# What distinguishes quantum thermodynamics from quantum statistical mechanics?

Yoram Alhassid asked the question at the end of my Yale Quantum Institute colloquium last February. I knew two facts about Yoram: (1) He belongs to Yale’s theoretical-physics faculty. (2) His PhD thesis’s title—“On the Information Theoretic Approach to Nuclear Reactions”—ranks among my three favorites.1

Over the past few months, I’ve grown to know Yoram better. He had reason to ask about quantum statistical mechanics, because his research stands up to its ears in the field. If forced to synopsize quantum statistical mechanics in five words, I’d say, “study of many-particle quantum systems.” Examples include gases of ultracold atoms. If given another five words, I’d add, “Calculate and use partition functions.” A partition function is a measure of the number of states, or configurations, accessible to the system. Calculate a system’s partition function, and you can calculate the system’s average energy, the average number of particles in the system, how the system responds to magnetic fields, etc.

My colloquium concerned quantum thermodynamics, which I’ve blogged about many times. So I should have been able to distinguish quantum thermodynamics from its neighbors. But the answer I gave Yoram didn’t satisfy me. I mulled over the exchange for a few weeks, then emailed Yoram a 502-word essay. The exercise grew my appreciation for the question and my understanding of my field.

An adaptation of the email appears below. The adaptation should suit readers who’ve majored in physics, but don’t worry if you haven’t. Bits of what distinguishes quantum thermodynamics from quantum statistical mechanics should come across to everyone—as should, I hope, the value of question-and-answer sessions:

One distinction is a return to the operational approach of 19th-century thermodynamics. Thermodynamicists such as Sadi Carnot wanted to know how effectively engines could operate. Their practical questions led to fundamental insights, such as the Carnot bound on an engine’s efficiency. Similarly, quantum thermodynamicists often ask, “How can this state serve as a resource in thermodynamic tasks?” This approach helps us identify what distinguishes quantum theory from classical mechanics.

For example, quantum thermodynamicists found an advantage in charging batteries via nonlocal operations. Another example is the “MBL-mobile” that I designed with collaborators. Many-body localization (MBL), we found, can enhance an engine’s reliability and scalability.

Asking, “How can this state serve as a resource?” leads quantum thermodynamicists to design quantum engines, ratchets, batteries, etc. We analyze how these devices can outperform classical analogues, identifying which aspects of quantum theory power the outperformance. This question and these tasks contrast with the questions and tasks of many non-quantum-thermodynamicists who use statistical mechanics. They often calculate response functions and (e.g., ground-state) properties of Hamiltonians.

These goals of characterizing what nonclassicality is and what it can achieve in thermodynamic contexts resemble upshots of quantum computing and cryptography. As a 21st-century quantum information scientist, I understand what makes quantum theory quantum partially by understanding which problems quantum computers can solve efficiently and classical computers can’t. Similarly, I understand what makes quantum theory quantum partially by understanding how much more work you can extract from a singlet $\frac{1}{ \sqrt{2} } ( | 0 1 \rangle - |1 0 \rangle )$ (a maximally entangled state of two qubits) than from a product state in which the reduced states have the same forms as in the singlet, $\frac{1}{2} ( | 0 \rangle \langle 0 | + | 1 \rangle \langle 1 | )$.

As quantum thermodynamics shares its operational approach with quantum information theory, quantum thermodynamicists use mathematical tools developed in quantum information theory. An example consists of generalized entropies. Entropies quantify the optimal efficiency with which we can perform information-processing and thermodynamic tasks, such as data compression and work extraction.

Most statistical-mechanics researchers use just the Shannon and von Neumann entropies, $H_{\rm Sh}$ and $H_{\rm vN}$, and perhaps the occasional relative entropy. These entropies quantify optimal efficiencies in large-system limits, e.g., as the number of messages compressed approaches infinity and in the thermodynamic limit.

Other entropic quantities have been defined and explored over the past two decades, in quantum and classical information theory. These entropies quantify the optimal efficiencies with which tasks can be performed (i) if the number of systems processed or the number of trials is arbitrary, (ii) if the systems processed share correlations, (iii) in the presence of “quantum side information” (if the system being used as a resource is entangled with another system, to which an agent has access), or (iv) if you can tolerate some probability $\varepsilon$ that you fail to accomplish your task. Instead of limiting ourselves to $H_{\rm Sh}$ and $H_{\rm vN}$, we use also “$\varepsilon$-smoothed entropies,” Rényi divergences, hypothesis-testing entropies, conditional entropies, etc.

Another hallmark of quantum thermodynamics is results’ generality and simplicity. Thermodynamics characterizes a system with a few macroscopic observables, such as temperature, volume, and particle number. The simplicity of some quantum thermodynamics served a chemist collaborator and me, as explained in the introduction of https://arxiv.org/abs/1811.06551.

Yoram’s question reminded me of one reason why, as an undergrad, I adored studying physics in a liberal-arts college. I ate dinner and took walks with students majoring in economics, German studies, and Middle Eastern languages. They described their challenges, which I analyzed with the physics mindset that I was acquiring. We then compared our approaches. Encountering other disciplines’ perspectives helped me recognize what tools I was developing as a budding physicist. How can we know our corner of the world without stepping outside it and viewing it as part of a landscape?

1The title epitomizes clarity and simplicity. And I have trouble resisting anything advertised as “the information-theoretic approach to such-and-such.”

This entry was posted in Real science, Reflections, The expert's corner, Theoretical highlights by Nicole Yunger Halpern. Bookmark the permalink.

I'm a theoretical physicist and an ITAMP Postdoctoral Fellow at the Harvard-Smithsonian Institute for Theoretical Atomic, Molecular, and Optical Physics (ITAMP). Catch me at ITAMP, Harvard physics, or MIT. Before moving here, I completed a physics PhD at Caltech's Institute for Quantum Information and Matter, under John Preskill's auspices. I write one article per month for Quantum Frontiers. My research consists of what I call "quantum steampunk" (https://quantumfrontiers.com/2018/07/29/so-long-and-thanks-for-all-the-fourier-transforms/): I re-envision 19th-century thermodynamics with 21st-century quantum information theory, and I use the combination as a new lens through which to view various fields of science.

## 7 thoughts on “What distinguishes quantum thermodynamics from quantum statistical mechanics?”

1. Yes- this explanation does show the value of studying in a liberal arts community like the one I was in while doing my Masters in Quality Systems (MSQS) at udallas.edu in the 1970s. I have been endeavoring to apply quantum mechanics to sociology and named it societal mechanics. Since 2011 my efforts have not been taken up or confirmed by anyone else as a viable field of research and study. This article is very interesting from my point of view to understand how many more links are there to our understanding any field of inquiry. Thanks for sharing your insights and interests.

2. Very nice post as always. One small request. Please change the font used for the main text. Its very thin and its very hard to make out the letters from the background sometimes. Thanks in advance!

3. Great post Nicole. The question is valuable to ask even if you drop the word ‘quantum’ from it – since people often conflate thermodynamics with statistical mechanics. Statistical mechanics is basically a hidden variable theory (atoms) for thermodynamics (phenomenological / operational). Boltzmann was strongly attacked by defenders of ‘pure thermodynamics’ on this very distinction.

• Agree with David Jennings that this is a fine post on an wonderfully interesting topic … for which, thank you Nicole!

Nicole’s essay was followed two days later by a fresh version (v3) of Philipp Strasberg’s arXiv preprint “An operational approach to quantum stochastic thermodynamics” (arXiv:1810.00698v3).

Read together, Nicole’s and Philip’s writings suggest multiple student-friendly opportunities for research. For example, advances in humanity’s collective understanding of thermodynamics commonly are stimulated by the investigation of new classes of dynamical systems … dynamical systems that satisfy thermodynamic postulates (at least putatively).

In my own weekly survey of the arXiv’s ‘quant-ph’ preprints, it appears that about 1/3 of all ‘quant-ph’ preprints are concerned with novel thermodynamical behaviors, as physically instantiated on physical systems as diverse as (e.g.) trapped ions, sonic black holes, Josephson junctions, NV centers, and as mathematically instantiated on dynamical state-spaces as diverse as synthetic dimensions, synthetic gauge fields, topological field theories, and tensor networks (etc).

Michael Nielsen has celebrated this 21st century flowering of physical and mathematical thermodiversity in an on-line essay “The varieties of material existence” (September 19, 2018). To adapt a phrase of Feynman (from the year 1965) we are all of us fortunate— young researchers especially — that “the age in which we live is the one in which we are discovering inventing the fundamental synthetic laws of nature.”

Thank you, again, for yet another thoughtful and timely essay, Nicole!

4. Great post. In addition to the physics I enjoyed meeting here Yoram Elhassid. We were both undergraduates at the Hebrew University of Jerusalem in roughly the same time. I also liked very much the last paragraph about studying physics in a liberal-arts college.