Dynamical Landauer Principle: Quantifying Information Transmission by Thermodynamics

TL;DR

The paper establishes a quantitative dynamical equivalence between classical information transmission and energy transfer in quantum channels, extending Landauer's principle to a dynamical setting. It introduces one-shot Θ-assisted classical capacity $C_{\Theta,(1)}^\epsilon(\mathcal{N})$ and a genuinely transmitted energy measure $W_{\rm corr|\Theta,(1)}^\epsilon(\mathcal{N})$, proving a tight inequality that ties these two figures of merit up to one-shot error terms. In the asymptotic iid limit, the result becomes exact: $(k_B T\ln 2) C_\Theta(\mathcal{N}) = W_{\rm corr|\Theta}(\mathcal{N}) = W_{\Phi|\Theta}(\mathcal{N})$, providing a thermodynamic interpretation of the HSW theorem and implying a dynamical form of Landauer's principle where transmitting $n$ bits requires at least $n k_B T\ln 2$ energy. The work also shows no-go results under non-equilibrium constraints, indicating that generating informational non-equilibrium does not enhance asymptotic classical communication, and outlines rich avenues for future exploration at the intersection of thermodynamics, information theory, and quantum dynamics.

Abstract

Energy transfer and information transmission are two fundamental aspects of nature. They are seemingly unrelated, while recent findings suggest that a deep connection between them is to be discovered. This amounts to asking: Can we phrase the processes of transmitting classical bits equivalently as specific energy-transmitting tasks, thereby uncovering foundational links between them? We answer this question positively by showing that, for a broad class of classical communication tasks, a quantum dynamics' ability to transmit $n$ bits of classical information is equivalent to its ability to transmit $n$ units of energy in a thermodynamic task. This finding not only provides an analytical correspondence between information transmission and energy extraction tasks, but also quantifies classical communication by thermodynamics. Furthermore, our findings uncover the dynamical version of Landauer's principle, showing the strong link between transmitting information and energy. In the asymptotic regime, our results further provide thermodynamic meanings for the well-known Holevo-Schumacher-Westmoreland Theorem in quantum communication theory.

Figures (3)

• Figure 1: Main question. We aim to seek the foundational relations between transmitting classical information and work-like energy.
• Figure 2: Operational tasks. (a) The one-shot $\Theta$-assisted classical capacity describes a communication task where the sender encodes classical data $\{m\}_{m=0}^{M-1}$ into states $\{\rho_m\}_{m=0}^{M-1}$. After transmission via $\mathcal{N}$ assisted by some $\Pi\in\Theta$, the receiver decodes the transmitted classical data by a measurement $\{E_m\}_{m=0}^{M-1}$. (b) The one-shot $\Theta$-assisted $\epsilon$-deterministic genuinely transmitted energy measures the energy that can only be transmitted by $\mathcal{N}$. The referee prepares $\eta_{AB}$ and distributes it to the sender and receiver. The sender sends their part $A$ to the receiver via $\Pi_M(\mathcal{N})$, a classical channel induced by $\mathcal{N}$ and $\Pi\in\Theta$. Then, the receiver extracts work from the bipartite correlation. This can be done by changing Hamiltonians without changing states (i.e., quenches) to make the bipartite output locally thermal and extracting work from that bipartite state (detailed in Appendix II).
• Figure 3: Main result. The abilities to transmit information and energy are equivalent for quantum dynamics.

Theorems & Definitions (1)

• Theorem 1