The result in the previous section shows that to bring a system into a non-equilibrating state, it is necessary to have at our disposal a system which can withstand equilibration to a sufficient degree. It is natural to ask, whether in the process of bringing the systemQ out of equilibrium, the system R is automatically brought into equilibrium, i.e., whether the resilience onR is spent in the process. We can thus ask whether a relation of the form
∆RQ≤ −∆RR (8.24)
holds true. In this case, we would obtain a "second law of equilibration", since the resilience to equilibrate could never be created but only re-shuffled. In general, this re-shuffling would
diminish the resilience, so that after sufficiently long time, all systems would have small resilience and equilibrate. Let us first discuss a case in which this relation holds. Suppose that after the application of the mapΛ the systems Q and R are uncorrelated, i.e.,
ρQR=ρQ⊗ρR (8.25)
and that ωH˜QR =ωH˜Q⊗ωH˜R. We can now use the following property of the resilience.
Lemma 8.3 (Super-additivity on product-states). For any bipartite, non-interacting system we have
R(ρQ⊗ρR, ˜HQR) ≥ R(ρQ, ˜HQ) + R(ρR, ˜HR). (8.26)
Proof. A direct calculation using the formulation of ωH˜QRas time-averages yields
ωH˜QR◦ωH˜Q⊗ωH˜R =ωH˜Q⊗ωH˜R◦ωHQR. (8.27)
But clearly, ωH˜QR◦ωH˜Q⊗ωH˜R = ωH˜Q⊗ωH˜R. Using the data-processing inequality
and the fact that maximally mixed states are product-states, we then get
R(ρQ⊗ρR, ˜HQR) =D2 ωH˜QR(ρQ⊗ρR)k1/dQR ≥D2 ωH˜Q⊗ωH˜R◦ωH˜QR(ρQ⊗ρR)k1/dQR =D2 ωH˜QR◦ωH˜Q⊗ωH˜R(ρQ⊗ρR)k1/dQR =D2 ωH˜Q⊗ωH˜R(ρQ⊗ρR)k1/dQR = R(ρQ, ˜HQ) + R(ρR, ˜HR).
Using this Lemma we now get
R(σQ, HQ) + R(σR, HR) = R(σQR, HQR)
≥ R(ρQ⊗ρR, ˜HQR)
≥ R(ρQ, ˜HQ) + R(ρR, ˜HR). (8.28)
Hence, (8.24) is fulfilled. However, the assumption that ρQRis uncorrelated is in general
unjustified. Indeed, we will now see that (8.24) can be violated in an extreme way even if the correlations are arbitrarily small in terms of the mutual information.
Theorem 8.4 (No second law of equilibration). Consider a family ofn-partite many-body systems with increasing n. For large enough n, there are stationary states σ(n)Q and initial and final non-interacting HamiltoniansHQR = H˜QRsuch that for any e>0 there
exists a corresponding stationary resource state σR(n)and a mixture of unitariesΛ(n), such that
(1) The resulting trajectory ρQ(t)onQ does not equilibrate.
(2) The state of the resource remains unchanged upon application of the channel:
ρ(n)R :=TrQ(Λ(n)(σQ(n)⊗σR(n))) =σR(n).
(3) The change in resilience of the systems∆R(n)Q diverges withn→∞.
(4) The correlation betweenR and Q as measured by the mutual information remain ar- bitrarily small:
A Q UA N T U M O F T H E R M O DY N A M I C S 93
Let us discuss the implication of this theorem. Put into words, it states that we can find arbitrary large many-body systemsQ and R and operations on RQ with the following properties: i)Q is initially stationary, ii) Q finally does not equilibrate, iii) the state on R does not changein the process, i.e., in the words of chapter 2 it is a catalyst, iv) arbitrary little correlations between Q and R are established (as measured by the mutual informa- tion). We thus conclude that it is, in principle, not necessary to spend the resilience inR in order to bringQ to a state that does not equilibrate.
Now suppose that Alice initially hasM copies QjofQ which are initially uncorrelated.
Then she can repeat this procedure: According to the theorem, she can use a single system R to bring all of the systems Qj out of equilibrium by first applying the channelR to R
andQ1, then toR and Q2etc. In other words, it is, in principle, possible to bring arbitrary
many systems out of equilibrium without spending any resilience to equilibrate.
How is this result compatible with theorem 8.1? After all it seems that the total change of resilience of all theM systems Qjis arbitrarily large. However, this is not true. In the
process of bringing all the systemsQjout of equilibrium, these systems become correlated
with each other. Furthermore due to the fact that the systems Qj are copies of the same
system, the total Hamiltonian HQ1···QM is non-interacting and has huge degeneracies in
the spectrum. This results in the fact that time-averaging all the copiesQ1. . . QMcreates
a large amount of correlations and we have
ωHQ1···QM 6=ωHQ1⊗ωHQ2⊗ · · · ⊗ωHQM. (8.30)
This in turn implies that
∆RQ1···QM 6=
∑
j∆RQj =M∆RQ1. (8.31)
Put in different words: It makes a huge difference if one tries to bring out of equilibrium many non-interacting systems, considered as one large system, which is easy in the sense of theorem 8.4, or to bring out of equilibrium one large, interacting system, which is extremely difficult as shown by theorem 8.1.
It is important to note that theorem 8.4 is an "in principle" result, since we allowed for arbitrary channelsΛ that can be applied with arbitrary precision. Considering the results in chapter 7, it seems plausible that essentially any small perturbance or imperfection in the implementation of the operationΛ would result in a state on Q (or the Qj) that equilibrates. Similarly, if there would be any, even arbitrarily small, interaction between the systemsQj
it would not be possible to bring any of the M systems out of equilibrium with a single copy ofR. Nevertheless, it shows that a "second law of equilibration" in the sense of (8.24) cannot be proven in general. Importantly, theorem 8.4 is independent of the measure that is used to quantify the resilience to equilibration. Thus, the violation of the "second law of equilibration" is not an artefact of the choice of equilibration measure.
Proof of theorem 8.4. The proof of theorem 8.4 rests on a result by Markus P. Müller, which states that a state σ can be brought to a state ρ with the help of a catalyst that can become correlated to the system if and only if the von Neumann entropy increases:S(ρ) ≥
S(σ)(see chapter 5 for a thorough discussion of catalysts that can be come correlated with
a system). In the form that is relevant for the purpose here, we can state it in the following way.
Theorem 8.5 (von Neumann entropy characterizes correlated catalytic transitions [164]). For any two finite-dimensional states σQand ρQ of same dimension and withS(σQ) ≤
S(ρQ), any δI>0 and any e>0, there exists a finite-dimensional state σRand a mixture
of unitariesΛ such that
(a) The channel produces the state ρQonQ to accuracy e:
ρQ−TrR(Λ(σQ⊗σR)) 1<e,
(b) The state onR afterΛ coincides with the state in which it was originally given:
σR=TrQ(Λ(σQ⊗σR)) =: ρR,
(c) The mutual information betweenR and Q afterΛ has acted is upper bounded by δI: D1(Λ(σQ⊗σR)kρQ⊗ρR) ≤δI.
The Hilbert-space dimension ofR may in general depend on both e and δI.
Note that in the theorem we can choose the eigenbasis of σRas we wish. In particular, if
we pick some arbitrary HamiltonianH(n)R onR, we can always choose the state σRto be di-
agonal in the energy basis. In the following construction, the initial and final Hamiltonians coincide, but they can also be chosen differently.
Consider a sequence ofn-partite systems with Hamiltonians HQ(n)of dimensiondnand the sequence of states
ρ(n)Q :=a|ΨihΨ|Ψ+ (1−a) Π
dn−2. (8.32)
Here,Ψ = √1
2(|E1i + |E2i) with |E1iand |E2itwo arbitrary energy eigenstates of
HQ(n) andΠ the projector onto the subspace orthogonal to |E1i and |E2i. These states
clearly do not equilibrate, since there are persistent Rabi-oscillations with amplitudea and frequencyE2−E1. Furthermore, the entropy of these states is given by
Sρ(n)Q =aS(|ΨihΨ|) + (1−a)S Π d−2 +H2(a) = (1−a)n log(d) + (1−a)log(1−2d−n) +H2(a) ≈ (1−a)n log(d) +H2(a), (8.33)
whereH2(a) = −a log(a) − (1−a)log(1−a)denotes the binary entropy and the last
equation holds with an error exponentially small in the system size. Thus the entropy diverges with the system size. The effective dimension approaches a constant, on the other hand: deff ρ(n)Q , HQ(n) = 1 a2+(1−a)2 dn−2 ≤ 1 a2. (8.34)
The states ρ(n)Q will be the final states on the systemQ. Let us now construct a sensible class of initial states. From theorem 8.5, we see that it is sufficient to have any states σQ(n) that are stationary and fulfill
• Sufficiently small entropy:S(σQ(n)) <S(ρ(n)Q ),
• Diverging effective dimension:deff(σ (n) Q , H
(n)
Q )diverges with the system size.
To achieve this, we can simply consider any micro-canonical state with a microcanonical window of dimensiondγnfor some γ < a. Such a state has both entropy and effective
dimension given by γnlog(d). The constant γ depends on the effective temperature of the micro-canonical state, but as long as γ<a, both conditions are fulfilled. Another possible choice is given by stationary states with a finite correlation length with entropy-density smaller than a log(d), since it has been proven that states with finite correlation length have an effective dimension that diverges with the system-size (see section 7.5.2).
To complete this section, let us also discuss the behaviour of the resilience of R as a function of the correlations. We will see that the resilience ofR has to diverge as e→0. To understand this, suppose that this was not true. Then there would exist constantsC(n) such that
R(σR(n), HR(n)) ≤C(n) for all e≥0. (8.35)
Let us now fix a system-size and drop then-labels everywhere for notational simplicity. Then we can use one system σRto bringm identical systems Qj to a final state ρQ1···Qm
that does not equilibrate. Using theorem 8.1 we then have
A Q UA N T U M O F T H E R M O DY N A M I C S 95
Now, all the systemsQjhave pair-wise correlations that are bounded by e in terms of mu-
tual information, since the correlations have been established throughR and R is correlated only by an amount e with each of the systems. Thus I(Qi: Qj) ≤e. We can now take the
limit e→0. Since the mutual information vanishes only for product-states we then obtain lim
e→0
ρQ1···Qm =ρQ1⊗ · · · ⊗ρQm. (8.37)
We can now use the super-additivity of the resilience on product-states to obtain
C≥m R(ρQ1, HQ1) − R(σQ1, HQ1) . (8.38)
Sincem is arbitrary we obtain a contradiction.