goldstein et al. (2013); Ikeda et al. (2015), the foundation of the second law in isolated quantum systems has not been fully addressed.

In this paper, we rigorously prove that the entropy production saturates at some value in the long time regime, for a broad class of initial states including pure states. We note that the entropy production, the sum of the change in von Neumann entropy of a subsystem and the heat absorbed from a heat bath, can change in time, though the von Neumann entropy of the total system is invariant under unitary dynamics Nielsen and Chuang (2000). In general, the saturation value of the entropy production depends on both the initial state and the Hamiltonian, and could be negative. We then prove that the saturation value is non-negative if the system satisfies the strong ETH, which leads to the second law of thermodynamics in the long time regime. Our results provide a quantum foundation to observation by Clausius.

We note that in our previous work Iyoda et al. (2017), we have proved the second law for pure quantum states in the short time regime, on the basis of the Lieb-Robinson bound Lieb and Robinson (1972); Nachtergaele and Sims (2006); Hastings and Koma (2006). However, the previous result cannot apply to the long time regime. In the present work, we have adopted a completely different technique from the Lieb-Robinson bound, and have proved the second law in the long time regime (see Fig. 1).

The rest of this paper is organized as follows. In Sec. II, we formulate our setup and make a brief review on equilibration in isolated quantum systems. In Sec. III, we prove the main theorem, which states that the entropy production takes a constant value for most times. In Sec. IV, we derive a lower bound of the saturation value of the entropy production. We also show that the saturation value is non-negative if the system satisfies the ETH. In Sec. V, we confirm our theory by numerical simulation of hard-core bosons. In Sec. VI, we summarize our results. In Appendix A, we make a remark on the effective dimension.

Ii Setup and notation

We consider a quantum many-body system described by a finite-dimensional Hilbert space. The total system is divided into two parts, a small system and a large heat bath . Their Hilbert space dimensions are denoted by and , respectively. The total Hamiltonian is given by

 H=HS+HB+HI, (1)

where is the Hamiltonian of system , is the Hamiltonian of bath , and is the interaction between system and bath . We write the spectrum decomposition of as

 H=∑jEjPj, (2)

where is an eigen-energy ( for ), and is the projector onto the eigenspace of with energy eigenvalue .

We assume that the initial correlation between system and bath is zero. The initial density operator of the total system is then given by

 ρ(0)=ρS(0)⊗ρB(0), (3)

where and are respectively the initial density operators of system and bath . We do not assume that is the canonical ensemble.

The total system obeys the unitary time evolution generated by the total Hamiltonian . The state at time is written by . Note that we set . We also write the time-averaged state by

 ω:=limτ→∞1τ∫τ0ρ(t)dt, (4)

which is called the diagonal ensemble. Because of the recurrence property of the unitary evolution, an isolated quantum system cannot converge to an exact stationary state. However, a system can relax to an effective stationary state most of the time between recurrences. In fact, it has been proved in Refs. Linden et al. (2009); Short and Farrelly (2012) that for any operator , and for any ,

 limτ→∞Probt∈[0,τ][|Tr[Oρ(t)]−Tr[Oω]|≥ϵ∥O∥]≤DGϵ2Deff. (5)

Here, is the uniform measure over , and is the operator norm of . is the degeneracy of the most degenerate energy gap, which is given by with . If the total Hamiltonian has no degenerate energy gap, . is the effective dimension of the initial state , which is defined as . If the Hamiltonian has no degeneracy, the effective dimension is equal to the inverse of the purity of the diagonal ensemble, i.e., . Inequality (5) means that observables equilibrate to their diagonal ensemble averages if is sufficiently large.

To quantify the distance between two density operators, we use the trace distance . When it is small, we hardly distinguish two states. We also use the local distance , where we write for the reduced density operator. We regard that system equilibrates to the diagonal ensemble when the local distance is small most of the time. It has been shown in Refs. Linden et al. (2009); Short and Farrelly (2012) that for any positive number ,

 limτ→∞Probt∈[0,τ][DS(ρ(t),ω)≥ϵ]≤12√D2SDGϵ2Deff. (6)

This inequality implies that system equilibrates to the diagonal ensemble if is sufficiently large compared to the Hilbert space dimension of system .

The effective dimension of the initial product state is bounded from below by the purity of the initial state of bath :

 Deff≥1DETr[ρB(0)2], (7)

where is the maximum energy degeneracy of the total Hamiltonian . The proof of inequality (7) is the following. We denote by an orthonormal basis of the degenerate eigenspace of energy . Then, we have

 1Deff =∑j(Tr[PjρS(0)⊗ρB(0)])2 (8) =∑j⎛⎜⎝∑|Ek⟩∈Kj⟨Ek|ρS(0)⊗ρB(0)|Ek⟩⎞⎟⎠2 (9) ≤∑j|Kj|∑|Ek⟩∈Kj⟨Ek|ρS(0)⊗ρB(0)|Ek⟩2 (10) ≤DE∑j∑|Ek⟩∈Kj⟨Ek|ρS(0)⊗ρB(0)|Ek⟩2 (11) ≤DETr[(ρS(0)⊗ρB(0))2] (12) ≤DETr[ρB(0)2], (13)

where we used to obtain the last line.

When the initial state of bath is the microcanonical ensemble , where is the projector onto the Hilbert space of the energy shell with (see Sec. IV.2 for more details), or the canonical ensemble , then the purity is exponentially small in the size of bath (see Appendix A). Thus is exponentially large in the size of bath . Furthermore, when the initial state of bath is randomly taken from the Hilbert space of the energy shell according to the Haar measure, typically becomes exponentially large in the size of bath (see again Appendix A). Therefore, system equilibrates to the diagonal ensemble if the initial state of bath is the microcanonical ensemble, the canonical ensemble, or a typical pure state, given that bath is sufficiently large.

Finally, we define the entropy production from time to by

 ⟨σ(t)⟩=ΔSS(t)−βQ(t), (14)

where is the change in the von Neumann entropy of system with , and is the heat absorbed by system . If the initial state of bath is the canonical ensemble , the entropy production is always non-negative:  Sagawa (2012), which is an expression of the second law of thermodynamics. This inequality results from the non-negativity of the quantum relative entropy. When the initial state of bath is not the canonical ensemble, the entropy production is not necessarily non-negative. In this paper, we discuss the behavior of the entropy production and the validity of the second law for general initial states.

Iii Saturation of the entropy production

In this section, we discuss our main topic: saturation of the entropy production under reversible unitary dynamics. In Sec. II, we discussed that system equilibrates to the diagonal ensemble if the effective dimension is sufficiently large, i.e., inequalities (5) and (6). However, the entropy production cannot be described as the expectation value of any instantaneous observable. Therefore, we cannot directly apply the argument in Sec. II to the entropy production. Instead, we evaluate the difference between the entropy production at time and the saturated entropy production by using inequalities (5) and (6).

We now formulate our theorem rigorously. We assume that the initial state of the total system is given by a product state (3). The saturated entropy production is defined as

 ⟨σ∞⟩:=S(ωS)−S(ρS(0))−βTr[HB(ρ(0)−ω)], (15)

where is defined in Eq. (4), and . If the difference is small most of the time, the entropy production saturates at . In fact, we show the following theorem.

Theorem 1 (Saturation of the entropy production).

For any ,

 limτ→∞Probt∈[0,τ][|⟨σ(t)⟩−⟨σ∞⟩| ≤ϵlog(DS−1)+H2(ϵ)+ϵβ∥HS+HI∥]

where .

Proof.

From the definitions of and , we have

 |⟨σ(t)⟩−⟨σ∞⟩| ≤|S(ρS(t))−S(ωS)|+β|Tr(HBρ(t)−ω)|. (17)

By applying the Fannes-Audenaert inequality Audenaert (2007), the first term on the right-hand side is bounded by

 |S(ρS(t))−S(ωS)| ≤DS(ρ(t),ω)log(DS−1)+H2(DS(ρ(t),ω)). (18)

Using inequality (6), we evaluate , and then we have for any ,

 limτ→∞Probt∈[0,τ][|S(ρS(t))−S(ωS)|≤ϵlog(DS−1)+H2(ϵ)]≥1−12√D2SDGϵ2Deff. (19)

Since the expectation value of the total Hamiltonian is the same for both the state and the diagonal ensemble , we have

 Tr(HB(ρ(t)−ω))=−Tr[(HS+HI)(ρ(t)−ω)]. (20)

By using inequality (5), we have for any ,

 limτ→∞Probt∈[0,τ][β|Tr(HB(ρ(t)−ω))|≤ϵβ∥HS+HI∥] ≥1−DGϵ2Deff. (21)

By combining inequality (19) with (21), we finally obtain inequality (LABEL:eq:saturation). ∎

We have made no assumption on the initial state of bath ; Theorem 1 is applicable not only to the canonical ensemble, but also to an arbitrary initial state. From Theorem 1 we find that a sufficient condition of the saturation is just the large effective dimension of the initial state. As we remarked in Sec. II, the effective dimension is exponentially large with the size of bath , when the initial state of bath is the microcanonical ensemble, the canonical ensemble, or a typical quantum state. From Theorem 1, the entropy production saturates at for these cases.

We remark that the time scale, which makes the long time limit in Theorem 1 meaningful, is expected to be doubly exponentially large with respect to the size of the total system for non-integrable systems. This is much longer than the time scale determined by the Lieb-Robinson bound. However, it has been observed numerically and analytically that the informational quantities, such as the entanglement entropy, ballistically spread and saturate at around the Lieb-Robinson time for both of integrable and non-integrable systems  Calabrese and Cardy (2005); De Chiara et al. (2006); Läuchli and Kollath (2008); Kim and Huse (2013). As a result, the entropy production, defined with the von Neumann entropy, is expected to relax to the long-time average at around the Lieb-Robinson time, as numerically observed in Ref. Iyoda et al. (2017) and also in Sec. V. Therefore, we expect that in practical situations, there is no intermediate time regime between the saturation of the entropy production and the Lieb-Robinson time. However, for the moment, it seems to be formidably difficult to prove it rigorously for general situations.

As mentioned before, it has been shown that the entropy production is always non-negative when the initial state of bath is the canonical ensemble Sagawa (2012). In this case, the entropy production increases and saturates at a non-negative value, which is consistent with the second law of thermodynamics.

On the other hand, the saturation value is not necessarily non-negative for a general initial state. In fact, is determined by the diagonal ensemble , which in general depends on the details of the initial state. For this reason, it is difficult to prove the non-negativity of for completely general situations.

If the total system satisfies the strong ETH, does not depend on the details of the initial state, but only depends on the initial average energy. We note that the strong ETH is believed to be true for non-integrable systems Rigol et al. (2008); Biroli et al. (2010); Kim et al. (2014); Beugeling et al. (2014); Garrison and Grover (2015) (see also Gogolin and Eisert (2016) for definitions of integrability in quantum systems). In such a case, the diagonal ensemble of a general initial state is the same as that of the initial canonical ensemble, and therefore takes the same value as the canonical case, where is non-negative. This is the scenario that we will prove in the next section by assuming the strong ETH.

Iv Non-negativity of the saturated entropy production

We now discuss the non-negativity of the saturated entropy . Specifically, we utilize the ETH Jensen and Shankar (1985); Rigol et al. (2008); Berry (1977); Peres (1984); Deutsch (1991); Srednicki (1994); Biroli et al. (2010); Kim et al. (2014); Beugeling et al. (2014); Garrison and Grover (2015), which is sufficient to show the non-negativity of .

In Sec. IV.1, we prove a general lower bound of without using the ETH. Then, in Sec. IV.2, we show that the lower bound approaches to zero in the large bath limit, if the system satisfies the strong ETH. In Sec. IV.3, we discuss consequences of the weak ETH.

iv.1 Lower bound of the saturation value

First, we derive a lower bound of the saturated entropy production by comparing the saturation value for a general initial state of the bath, , with that for the canonical initial state, . For this purpose, we choose the canonical ensemble such that for a given . We denote the diagonal ensemble of by , and the local trace distance between and by .

Let us first remark on a special case where the deviation vanishes as in the thermodynamic limit, and the interaction energy is negligible. In such a case, the system equilibrates to the same state as the case that the bath is initially in the canonical ensemble . In this case, the saturation value of the entropy production for a general initial state takes the same value as the canonical initial state, and therefore it is non-negative. In Sec. IV.2, we will show that the deviation indeed vanishes, , when the strong ETH holds for the total system.

For general quantum systems, the deviation does not necessarily vanish even in the thermodynamic limit. By keeping finite, and also by taking the interaction energy into consideration, we have the following lower bound of .

Theorem 2 (A lower bound of the saturation value).

The saturated entropy production satisfies

 ⟨σ∞⟩≥−δEP. (22)

Here, we defined the error term as

 δEP:=δeqlog(DS−1)+H2(δeq)+2δeqβ∥HS∥+βδEI, (23)

where is the change in the interaction energy defined as

 δEI: =|Tr[HI(ρS(0)⊗ρB(0)−ω)]
Proof.

The entropy production with the initial canonical state , corresponding to the diagonal ensemble , is given by

 ⟨σcan∞⟩ :=S(ωcanS)−S(ρS(0))−βTr[HB(ρcanB−ωcan)]. (25)

The deviation between and is

 ⟨σ∞⟩−⟨σcan∞⟩ =S(ωS)−S(ωcanS) (26)

By applying the Fannes-Audenaert inequality, the difference of the von Neumann entropies on the right-hand side of Eq. (26) is evaluated as

 (27)

The total average energy is the same for both the initial state and the diagonal ensemble. Therefore, we have

 Tr[HB(ω−ρS(0)⊗ρB(0))] =−Tr[(HS+HI)(ω−ρS(0)⊗ρB(0))], (28) Tr[HB(ωcan−ρS(0)⊗ρcanB)] =−Tr[(HS+HI)(ωcan−ρS(0)⊗ρcanB)]. (29)

From inequalities (28) and (29), we can evaluate as

 |Tr[HB(ω−ωcan)]|≤2∥HS∥δeq+δEI. (30)

From inequalities (30) and (27), we have

 |⟨σ∞⟩−⟨σcan∞⟩| ≤δeqlog(DS−1)+H2(δeq)+2β∥HS∥δeq+βδEI. (31)

By combining inequality (31) and , we prove the theorem. ∎

The first three terms on the right-hand side of Eq. (23) vanishes in the limit of . The last term on the right-hand side of Eq. (23) vanishes (i.e., ) if

 [HS+HB,HI]=0, (32)

which means that the sum of the energies of system and bath is a conserved quantity, and the interaction energy does not change during the dynamics. The same condition as Eq. (32) is also assumed in the thermodynamic resource theory Horodecki and Oppenheim (2013); Brandão et al. (2015).

We note that Eq. (32) does not necessary imply that the interaction itself is weak. For example, the Jaynes-Cummings model with the resonance condition satisfies Eq. (32) even when the interaction is strong Jaynes and Cummings (1963). Although a general quantum system does not necessarily satisfy the condition (32), we have numerically shown that the contribution from is negligibly small in some non-integrable models (see Sec. V and Ref. Iyoda et al. (2017)). In general, however, it is still unclear under what condition the interaction term is negligible without weak coupling limit, which is a future issue.

iv.2 Non-negativity from the strong ETH

We next discuss the initial state independence (i.e., ) on the basis of the strong ETH, which states that every energy eigenstate represents thermal equilibrium. There has not yet been a mathematical proof of the strong ETH, while it has been confirmed by a lot of numerical experiments Rigol et al. (2008); Biroli et al. (2010); Kim et al. (2014); Beugeling et al. (2014); Garrison and Grover (2015). In this section, we just assume the strong ETH as a starting point of our argument.

To state the strong ETH rigorously, we focus on the situation that the total system has the well-defined thermodynamic limit, when only bath becomes large and system is kept small. Let be the total system size (i.e., the number of lattice sites or particles). For simplicity, we also assume that Hamiltonian has no degeneracy (i.e., ).

The energy shell with energy density and width is defined as the following set of indexes of energy eigenvalues of the total Hamiltonian :

 MuN,Δ:={j:∣∣Ej−uN∣∣≤Δ}. (33)

Here, can be arbitrary taken within the order of with . The microcanonical ensemble of the energy shell is denoted by . We note that in rigorous statistical mechanics Ruelle (1999), the energy shell is defined as , where exists and is finite, and with . With this definition, many of the fundamental properties in standard statistical mechanics, such as Boltzmann’s formula and the equivalence of ensembles, have been rigorously proved Ruelle (1999); Tasaki (2016b). Our definition of the energy shell (33) is regarded as a special case of the above standard definition, by taking and (i.e., ).

In the present context, the strong ETH of the total system is defined as follows.

Assumption 1 (Strong ETH).

We say that the total system satisfies the strong ETH, if for any ,

 DS(|Ej⟩⟨Ej|,ρmc)<ϵfor any j∈MuN,Δ (34)

holds for sufficiently large .

From the non-degeneracy of the energy eigenvalues, the diagonal ensemble can be represented as

 ω=∑j|Ej⟩⟨Ej|ρS(0)⊗ρB(0)|Ej⟩⟨Ej|. (35)

As shown below, by using the strong ETH, we can show the initial state independence of in the thermodynamic limit. To apply the strong ETH, we need to make an additional assumption on the initial energy distribution.

Assumption 2 (Energy distribution of the initial state).

Let be the projection operator onto the subspace corresponding to the outside of the energy shell: . We assume that for any ,

 Tr[PoutρS(0)⊗ρB(0)] <ϵ (36)

holds for sufficiently large .

This assumption implies that the energy distribution of the initial state is narrow and centered around . This is a reasonable assumption if we take the width of the energy shell within the order of with . In fact, for the case of the canonical ensemble, the standard deviation of the energy distribution is the order of  Tasaki (2016b).

If the energies of the system and the interaction are small enough than the energy of the bath, the energy distribution of the total initial state is similar to that of the bath. We can thus expect that Assumption 2 would hold when the bath is initially in the microcanonical ensemble, the canonical ensemble, or a pure state in the energy shell.

We now evaluate the local trace distance between and :

Lemma 1 (Initial state independence from the strong ETH).

Under Assumptions 1 and 2, for any ,

 DS(ω,ρmc)<ϵ (37)

holds for sufficiently large .

Proof.

From the triangle inequality, we find

 (38)

where we defined the diagonal ensemble projected onto the energy shell by . From Assumption 1, for any ,

 DS(~ω,ρmc)<ϵ2 (39)

holds for sufficiently large . On the other hand, from Assumption 2, for any ,

 DS(ω,~ω)<ϵ2 (40)

holds for sufficiently large . By summing up these two terms, we obtain Eq. (37). ∎

This result means that the system reaches an equilibrium state independent of the initial state, if Assumptions 1 and 2 are satisfied. Then, by combining Lemma 1 with Theorem 2, we obtain the non-negativity of the saturation value in the thermodynamic limit.

Theorem 3 (Non-negativity from the strong ETH).

We assume that , along with Assumptions 1 and 2. For any ,

 ⟨σ∞⟩≥−ϵ (41)

holds for sufficiently large .

Proof.

In Eq. (23), vanishes from Lemma 1, and vanishes from . ∎

We note that Theorem 3 is valid as long as its assumptions are satisfied, regardless of the initial state of the bath. When commutator is not equal to , an negative error term defined in Eq. (LABEL:eq:def_dei) is just added to the right-hand side of inequality (41). However, is expected to be small for some cases, as will be numerically demonstrated in Sec. V.

We have not mentioned the size dependence of the error term in inequality (41), which generally depends on the nontrivial scaling in inequalities (34) and (36). We will numerically show in Sec. V that becomes non-negative without error, even with a bath of sites.

By combining the saturation of the entropy production (Theorem 1) with the non-negativity (Theorem 3), we find that

 ⟨σ(t)⟩≳0 (42)

holds for a typical time . This inequality represents the second law of thermodynamics at late times for general initial states. We note that our theory implies a stronger statement than just the non-negativity of the entropy production: it saturates at a non-negative value.

iv.3 Non-negativity from the weak ETH

Non-negativity of the saturated entropy production can also be shown by using the weak ETH Biroli et al. (2010); Iyoda et al. (2017); Mori (2016) instead of the strong ETH. In fact, if the initial state has a sufficiently large effective dimension, we can prove the initial-state independence of .

In the present context, we formulate the weak ETH as the following large-deviation type bound.

Assumption 3 (Weak ETH).

We say that the total system satisfies the weak ETH, if for any , there exists a constant such that

 Probj∈MuN,Δ[DS(|Ej⟩⟨Ej|,ρmc)]>ϵ]

where represents the probability in the uniform distribution on the set of the energy eigenstates in the energy shell.

This form of the weak ETH has rigorously been proved for general lattice systems with translation invariance, including integrable systems Tasaki (2016a); Mori (2016). We note that the proof in Ref. Tasaki (2016a); Mori (2016) has been presented with the choice of the energy shell with , while it is straightforward to generalize the proof to the case of our choice of the energy shell . Under the weak ETH, a sufficient condition for the initial state independence is that the initial state has an exponentially large effective dimension:

Assumption 4 (Large effective dimension).

We assume that for any , the effective dimension of the initial state satisfies

 Deff>DmcBe−γϵN~ϵ (44)

for sufficiently large .

Then, we evaluate the local trace distance between the diagonal ensemble and the microcanonical ensemble .

Lemma 2 (Initial state independence from the weak ETH).

Under Assumptions 2, 3, and 4, for any ,

 DS(ω,ρmc)<ϵ (45)

holds for sufficiently large .

Proof.

From Assumption 2, vanishes in large . By using the Schwarz inequality and Assumption 3, we have for any ,

 DS(~ω,ρmc)<ϵ+√DmcBe−γϵNDeff. (46)

Therefore, by using Assumption 4, we prove Lemma 2. ∎

Lemma 2 means that the diagonal ensemble is locally thermal if the initial state has the exponentially large effective dimension. Then we can prove the non-negativity of the saturation value in the same manner as is the case for the strong ETH.

Theorem 4 (Non-negativity from the weak ETH).

We assume that , along with Assumptions 2, 3, and 4. For any ,

 ⟨σ∞⟩≥−ϵ (47)

holds for sufficiently large .

We note that the validity of Assumption 4 is highly nontrivial in practice. On the other hand, an advantage of the argument by the weak ETH is that it is applicable to both integrable and non-integrable systems.

V Numerical simulation

To confirm the validity of our theory, we performed a numerical simulation by using exact diagonalization. As a non-integrable model, we consider a one-dimensional hard-core bosons with nearest- and next-nearest-neighbor hoppings and interactions, which satisfies the strong ETH Kim et al. (2014). System is a single site and bath consists of sites. The annihilation (creation) operator of a hard-core boson at site is written as (), which satisfies the commutation relations for , , and . The Hamiltonians are then given by

 HS =ωc†0c0, (48) HI =−γI(c†0c1+c†1c0), (49) HB =ωL∑i=1c†ici +L−1∑i=1[−γ(c†ici+1+c†i+1ci)+gc†icic†i+1ci+1] +L−2∑i=1[−γ′(c†ici+2+c†i+2ci)+g′c†icic†i+2ci+2], (50)

where is the on-site potential, is the hopping rate between system and bath , and are the hopping rates in bath , and and are the repulsion energies. We assume the open boundary condition. We note that in this model. We set the initial number of bosons in bath by . The initial state is a product state of system and bath , where the state of system is given as , and the state of bath is an energy eigenstate .

Figure 3 shows of the initial state , which implies that increases with at all energy scales. We note that an energy eigenstate is not necessarily typical in the Hilbert space of the energy shell, and the amount of is nontrivial. The numerical result in Fig. 3 suggests that fluctuations of observables around the diagonal ensemble averages are negligible, and the entropy production will indeed saturate.

Figure 4 shows the time dependence of the entropy production. We set the inverse temperature as , and choose an energy eigenstate such that its inverse temperature is closest to in the energy shell. Here, the inverse temperature of is given by the solution of . We set parameters as , and , or .

As shown in Fig. 4, the entropy production gradually increases up to , oscillates in the medium regime , and then saturates in the long time regime . This behavior is consistent with Theorem 1. The saturation value in the long time regime is positive for all the cases of . This implies that the error term in Theorem 3 is negligible in this example. The error term is also small in a two dimensional hard-core bosons with nearest-neighbor repulsion Iyoda et al. (2017).

We note that the saturation value is independent of the interaction rate, which implies that the interaction strength affects the entropy production only in the short time regime. The entropy production is also non-negative in the short time regime, which is out of the scope of the present theory, but can be explained by our previous theorem based on the Lieb-Robinson bound Iyoda et al. (2017).

Vi Conclusion

In this paper, we have discussed the second law of thermodynamics for general quantum states in the long time regime. We have proved Theorem 1, which states that the entropy production saturates for most times [inequality (LABEL:eq:saturation)]. The saturation value (15) is given by the entropy production with respect to the diagonal ensemble . We have then derived the lower bound (22) of the saturated entropy production. If the diagonal ensemble is independent of the parameters of the initial state except for the average energy, the obtained bound (22) implies the non-negativity of the saturated entropy production. This scenario can be justified by assuming the ETH. We have shown the initial state independence (Lemma 1) and the non-negativity of the saturation value (Theorem 3) under the strong ETH (Assumption 1) and an additional assumption on the initial energy (Assumption 2). We have also shown the non-negativity from the weak ETH (Theorem 4).

Our results have clarified that thermodynamic irreversibility can emerge from unitary dynamics of quantum many-body systems in the long time regime. We emphasize that, if the initial state of the bath is a pure state, the fluctuation theorem holds only in the short time regime, but breaks down in the long time regime Iyoda et al. (2017). This is because non-thermal quantum fluctuations of the initial state affect higher-order fluctuations of the entropy production in the long time regime. In contrast, as shown in the present work, the second law is robust in the long time regime, because the second law only concerns the first cumulant (i.e., the average) of the entropy production. More detailed investigation of the time scale of saturation of the entropy production is a future issue.

Acknowledgements.
We are grateful to Jordan M. Horowitz for his careful reading of the manuscript. The authors also thank Takashi Mori for valuable discussions. K.K. are supported by JSPS KAKENHI Grant Number JP17J06875. E.I. and T.S. are supported by JSPS KAKENHI Grant Number JP16H02211. E.I. is also supported by JSPS KAKENHI Grant Number JP15K20944. T.S. is also supported by JSPS KAKENHI Grant Number JP25103003.

Appendix A Lower bounds of the effective dimension

We show lower bounds of effective dimensions for the three cases that the initial state of bath is the microcanonical ensemble , the canonical ensemble , and a typical pure state in the Hilbert space of the energy shell. In these cases, the effective dimensions are exponentially large with respect to the size of bath .

a.1 Microcanonical ensemble

The purity of the microcanonical ensemble is given by

 Tr[(ρmcB)2] =1DmcB=e−NBs(u), (51)

where is the dimension of the energy shell, is the size of bath , and is the entropy density as a function of energy density . From inequality (7), we have the lower bound

 Deff≥eNBs(u)DE. (52)

Since approaches an -independent function in the large bath limit Ruelle (1999), we find that the right-hand side of inequality (52) becomes exponentially large in .

a.2 Canonical ensemble

The purity of the canonical ensemble is given by

 Tr[(ρcanB)2] =Tr[e−2βHB](Tr[e−βHB])2 (53) =exp[−2βNB(f(2β)−f(β))], (54)

where we defined the free energy density by . From inequality (7), we have the lower bound

 Deff≥exp[2βNB(f(2β)−f(β))]DE. (55)

Since approaches an -independent function in the large bath limit, we find that the right-hand side of inequality (55) becomes exponentially large in .

a.3 Typical pure state

We randomly sample a typical pure state from the Hilbert space of the energy shell of bath with respect to the Haar measure. From Theorem 2 of Ref. Linden et al. (2009), the probability that the effective dimension is smaller than is bounded from above:

 Probψ[Deff

This inequality implies that the effective dimension of a typical pure state becomes exponentially large with increasing .

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