Dr. Uttam Singh presents a primer on the resource theory of classical Thermodynamics along with some of the challenges it poses.
Classical thermodynamics is a remarkably successful macroscopic phenomenological theory that has profoundly transformed our everyday lives since its inception—through innovations such as heat engines, refrigerators, power plants, and in the understanding of the biological processes. The laws of thermodynamics have stood the test of time within their domain of applicability (thermodynamic limit): namely, in systems composed of a large number of particles and large volumes, where their ratio remains finite. For a long time, the technologies leveraging thermodynamics comfortably operated within this well-understood regime. However, with the rapid pace of technological breakthroughs and our growing ability to manipulate microscopic systems, we now encounter scenarios where the traditional thermodynamic limit breaks down. In these exciting frontiers, understanding energy exchanges in small quantum systems—and the intrinsic randomness that comes with them—demands a new kind of thermodynamic framework, one that reaches beyond the boundaries of classical thermodynamics1. In response to this challenge, recent years have witnessed a surge of groundbreaking efforts to build such a framework—ushering in the vibrant and rapidly evolving field now known as quantum thermodynamics.
Quantum thermodynamics is still a theory in the making, and several competing approaches are currently being explored2. Among them, one particularly compelling framework is the resource theory of quantum thermodynamics. This approach is built on a simple yet powerful idea: any quantum state from which no work can be extracted is considered useless or free, and any operation that only maps such useless states to other useless states is itself deemed free3. These free operations—referred to as thermal operations—set the boundaries for what state transformations are allowed, and these boundaries effectively become the laws of quantum thermodynamics. To illustrate, classical thermodynamics tells us that the free energy of a system can never increase—this is one formulation of the second law. But in the quantum realm, things get more intriguing: instead of a single second law, quantum thermodynamics demands the simultaneous decrease of an entire family of free energies, revealing a far richer and more nuanced structure4.
While the resource theory of thermodynamics successfully lays down the foundational laws of quantum thermodynamics and provides a solid framework for defining the amount of work that can be extracted from quantum systems, it also comes with a few subtle challenges. Addressing these challenges promises not only to deepen and refine our understanding of quantum thermodynamics, but also to pave the way for practical, miniature thermal machines—a thrilling possibility in light of the rapid technological advances unfolding in the quantum realm. In what follows, we discuss three noteworthy such subtleties.
Absence of heat bath
Thermal operations on a quantum state are defined through a three-step process:
1. preparing an infinite-capacity heat bath,
2. applying a joint energy-preserving unitary operation on the system and the bath, and
3. discarding (or partially tracing out) the bath. While elegant in theory, this framework hinges on an idealization—an infinite-capacity heat bath—that does not exist in practice, since every quantum system is fundamentally out of equilibrium. To overcome this limitation, we have taken important first steps by developing a thermodynamic framework that operates without relying on heat baths5. Though much work remains to fully realize a complete theory for such scenarios, these developments mark a promising direction in building a more practical foundation for quantum thermodynamics.
Unknown input state
Preparing a desired quantum state is inherently a noisy process—after all, achieving perfect isolation from the surrounding environment is virtually impossible in any real experimental setup. This unavoidable noise contaminates quantum state preparation. To make matters even more challenging, quantum states cannot be verified without performing measurements, and those very measurements destroy the state. As a result, the input quantum state is almost always partially unknown. This leads to a fundamental question: how can we assess the usefulness of a quantum state when we do not fully know what it is? More specifically, can we extract any work from an unknown quantum state?
Unknown input state
Preparing a desired quantum state is inherently a noisy process—after all, achieving perfect isolation from the surrounding environment is virtually impossible in any real experimental setup. This unavoidable noise contaminates quantum state preparation. To make matters even more challenging, quantum states cannot be verified without performing measurements, and those very measurements destroy the state. As a result, the input quantum state is almost always partially unknown. This leads to a fundamental question: how can we assess the usefulness of a quantum state when we do not fully know what it is? More specifically, can we extract any work from an unknown quantum state?
Efficient thermal operations
The energy-preserving unitaries central to thermal operations may not always be efficient to implement—that is, the number of one- and two-qubit gates required to realize them might grow faster than any polynomial in the number of input qubits. To build a computational theory of quantum thermodynamics, it is crucial to define what constitutes efficient thermal operations. This is a challenging question, but answering it could unlock a computational understanding of extractable work and even inspire quantum cryptographic primitives grounded in the hardness of thermodynamic transformations. We at CQST (https://cqst.iiit.ac.in/) are actively exploring these directions and have obtained some promising initial results, though a complete picture of these notions remains an open and exciting frontier.
This article was initially published in the March ’25 edition of TechForward Dispatch
References
1. J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom heat engine, Science 352, 325 (2016).
2. F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, eds., Thermodynamics in the Quantum Regime, Fundamental Theories of Physics (Springer, Cham, Switzerland, 2018).
3. M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics, Nat.Commun. 4, 2059 (2013).
4. F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. U.S.A. 112, 3275 (2015).
5. Swati, U. Singh, and G. Chiribella, A resource theory of activity for quantum thermodynamics in the absence of heat baths, arXiv:2304.08926.
6. Shantanav Chakraborty, Siddhartha Das, Arnab Ghorui, Soumyabrata Hazra, Uttam Singh, Sample Complexity of Black Box Work Extraction, arXiv:2412.02673.
May 2025