An improved Landauer Principle with finite-size corrections (1306.4352v3)

Abstract: Landauer's Principle relates entropy decrease and heat dissipation during logically irreversible processes. Most theoretical justifications of Landauer's Principle either use thermodynamic reasoning or rely on specific models based on arguable assumptions. Here, we aim at a general and minimal setup to formulate Landauer's Principle in precise terms. We provide a simple and rigorous proof of an improved version of the Principle, which is formulated in terms of an equality rather than an inequality. The proof is based on quantum statistical mechanics concepts rather than on thermodynamic argumentation. From this equality version, we obtain explicit improvements of Landauer's bound that depend on the effective size of the thermal reservoir and reduce to Landauer's bound only for infinite-sized reservoirs.

• The paper rigorously reformulates Landauer's Principle within quantum statistical mechanics by incorporating correlations and free energy changes.
• The study derives finite-size corrections, demonstrating that heat dissipation surpasses the traditional Landauer bound in small reservoirs.
• The findings offer key insights for designing energy-efficient nanoscale and quantum information processing devices.

An Improved Landauer Principle with Finite-Size Corrections

This paper addresses some fundamental aspects of the thermodynamic principles governing information processing, specifically focusing on Landauer's Principle. This principle, originally proposed by Rolf Landauer, asserts that erasing information from a computational memory has an inevitable thermodynamic cost. Specifically, the erasure operation is associated with an entropy increase in the environment, leading to dissipation of heat quantified by the well-known Landauer bound: $\Delta Q \geq \Delta S/\beta$, where $\Delta S$ is the entropy decrease and $\beta$ the inverse temperature of the environment.

The paper makes two primary contributions to our understanding of this principle. Firstly, it provides a rigorous mathematical formulation of Landauer's Principle within the framework of quantum statistical mechanics. This formalization results in a sharpened version of the principle expressed as an equality that incorporates additional terms representing correlations and free energy changes in the reservoir. The derived equality is $$\beta \Delta Q = \Delta S + I(S':R') + D(\rho'_R\|\rho_R)$$, where $I(S':R')$ denotes the mutual information between the system and the reservoir after the process, and $D(\rho'_R\|\rho_R)$ is the relative entropy quantifying the change in the reservoir's free energy. This formulation underscores the thermodynamic cost associated with correlation changes and reservoir fluctuations during the erasure process.

Secondly, the authors explore finite-size corrections to Landauer's bound. In practical scenarios where the reservoir assisting in the erasure process is of finite dimensions, the authors reveal that the heat dissipation exceeds the Landauer bound, achieving values significantly higher for small systems. They present an explicit bound $\beta \Delta Q \geq \Delta S + M(\Delta S, d)$, where the function $M(\Delta S, d)$ quantifies the improvement in the bound, primarily dependent on the size $d$ of the reservoir. This result is critical in understanding thermodynamic limits in nanoscale and quantum information processing where reservoirs are not infinitely large.

The mathematical results established in this paper have far-reaching implications. The recognition that the thermodynamic cost of information erasure is influenced by correlations and finite-size effects is vital for the design and operation of future low-power computational devices. Additionally, the explicit trajectory-dependent corrections presented could inform the modeling and evaluation of energy costs in emerging quantum technologies.

In closing, while the paper provides comprehensive insights into the energetics of information processing, it also raises important questions about the nature of thermodynamic irreversibility at microscopic scales and presents potential directions for future research, including the necessity to explore one-shot or finite-sample settings as well as infinite-dimensional generalizations. This nuanced understanding of Landauer's Principle can serve as a foundational element in developing efficient algorithms and architectures for quantum information processing and nanoscale engineering in the future.