Relative thermalization

Lídia del Rio, Adrian Hutter, Renato Renner, and Stephanie Wehner

Phys. Rev. E 94, 022104 – Published 3 August 2016

Abstract

Locally thermal quantum systems may contradict traditional thermodynamics: heat can flow from a cold body to a hotter one, if the two are highly entangled. We show that to recover thermodynamic laws, we must use a stronger notion of thermalization: a system $S$ is thermal relative to a reference $R$ if $S$ is both locally thermal and uncorrelated with $R$ . Considering a general quantum reference is particularly relevant for a thermodynamic treatment of nanoscale quantum systems. We derive a technical condition for relative thermalization in terms of conditional entropies. Established results on local thermalization, which implicitly assume a classical reference, follow as special cases.

Received 14 November 2015
Revised 4 March 2016

DOI:https://doi.org/10.1103/PhysRevE.94.022104

Physics Subject Headings (PhySH)

Entropy Thermodynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

Lídia del Rio^1,2,*, Adrian Hutter^2,3,4, Renato Renner², and Stephanie Wehner^4,5

¹School of Physics, University of Bristol, BS8 1TL Bristol, United Kingdom
²Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
³Department of Physics, University of Basel, 4056 Basel, Switzerland
⁴Centre for Quantum Technologies, National University of Singapore, 117543 Singapore
⁵QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands

^*lidia.delrio@bristol.ac.uk

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Vol. 94, Iss. 2 — August 2016

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Figure 1
Anomalous heat flow. If two thermal bodies are put in contact, heat normally flows from the hotter body to the colder one. However, it could be that the two systems are correlated, while still presenting local thermal states. If those correlations are strong enough (for instance if they are highly entangled), heat may flow from the colder to the hotter body. There is no contradiction with the second law, if one formulates it in terms of relative thermalization, because the two bodies are not thermal relative to each other.
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Figure 2
Local and relative thermalization. The same system $S$ may be thermalized relative to a reference $R^{'}$ but not another, $R$ . For instance, imagine that $R$ and $R^{'}$ are the memories of two observers who measured $S$ . In the case of state $ρ_{S R R^{'}} = (\sum_{i = 1}^{|Ω|} \frac{1}{|Ω|} {T r}_{E} {| i 〉 〈 i |}_{Ω} \otimes {| i 〉 〈 i |}_{R}) \otimes {| 0 〉 〈 0 |}_{R^{'}}, S$ is locally thermalized, with $ρ_{S} = π_{S}$ . Here, the observer with memory $R^{'}$ measured a few macroscopic parameters of $S \otimes E$ , enough only to determine the subspace $Ω$ ; the reduced state of $S \otimes R^{'}$ is precisely $π_{S} \otimes ρ_{R^{'}}$ . A second observer, with more precise measurement instruments, may determine the exact state $| i 〉$ of $S \otimes E$ , and write it down on the memory $R$ . Although $S$ is locally thermalized, $S$ and $R$ are classically correlated. In a more critical example, suppose that the global state is $ρ_{S R R^{'}} = {T r}_{E} {| Ψ 〉 〈 Ψ |}_{Ω R} \otimes ρ_{R^{'}}$ , where $| ψ 〉$ is entangled between $Ω$ and $R, {| Ψ 〉}_{Ω R} = {|Ω|}^{- \frac{1}{2}} \sum_{i = 1}^{|Ω|} {| i 〉}_{Ω} \otimes {| i 〉}_{R}$ . Here again $S$ is locally thermalized, and also thermalized towards $R^{'}$ , but it is entangled with $R$ . This difference has actual physical consequences: in that limit, a joint evolution of $S$ and $R^{'}$ will likely increase the entropy of the reference $R^{'}$ , because $π_{S}$ is a very mixed state. On the other hand, no global evolution of $S$ and $R$ can increase the entanglement between the two, and therefore the entropy of $R$ |not the typical effect of a thermal bath. In other words, $S$ acts as a source of random noise towards $R^{'}$ , but not towards $R$ .
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Figure 3
Setting and nature of our results. A system $S \otimes E$ is subject to a physical constraint $Ω$ . This system may be correlated with a reference $R$ , in an arbitrary global state $ρ_{Ω R}$ . We let $ρ_{Ω R}$ evolve under a unitary $U_{Ω}$ acting on $Ω$ , and then ask if $S$ is (approximately) thermalized relative to the reference $R$ (a). We derive an entropic condition on $ρ_{Ω R}$ and $Ω$ , (1), that guarantees that most unitaries, according to the Haar measure, lead to relative thermalization. This means that, under that condition, if all we know about $U_{Ω}$ is that it is a unitary in $Ω$ , it is highly likely, from our point of view, that $U_{Ω}$ will thermalize $S$ relative to $R$ . Usually, though, we know more about $U_{Ω}$ , for instance, that it is induced by a given local Hamiltonian. As the set of all unitaries in $Ω$ is full of operators that are unrelated to our physical setting (like nonlocal evolutions, ruled out by our knowledge), it is desirable to obtain similar probabilistic statements about smaller sets that still contain $U_{Ω}$ , like those generated by local interactions (b). This is possible, because the decoupling approach [12, 13, 14, 15] used to obtain our results is very general, and can be applied to more physical sets of unitaries, consisting of local two-body interactions [16, 17, 18], or time-independent Hamiltonians [3, 4, 16, 18, 19, 20]. See the Methods Summary for further discussion.
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Figure 4
Application to spin systems. Consider a system of $N$ weakly interacting spins, subject to the Hamiltonian $\hat{H} = {\hat{H}}_{0} + \hat{V}$ , where ${\hat{H}}_{0} = J \sum_{i} {| ↑ 〉 〈 ↑ |}_{i}$ and $\hat{V}$ is a random nearest-neighbor perturbation that conserves the total spin (with $| \hat{V} | ≪ | {\hat{H}}_{0} |$ ); this system is also studied in the preprint version of Ref. [1]. We select $α N$ of those spins to be our subsystem $S$ , while the remaining $(1 - α) N$ spins are called the environment $E$ . In addition, the spins of $S \otimes E$ may be correlated with a reference spin system $R$ . We want to study thermalization of $S$ relative to $R$ , for an arbitrary initial state $ρ_{S E R}$ . Note that the energy subspaces of $S \otimes E$ are invariant under time evolution ruled by $\hat{H}$ ; therefore we will look at states that lie in one of these invariant subspaces. For mixtures and superpositions over different subspaces, the results follow by linearity. Each energy shell ${Ω_{k}}$ is generated by states with a fixed number $k$ of spins up, $Ω_{k} = span {{| Ψ 〉}_{S E} : {\hat{H}}_{0} | Ψ 〉 = k J | Ψ 〉}$ . The initial state is $ρ_{S E R}$ , with $ρ_{S E} \in E n d (Ω_{k})$ for some $k$ . We apply the weaker condition for relative thermalization, $H^{ɛ} {(Ω | R)}_{ρ} > {log}_{2} | S | - {log}_{2} | Ω |$ . The dimension of $S$ is $2^{α N}$ , while $| Ω_{k} | = (\begin{matrix} N \\ k \end{matrix})$ . For large $N, {log}_{2} (\begin{matrix} N \\ k \end{matrix}) \approx N h (k / N)$ , where $h (p) = - p {log}_{2} p - (1 - p) {log}_{2} (1 - p)$ is the binary entropy of $p$ . We obtain the following result: if $H^{ɛ} {(Ω | R)}_{ρ} > N [2 α - h (k / N)]$ , then $S$ will be $δ$ -thermalized relative to $R$ after most evolutions. Conversely, if $H^{ɛ} {(Ω | R)}_{ρ} < - H^{1} {(E)}_{π}$ , then no evolution in $Ω_{k}$ leads to relative thermalization of $S$ (Theorem t2; here $H^{1} {(E)}_{π} = {log}_{2} [S u p p [H y p e r g e o m e t r i c D i s t r i b u t i o n [α N, k N, N]]]$ ).
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Figure 5
Proof steps for Lemma t24. We start from $H_{min}^{ɛ} {(A | B)}_{ρ}$ and bound it successively until we end up with the generalized smooth entropy for the same state. Along the way we need to extend our state $ρ$ to a larger Hilbert space (below).
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