Work Extraction from a Single Energy Eigenstate

We first formulate work extraction from a quantum many-body system by a cyclic operation Pusz1978 ; Lenard1978 . We drive the system by the time-dependent Hamiltonian $H(t)$ from $t=0$ to $t=\tau$. The corresponding time evolution is represented by the unitary operator $U=\mathcal{T}\exp\left(-i\int_{0}^{\tau}H(t)dt\right)$, where $\mathcal{T}$ is the time-ordering operator. We assume that the initial and the final Hamiltonians are the same: $H(0)=H(\tau)=:H_{\mathrm{I}}$.

The average work extraction from the state $\rho$ during this protocol is

We focus on work extraction from a single energy eigenstate of $H_{\mathrm{I}}$. Since an energy eigenstate is not passive, $W$ can be positive for some unitary operations. Given that, in the following, we focus on whether we can extract positive work by quench protocols of local Hamiltonians, and numerically investigate the fundamental difference between integrable and non-integrable systems.

As a simple model that we can control integrability, we study the quantum Ising model with a transverse field $g$, longitudinal field $h$, nearest-neighbor coupling $J$, and next nearest-neighbor coupling $J^{\prime}$. The Hamiltonian is given by

where $\sigma_{i}^{x},\sigma_{i}^{z}$ are the local Pauli operators on the site $i$, and $N$ is the number of spins. We adopted the periodic boundary condition. This model is integrable when $g=0$ or $h=J^{\prime}=0$. In non-integrable cases, the level spacing statistics can be well fitted by the Wigner-Dyson distribution (see the Appendix A for $N=18,J=1,J^{\prime}=0,g=\frac{\sqrt{5}+5}{8},2.5\times 10^{-2}\lesssim h\lesssim 1.5$). This model is also known to satisfy the strong ETH in the non-integrable parameter region Kim2014 ; Dymarsky2018 .

We next describe the quench protocol in our numerical simulation. Let $H_{\mathrm{I}}$ be the initial Hamiltonian, and suppose that the system is initially prepared in its eigenstate $\ket{j}$ with energy $E_{j}$. We perform the following work extraction protocol: (i) At initial time $t=0$, the longitudinal field and transverse field are suddenly quenched ($h_{\mathrm{I}}\to h_{\mathrm{Q}},g_{\mathrm{I}}\to g_{\mathrm{Q}}$). The post-quench Hamiltonian is denoted by $H_{\mathrm{Q}}$. (ii) The system evolves from time $0$ to $\tau$ according to the new Hamiltonian $H_{\mathrm{Q}}$. (iii) At time $t=\tau$, we again quench the fields in the backward direction ($h_{\mathrm{Q}}\to h_{\mathrm{I}},g_{\mathrm{Q}}\to g_{\mathrm{I}}$) and restore the Hamiltonian to $H_{\mathrm{I}}$.

The average work extraction from eigenstate $\ket{j}$ during this protocol is given by

We then consider the following four cases of the integrability: (a) $H_{\mathrm{I}}$ is non-integrable and $H_{\mathrm{Q}}$ is non-integrable; (b) $H_{\mathrm{I}}$ is integrable and $H_{\mathrm{Q}}$ is non-integrable; (c) $H_{\mathrm{I}}$ is non-integrable and $H_{\mathrm{Q}}$ is integrable; (d) $H_{\mathrm{I}}$ is integrable and $H_{\mathrm{Q}}$ is integrable.

We note the definition of the inverse temperature of an energy eigenstate. The inverse temperature $\beta_{j}$ associated with an energy eigenstate $\ket{j}$ is defined through the equation $E_{j}=u_{\mathrm{Gibbs}}(\beta_{j})$, where $u_{\mathrm{Gibbs}}(\beta_{j})$ is the energy density of the Gibbs state at inverse temperature $\beta_{j}$. We cannot extract positive work form the Gibbs state with positive inverse temperature $\beta>0$, but can do from that with negative inverse temperature $\beta<0$.

We show our numerical result on the distribution of the extracted work over energy eigenstates. Figure 1 shows $W_{j}$ versus the eigenenergy $E_{j}$. Overall, the average work tends to be negative (positive) in the positive (negative) temperature region.

Specifically, in the case of (a), (b), and (c), where one or both of Hamiltonians $H_{\mathrm{I}}$ and $H_{\mathrm{Q}}$ are non-integrable, the distribution of the data points becomes narrower in the horizontal direction from $N=12$ to $18$. We thus expect that these data points converge to a single smooth curve in the thermodynamic limit $N\to\infty$. This is analogous to the strong ETH Jensen1985 ; Rigol2008 ; Steinigeweg2013 ; Steinigeweg2014 ; Beugeling2014 ; Kim2014 ; Khodja2015 ; Garrison2015 ; Mondaini2016 ; Dymarsky2018 ; Yoshizawa2017 . On the other hand, in the case of (d), the distribution of the data points is more spread out than the other cases and does not look like convergent to a single curve. This is analogous to the behavior of the ETH in integrable systems Rigol2008 ; Biroli2010 .

We also note that in the case of (a), (b), and (c), eigenstates with $W_{j}>0$ do not exist in the positive temperature region for $N=18$, whereas in the integrable case (d), there are work extractable eigenstates also in the positive temperature region.

To confirm the validity of the above argument more quantitatively, we systematically investigate the number of work-extractable energy eigenstates in the next subsection.

We next quantitatively study the system-size dependence of the number of energy eigenstates from which one can extract positive work. We focus on eigenstates at positive finite temperature $(\beta_{j}>0)$. In addition, we set a threshold for the inverse temperature, denoted as $\beta_{\mathrm{s}}>0$, because the finite-size effect is significant in small $\beta_{j}$. The set of energy eigenstates with $\beta_{j}\geq\beta_{\mathrm{s}}$ is denoted by $M_{\beta_{\mathrm{s}}}:=\{j:\beta_{j}\geq\beta_{\mathrm{s}}\}.$ We denote the number of energy eigenstates in $M_{\beta_{\mathrm{s}}}$ by $D_{\beta_{\mathrm{s}}}$. Let $D^{U}_{\mathrm{out}}$ be the number of eigenstates in $M_{\beta_{\mathrm{s}}}$ from which one can extract work greater than $N\delta$ (i.e., $D^{U}_{\mathrm{out}}:=\#\{j\in M_{\beta_{\mathrm{s}}}:W_{j}>N\delta\}$ ). We numerically investigate the system-size dependence of the ratio $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$.

Figure 2 shows the system-size dependence of $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ for several quench protocols. In the case of (a), (b), and (c), where one or both of Hamiltonians $H_{\mathrm{I}}$ and $H_{\mathrm{Q}}$ are non-integrable, $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ decays at least exponentially with $N$. On the other hand, $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ does not seem to be decreasing in the case of (d), where both $H_{\mathrm{I}}$ and $H_{\mathrm{Q}}$ are integrable. Especially, in the case of (a) and (b), $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ becomes exactly zero even at finite $N$, at which eigenstates with $W_{j}>0$ completely vanish. This is regarded as a stronger version of the second law at the level of individual energy eigenstates, which is analogous to the strong ETH.

Apparently, $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ does not seem to be decreasing in the case of (d), where both $H_{\mathrm{I}}$ and $H_{\mathrm{Q}}$ are integrable. However, as will be shown in inequality (8), which is applicable to this case, $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ must decrease at least exponentially. This is because the parameters in inequality (8) in our numerical setup are very small ($\beta_{\mathrm{s}}=0.02/J$ and $\delta=0.001J$), and therefore the decay rate is too small to be observed in our numerical result. We note that in the case of (d), $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}$ itself increases exponentially with $N$, because $D_{\beta_{\mathrm{s}}}$ itself increases exponentially. Therefore, $D^{U}_{\mathrm{out}}$ does not become exactly zero at finite $N$, which is analogous to the weak ETH in integrable systems Biroli2010 ; Yoshizawa2017 ; Tasaki2015 ; Mori2016 .

Table 1 summarizes whether the stronger version of the second law (i.e., $D^{U}_{\mathrm{out}}=0$ at finite $N$) holds or not in our numerical calculations. In the case of (c), whether this property holds is not very clear from Fig. 2 (c), due to the limitation of our numerical resource. However, we argue that $D^{U}_{\mathrm{out}}/D_{\beta_{\mathrm{s}}}=0$ holds at finite $N$ in this case too, from the following discussion and the numerical result in Fig. 3 (c).

We assume that Hamiltonian $H_{\mathrm{I}}$ satisfies the strong ETH for observable $H_{\mathrm{I}}(\tau):=e^{iH_{\mathrm{Q}}\tau}H_{\mathrm{I}}e^{-iH_{\mathrm{Q}}\tau}$, i.e.,

Here, $\braket{\cdots}^{\mathrm{MC}}_{E_{j}}$ represents the microcanonical average with respect to $H_{\mathrm{I}}$ around the energy $E_{j}$.

Under this assumption, the average work extracted from $\ket{j}$ is given by

Meanwhile, the average work from the Gibbs state of $H_{\mathrm{I}}$ at the mean energy $E_{j}$ is given by

where we used the strong ETH and the concentration of the energy in the Gibbs state Tasaki2018 . We thus obtain $W_{j}\simeq W_{\mathrm{Gibbs}}$, and the passivity of the Gibbs state Pusz1978 ; Lenard1978 (i.e., $W_{\mathrm{Gibbs}}\leq 0$) immediately leads to $W_{j}\lesssim 0$. Therefore, we conclude that $W_{j}\lesssim 0$ for all $j$ in the case of (a), (b), and (c), where the strong ETH holds.

We numerically confirm the strong ETH (4). We remark that in general $H_{\mathrm{I}}(\tau)$ is a non-local observable, which involves products of $\mathcal{O}(L)$ local operators. The strong ETH for such a non-local many-body observable has not been fully investigated so far.

We calculate the ETH indicator $r_{j}:=|\braket{j+1|H_{\mathrm{I}}(\tau)|j+1}-\braket{j|H_{\mathrm{I}}(\tau)|j}|/N$, which was introduced in Ref. Kim2014 . Here, $\ket{j}$ is the $j$th energy eigenstate in the increasing order of $E_{j}$ in the Hilbert space without dividing it into momentum sectors. If the strong ETH holds, all of $r_{j}$ decay exponentially with $N$. In fact, if there is an athermal eigenstate $\ket{j}$, the corresponding $r_{j}$ does not decay. However, even in such a case, the average of $r_{j}$ decays if the weak ETH holds.

Since we are interested in eigenstates at positive temperature, we remove eigenstates with $E_{j}/N>u_{\mathrm{Gibbs}}(\beta=0)$. We also remove eigenstates at the edge of the spectrum, because the density of states is too small there. Therefore, we calculated $r_{j}$ for eigenstates satisfying $E_{0}/2N\leq E_{j}/N\leq u_{\mathrm{Gibbs}}(\beta=0)$.

Figure 3 shows the system-size dependence of the average, the largest, the fifth largest, and the tenth largest values of $r_{j}$. The average of $r_{j}$ decays exponentially in all the cases, which implies the weak ETH. On the other hand, other values of $r_{j}$ decay only in the case of (a), (b), and (c), where one or both of Hamiltonians $H_{\mathrm{I}}$ and $H_{\mathrm{Q}}$ are non-integrable. These values do not decay in the case of (d) where both $H_{\mathrm{I}}$ and $H_{\mathrm{Q}}$ are integrable. Therefore, the strong ETH is valid for the case of (a), (b), and (c), which is consistent with Table 1.

We consider a quantum many-body system of $N$ spins, and denote the number of energy eigenstates of $H_{\mathrm{I}}$ satisfying $E_{j}\leq E$ by $D(E)$. We then assume that there exists an $N$-independent positive function $\sigma(u)$ such that

for any $u\in[u_{\mathrm{s}}-\delta,u_{\mathrm{s}}]$, where $u_{\mathrm{s}}:=u_{\mathrm{Gibbs}}(\beta_{\mathrm{s}})$, and $o(N)$ satisfies $o(N)/N\to 0$ in $N\to\infty$. From the Boltzmann formula Ruelle1999 , $\sigma(u)$ is nothing but the entropy density. We also assume that $\sigma^{\prime}(u_{\mathrm{s}})=\beta_{\mathrm{s}}>0$ and $\sigma^{\prime\prime}(u_{\mathrm{s}})<0$. These assumptions are indeed provable in rigorous statistical mechanics if the system is translation invariant and the interaction is local Ruelle1999 ; Tasaki2018 .

In this setup, we can show that for an arbitrary fixed unitary operator $U$ and arbitrary $\delta>0$, the inequality

holds. We show a derivation in the next subsection. Inequality (8) implies that the ratio of energy eigenstates, from which one can extract macroscopic work, decays at least exponentially with the system size. This bound holds for both integrable and non-integrable systems, as long as the aforementioned assumptions are satisfied.

We also remark a similarity between the above argument and the weak ETH, which states that the ratio of athermal eigenstates decays at least exponentially with the system size Yoshizawa2017 ; Tasaki2015 ; Mori2016 . Despite this similarity, the logic behind inequality (8) is different from that behind the weak ETH, as seen in the derivation of inequality (8).

We now derive inequality (8). We first consider a slightly different quantity from $D^{U}_{\mathrm{out}}$, denoted by $\tilde{D}^{U}_{\mathrm{out}}$, and prove a rigorous bound on $\tilde{D}^{U}_{\mathrm{out}}$. Then, inequality (8) is approximately derived as a corollary.

First of all, we define the stochastic work Esposito2009 ; Campisi2011 ; Sagawa2012 . We perform projective measurements of $H_{\mathrm{I}}$ before and after the operation $U$, which give outcomes $E_{j}$ and $E_{k}$, respectively. The stochastic work is defined as the difference between the measured energies: $w_{j,k}:=E_{j}-E_{k}$. If the initial density operator is $\rho$, the joint probability of observing such outcomes is $p(j,k):=|\braket{k|U|j}|^{2}\braket{j|\rho|j}$.

For a given positive constant $\delta$, the probability that we extract work larger than $N\delta$ is given by $P(w\geq N\delta):=\sum_{j,k}\theta(w_{j,k}-N\delta)p(j,k)$, where $\theta(\cdot)$ is the step function, i.e., $\theta(x)=0\ (x\leq 0)$ and $\ \theta(x)=1\ (x>0)$. Here, $P(w\geq N\delta)$ is the probability of extracting macroscopic work, as $N\delta$ is proportional to the system size. For an energy eigenstate, the probability of extracting macroscopic work is $P_{j}(w\geq N\delta)=\sum_{k}\theta(E_{j}-E_{k}-N\delta)|\braket{k|U|j}|^{2}$. If $P_{j}(w\geq N\delta)$ is sufficiently small for a small $\delta$, we cannot extract macroscopic work from the energy eigenstate $\ket{j}$.

We consider the number of energy eigenstates in $M_{\beta_{\mathrm{s}}}$, from which we can extract macroscopic work by a fixed operation $U$. We focus on the energy eigenstate $\ket{j}$ with which $P_{j}(w\geq N\delta)$ decays exponentially with $N$. In other words, its energy distribution after the protocol shows the large deviation behavior Touchette2009 , and the probability of observing lower energy than the initial state is negligibly small on a macroscopic scale. Then, we denote by $\tilde{D}^{U}_{\mathrm{out}}$ the number of energy eigenstates whose $P_{j}(w\geq N\delta)$ is greater than $e^{-N\beta_{\mathrm{s}}\delta/2}$:

We note that $e^{-N\beta_{\mathrm{s}}\delta/2}$ is negligibly small for large $N$, which corresponds to the second law as work extraction.

In this setup, we can rigorously prove the following inequality:

The proof of this inequality is as follows. From the definition of $\tilde{D}^{U}_{\mathrm{out}}$, we have

We can evaluate the sum in the right-hand side by using the definition of $P_{j}(w\geq N\delta)$ as

where we used $\sum_{j\in M_{\beta_{\mathrm{s}}}}|\braket{E_{k}|U|E_{j}}|^{2}\leq 1$ to obtain the fourth line. From the assumption (7), we obtain

Meanwhile, $\sigma^{\prime}(u_{\mathrm{s}})=\beta_{\mathrm{s}}$ and $\sigma^{\prime\prime}(u_{\mathrm{s}})<0$ imply $\sigma(u_{\mathrm{s}})>\sigma(u_{\mathrm{s}}-\delta)+\beta\delta$. Therefore, we have

By combining this inequality and inequality (15), we have

which implies inequality (10).

The definition of $\tilde{D}^{U}_{\mathrm{out}}$ is slightly different from $D^{U}_{\mathrm{out}}$. For the latter, we can only obtain an approximate inequality (8) as follows. If $P_{j}(w\geq N\delta)<e^{-N\beta_{\mathrm{s}}\delta/2}$, the average work extraction from $\ket{j}$ is bounded as

From this inequality, we have $\tilde{D}^{U}_{\mathrm{out}}\gtrsim D^{U}_{\mathrm{out}}$. Therefore, we obtain

From inequality (8), we can show that a positive amount of work cannot be extracted from any pure state if it has sufficiently large effective dimensions. We consider a general pure state in the Hilbert space of positive temperature region, i.e., $\ket{\psi}=\sum_{j\in M_{\beta_{\mathrm{s}}}}c_{j}\ket{j}$ with $\sum_{j\in M_{\beta_{\mathrm{s}}}}|c_{j}|^{2}=1$. We apply the general work extraction protocol in Sec. IV.2 to $\ket{\psi}$.

The probability of extracting macroscopic work from $\ket{\psi}$ is $P(w\geq N\delta)=\sum_{j\in M_{\beta_{\mathrm{s}}}}|c_{j}|^{2}P_{j}(w\geq N\delta)$. The Cauchy-Schwartz inequality gives

where we used inequality (10) to obtain the third line and defined the effective dimension $D_{\mathrm{eff}}:=1/\sum_{j\in M_{\beta_{\mathrm{s}}}}|c_{j}|^{4}$ Reimann2008 . From this inequality, the average work extraction from $\ket{\psi}$ is bounded as

where $w_{\mathrm{max}}$ is the maximum value of the stochastic work. $w_{\mathrm{max}}$ scales at most linearly with $N$.

Therefore, $W\lesssim N\delta$ holds for any unitary operation, if the initial state has sufficiently large effective dimensions: $D_{\mathrm{eff}}\gg D_{\beta_{\mathrm{s}}}e^{-N\beta_{\mathrm{s}}\delta/2}$. Because the effective dimension of a typical pure state with respect to the Haar measure is estimated as $D_{\mathrm{eff}}\simeq D_{\beta_{\mathrm{s}}}\gg D_{\beta_{\mathrm{s}}}e^{-N\beta_{\mathrm{s}}\delta/2}$ Linden2009 , we cannot extract macroscopic work from a typical pure state.