## Quantum Thermodynamics

In the last few years, quantum information theoretic ideas and methods have been used to study the foundations of thermodynamics in a new light, and to extend its laws to very small systems, into regimes where the thermodynamic limit does not yet apply [10,11,4].

One major idea is to reformulate thermodynamics as a resource theory: instead of analyzing how systems evolve in time (e.g. how gas molecules dynamically spread in a box), this approach views the laws of thermodynamics as consequences of an agent’s incomplete knowledge about (and control over) the physical system. For example, if we have a particle in a box, and we know whether a measurement would discover it in the left or the right half, we can insert a piston and extract work (as in a Szilard engine). On the other hand, if we don’t know, then trying to extract work will randomly lead to gain or loss.

Thus, the information that the agent has about the physical system plays a crucial role – but so do additional restrictions that arise from the laws of physics: for example, energy conservation and microscopic reversibility limit the actions that the agent may implement to control the system. We can formalize this setup within the mathematical framework of a *resource theory*, and use it to give rigorous answers to questions like the following: *Given any system in some quantum state, how much work can we reliably extract from it? What state transitions can we enforce? How can we derive the standard textbook “laws of thermodynamics” in the limit of large particle numbers, and how do they deviate for small or strongly correlated systems? What is the role of correlations or coherence in thermodynamics?*

It has been demonstrated that the laws of thermodynamics at the nanoscale differ significantly from those in the thermodynamic limit [10,11], but our group has shown [1,2,3] that this is surprisingly *not* the case if one allows correlations to build up between the thermal machine and its working medium: in this case, we recover the standard textbook rule that Helmholtz free energy uniquely determines the possibility of state transitions and the amount of reliably extractable work. Thus, small thermal machines may use clever “correlation engineering” to enhance their efficiency and to reduce quantum fluctuations, and they can consume stochastic independence as a resource [3]. We have examined and extended the corresponding mathematical results in the context of majorization theory, leading to new characterizations of entropy [2]. This may have independent applications, for example in entanglement theory where majorization plays a crucial role in the context of LOCC operations.

In current and future work, one of our goals is to improve our understanding of quantum coherence in this framework, and to compare it to other information-theoretic approaches based on Kolmogorov complexity [12].

A complementary approach, also to a large extent motivated by quantum information, has been to take a dynamical point of view, and to ask: how do closed quantum systems thermalize? That is, how do “large” quantum systems satisfy the predictions of thermodynamics, despite the fact that they evolve unitarily in time?

An exciting insight in the field has been that *entanglement* may play a crucial role here: if a quantum system is in a pure, but entangled state, then this means that it can be in a mixed state on a subsystem – and this mixed state can potentially resemble a statistical ensemble [8,9]. This is impossible in classical physics, where maximal knowledge of the full system always implies maximal knowledge of all its parts.

This raises several questions: first, under what conditions do quantum systems *equilibrate *for (most) long times? For which systems is the equilibrium state independent from the initial state, and when is it actually identically to a thermal state, as predicted by textbook thermodynamics? In [5], we have shown (based on earlier mathematical physics results) that unitary evolution on a large class of translation-invariant quantum system and initial states does in fact lead to thermal states (i.e. Gibbs states) on small subsystems. In another work [6], we have proven that integrability does not play as important a role as it was widely believed before: there are non-integrable systems that nevertheless do not thermalize, in the sense that they keep some memory on their initial state.

In ongoing work, we are exploring the Eigenstate Thermalization Hypothesis [13,14], a major open problem in the field (in a nutshell, it claims that individual eigenstates on many-body systems resemble, under some weak conditions, thermal states on subsystems). For example, we are currently exploring this hypothesis and its generalizations in translation-invariant fermionic models.

[1] M. P. Müller, *Correlating thermal machines and the second law at the nanoscale*, arXiv:1707.03451

[2] M. P. Müller and M. Pastena, *A generalization of majorization that characterizes Shannon entropy*, IEEE Trans. Inf. Th. **62**(4), 1711-1720 (2016). arXiv:1507.06900

[3]. M. Lostaglio, M. P. Müller, and M. Pastena, *Stochastic independence as a resource in small-scale thermodynamics**,* Phys. Rev. Lett. **115**, 150402 (2015). arXiv:1409.3258

[4] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N Yunger Halpern, *The resource theory of informational nonequilibrium in thermodynamics*, Phys. Rep. **583**, 1—58 (2015). arXiv:1309.6586

[5] M. P. Müller, E. Adlam, Ll. Masanes, and N. Wiebe, *Thermalization and canonical typicality in translation-invariant quantum lattice systems*, Commun. Math. Phys. **340**(2), 499—561 (2015). arXiv:1312.7420

[6] C. Gogolin, M. P. Müller, and J. Eisert, *Absence of thermalization in non-integrable systems*, Phys. Rev. Lett. **106**, 063001 (2011). arXiv:1009.2493

[7] M. P. Müller, D. Gross, and J. Eisert, *Concentration of measure for quantum states with a fixed expectation value*, Commun. Math. Phys. **303**(3), 785—824 (2011). arXiv:1003.4982

**Further reading:**

[8] S. Popescu, A. J. Short, and A. Winter, *Entanglement and the foundations of statistical mechanics*, Nat. Phys. **2**, 754 (2006).

[9] P. Reimann, *Foundation of Statistical Mechanics under Experimentally Realistic Conditions*, Phys. Rev. Lett. **101**, 190403 (2008). arXiv:0810.3092

[10] M. Horodecki and J. Oppenheim, *Fundamental limitations for quantum and nanoscale thermodynamics*, Nat. Comm. **4**, 2059 (2013). arXiv:1111.3834

[11] F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, *The second laws of quantum thermodynamics*, Proc. Natl. Acad. Sci. USA **112**(11), 3275—3279 (2015). arXiv:1305.5278

[12] S. Wolf, *An All-Or-Nothing Flavor to the Church-Turing Hypothesis*, arXiv:1702.00923.

[13] J. M. Deutsch, *Quantum statistical mechanics in a closed system*, Phys. Rev. A **43**, 2046 (1991).

[14] M. Srednicki, Chaos and Quantum Thermalization, Phys. Rev. E 50, 888 (1994). arXiv:cond-mat/9403051.

[15] M. Lostaglio, D. Jennings, and T. Rudolph, *Description of quantum coherence in thermodynamic processes requires constraints beyond free energy*, Nat. Comm.** 6**, 6383 (2015). arXiv:1405.2188