Minimum work principle in the time-average equilibration model

The work-cost is therefore minimized if the energy of the final state is minimized. If

β(N) > 0, this is the case if the von Neumann entropy is minimized, because we are

dealing with Gibbs states. Since the von Neumann entropy can only increase during a ther- modynamic protocol, we therefore conclude that the work-cost is minimized in a reversible process. We thus obtain again the minimum work-principle just as in phenomenological thermodynamics as long as the inverse temperature does not change sign during the pro- cess. This is the case as long as the final energy fulfills

Trω

β(N)H(1)



≤ 1

dTr(H(1)), (10.47) whered is the Hilbert-space dimension of the total Hamiltonian H(1). In a system-bath setting with a large bath, this is fulfilled as long as we do not have a work-cost that scales extensively with size of the bath. If we fix an initial and final Hamiltonian on the system, this is true as long as the bath is large enough, since the work cost is bounded bykH(0)k + kH(1)kbut the total energy diverges with the size of the bath.

Also note that if the final temperature in the reversible process β(1)is positive, then the initial temperature also must have been positive. This is due to the fact that the temperature

β(u)is a smooth function. Thus if it would change sign in the process it would have to take the value β(u) =0 in which case the entropy would take the maximum possible value and therefore cannot be preserved.

If we consider a cyclic process, H(0) = H(1)and the initial temperature is positive, the above arguments also imply that no positive work can be extracted since the system can only "heat up" in a cycle by creating entropy. We thus obtain the second law of thermody- namics in terms of work extraction. The above discussion is not very surprising, but mostly shows that our framework is adequate and reproduces standard results in the limit where they should apply. We will now go further and discuss the minimum work principle in the case of GGEs, starting with the time-average equilibration model.

10.4.2

Minimum work principle in the time-average equilibration model

Let us again fix a smooth trajectory of HamiltoniansH(u)and assume some initial equi- librium state ωt.a.(0). We already know from Lemma 10.2 that in the quasi-static limit,

the spectrum of the density matrix ωt.a.(u)is conserved. The question is whether this also

implies that the work-cost of the process is minimized. We will now see that this is in general only the case if the final state of the quasi-static process, ωt.a.(1)is a so-called

final Hamiltonian with energiesE(1)k. The state ωt.a.(1)is diagonal in this basis. Let

ωt.a.(1)↓denote the vector with the eigenvalues of ωt.a.(1)ordered in non-increasing or-

der. Then ωt.a.(1)is called passive if its eigenvalues decrease with increasing energies:

(ωt.a.(1))↓_k ≥ (ωt.a.(1))↓_l impliesEk(1) ≤ El(1)for allk and l. Passive states have the

property that their average energies can only be increased using arbitrary unitary opera- tions [236, 237] (however, also see the recent work [238]):

Tr(ρH) ≤Tr(UρU†H), ∀U ⇔ρ is passive w.r.t. H. (10.48)

With this definition at hand we can now show the following Lemma.

Lemma 10.7 (Passive quasi-static protocols are optimal). Given a smooth trajectory of HamiltoniansH(u)and an initial state ωt.a.(0), if the final state in the quasi-static realisa-

tion of the process is passive, then the work-cost is minimized in the quasi-static realisation of the protocol.

Proof. In the quasi-static limit, the final state ωt.a.(1)is related to the initial state by a

unitary operationW: ωt.a.(1) =Wωt.a.(0)W†. This follows since their spectra are identi-

cal. Now consider any discretization of the process with final state ω(N)_t.a. and note that the time-averaging process can be seen as a mixture of unitaries. Since ωta(j+1) is obtained

from ωta(j) by time-averaging and this holds for allj, the final state ωt.a.(N)is to the initial

state and to ωt.a.(1)by a mixture of unitaries:

ω(N)_t.a. =

∑

α

pαVαωt.a.(0)Vα†=

∑

pα(VαW†)ωt.a.(1)(VαW†)†, (10.49)

whereUαare some unitary matrices. Since ωt.a.(1)is passive, the state ω

(N)

t.a. can therefore

only have higher energy than ωt.a.(1).

The Lemma establishes that the minimum work principle holds if the final state in the quasi-static realisation is passive. Given any two HamiltoniansH(0)andH(1)one can in fact always construct a smooth trajectory such that the final state is passive (for an explicit construction, I refer to Ref. [3]). Furthermore, if the spectrum of the initial state is non- degenerate all such trajectories are equivalent in the sense that they have the same work- cost, since the ordering into a passive state is unique. However, in general, such trajectories require changing the Hamiltonian over time globally. Thus, in a system-bath setting it might be impossible to find a trajectory of the local Hamiltonian of the system so that the final state is passive and in principle it can become beneficial to implement a protocol rapidly instead of in a quasi-static way to extract the most work.

In the case of cyclic processes, the above considerations show that the optimal protocol is one where the final state is passive and has the same spectrum as the initial state. We then conclude from (10.48) that we can extract work in a cyclic protocol if and only if the initial state is non-passive. Thus non-passive states can be considered "non-equilibrium" states in the framework of the time-average equilibration model, even if they are diagonal in the energy eigenbasis: Work can be extracted from them, but only once, since they end up being passive.

Lemma 10.7 also can be seen as a generalisation of previous results in Ref. [250], where the minimum work principle was studied for cyclic unitary processes. In Ref. [250], the authors showed that the minimum work principle is valid if i) the initial state is passive with respect toH(0)andii) the trajectory of Hamiltonians does not induce level crossing. This means thatEi(0) ≥ Ej(0)then alsoEi(1) ≥ Ej(1)(due to the smoothness of the

trajectory, the labelling of the basis is fixed throughout the trajectory). Now, if the initial state is passive then the absence of level-crossings is equivalent to having a final passive state. Therefore the Lemma naturally generalizes this result from Ref. [250].

Finally, let us note that in the case of the Gibbs-equilibration model, the condition that the final temperature remains positive can be seen as enforcing passivity of the final state since Gibbs states are passive if and only if they have a positive temperature. We can then formulate the results of section 10.4.1 in complete analogy to Lemma 10.7:

A Q UA N T U M O F T H E R M O DY N A M I C S 129

Lemma 10.8 (Passive quasi-static protocols are optimal in the Gibbs equilibration model). Consider a smooth trajectory of HamiltoniansH(u)and an initial state ω_β(0)(H(0)). If the final state in the quasi-static realisation of process in the Gibbs equilibration model is passive, then the quasi-static realisation is optimal.

We can thus summarize that both for the time-average and the Gibbs equilibration model, we can check whether the minimum work-principle is fulfilled by calculating the final state of the quasi-static protocol. If it is passive, then the minimum work principle holds. Let us now connect the minimum work principle to the problem of work-extraction from non-equilibrium states and then turn to GGEs with an intermediate number of con- served quantities.