Fundamental Work Cost of Quantum Processes

Information-theoretic approaches provide a promising avenue for extending the laws of thermodynamics to the nanoscale. Here, we provide a general fundamental lower limit, valid for systems with an arbitrary Hamiltonian and in contact with any thermodynamic bath, on the work cost for the implementation of any logical process. This limit is given by a new information measure—the coherent relative entropy—which accounts for the Gibbs weight of each microstate. The coherent relative entropy enjoys a collection of natural properties justifying its interpretation as a measure of information and can be understood as a generalization of a quantum relative entropy difference. As an application, we show that the standard first and second laws of thermodynamics emerge from our microscopic picture in the macroscopic limit. Finally, our results have an impact on understanding the role of the observer in thermodynamics: Our approach may be applied at any level of knowledge—for instance, at the microscopic, mesoscopic, or macroscopic scales—thus providing a formulation of thermodynamics that is inherently relative to the observer. We obtain a precise criterion for when the laws of thermodynamics can be applied, thus making a step forward in determining the exact extent of the universality of thermodynamics and enabling a systematic treatment of Maxwell-demon-like situations.

Popular Summary

It is something every laptop owner experiences from time to time. Certain tasks, such as rendering graphics, cause the laptop to heat up and its battery to discharge rapidly. There is a general thermodynamic principle behind this behavior: Processing of information consumes energy from a “work reservoir” (e.g., a battery), which is dissipated as heat to the environment. While laptop electronics design can minimize this work consumption, researchers would like to know if there is a fundamental limit to how much it can be reduced. We have found that there is a limit that applies not only to computing but also to information processing in our brains, the replication of DNA, and even scenarios that can be described only within quantum mechanics.

The limit is derived using modern tools of quantum information theory. In particular, generalized entropy measures allow us to quantify the worst-case amount of work expended at the microscopic level. Remarkably, the limit is subjective. An observer who has additional prior knowledge about the input state can design a better implementation that exploits this knowledge to reduce the work expenditure. In other words, our results may be interpreted as a microscopic formulation of thermodynamics that is inherently relative to the observer. This subjectivity is similar to the theory of relativity, where the fundamental laws of physics describe notions such as time that are, in fact, dependent on the observer.

While we have found a fundamental limit, a lot of work needs to be done to actually reach this limit in practice. Current computing devices are still orders of magnitude away.

Figure 1

Implementation of a logical process $\mathcal{E}$ (any quantum process) using thermodynamic operations. The process acts on $X$ and has output on $X^{\prime }$, and is implemented by acting on the system and the battery with a joint Gibbs-preserving operation. The battery starts with a depletion state ${\lambda }_1$ and finishes with a depletion state ${\lambda }_2$. The overall extracted work is given by the difference ${\lambda }_1-{\lambda }_2$.

Figure 2

Macroscopic thermodynamics emerges from our framework when singling out a set of states that can be parametrized by continuous parameters to a good approximation and can be reversibly interconverted into one another. We consider the case of a textbook thermodynamic gas confined in a box, with a piston capable of furnishing work. In this setting, we recover the usual second law of thermodynamics, $dS\ge \delta Q/T$, relating the change in entropy, the dissipated heat, and the temperature.

Figure 3

The Maxwell demon concentrates all particles on one side of the box by opening the trap door at appropriate times. (a) A macroscopic observer describing only the gas sees its entropy decrease, in apparent violation of the macroscopic observer’s idea of the second law of thermodynamics. (b) The demon observes no entropy change, as the state of the gas is conditioned on his knowledge. By modeling his memory as an explicit system, originally in a pure state, we may understand his actions as simply correlating his memory with the state of the gas. In doing so, a macroscopic observer may be induced into witnessing a violation of a macroscopic second law. If the demon wishes to operate cyclically, he needs to reset his memory register back to a pure state, which costs work according to Landauer’s principle [2, 3]; any work he might have extracted using his scheme is paid back at this point.

Figure 4

Observers in thermodynamics. Alice has access to microscopic degrees of freedom of a gas, while Bob can only observe its coarse macroscopic properties, such as its temperature $T$, volume $V$, and pressure $p$. Alice describes the evolution of the gas using Gibbs-preserving maps, with a Gibbs state ${\mathrm{\Gamma }}_A^{\mathcal{A}}$ on the full state space of the many particles of the gas. On the other hand, Bob describes the gas using his own knowledge—for instance, the macroscopic variables $T$, $V$, $p$—which, in full generality, we can represent as a quantum state in a state space ${\mathcal{H}}_B$, which is obtained by applying a given mapping ${\mathcal{F}}_{A\to B}^{\mathcal{A}\to \mathcal{B}}(·)$ on Alice’s state. (For instance, this map may trace out the inaccessible microscopic information.) States of the gas described by Bob may be transformed to Alice’s picture by applying a suitable recovery map, such as the Petz map [77, 112, 113, 114, 115, 116]. Then, Alice’s ${\mathrm{\Gamma }}_A^{\mathcal{A}}$-preserving maps appear to Bob as ${\mathrm{\Gamma }}_B^{\mathcal{B}}$-preserving maps, where Bob’s ${\mathrm{\Gamma }}_B^{\mathcal{B}}$ operator is taken to be ${\mathrm{\Gamma }}_B^{\mathcal{B}}={\mathcal{F}}_{A\to B}^{\mathcal{A}\to \mathcal{B}}({\mathrm{\Gamma }}_A^{\mathcal{A}})$. Conversely, operations that preserve ${\mathrm{\Gamma }}_B^{\mathcal{B}}$ for Bob may be described by Alice as preserving ${\mathrm{\Gamma }}_A^{\mathcal{A}}$.