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Dynamical Landauer principle: Thermodynamic criteria of transmitting classical information

Chung-Yun Hsieh

TL;DR

The paper establishes a quantitative link between transmitting information and transmitting energy in quantum channels by developing a one-shot framework that bounds classical communication capacities with energy-transmission tasks. It introduces entropic bounds based on $D_0^\epsilon$ and $D_h^\omega$, and defines energy-transmission figures $W_{corr|\Theta,(1)}^{\omega}$ and $W_{\Phi|\Theta,(1)}^{\epsilon}$ that bound the one-shot capacity via $C_{\Theta,(1)}^\epsilon$. A dynamical version of Landauer's principle is derived, showing that transmitting $n$ bits necessitates at least $n\times k_B T\ln 2$ of work-like energy in the one-shot scenario, with asymptotic results recovering the HSW theorem and revealing no-go results for leveraging informational non-equilibrium to boost asymptotic communication. The framework further interprets thermodynamic aspects of channel capacities and establishes rigorous bounds linking information processing and energy exchange, offering a thermodynamic perspective on classical communication limits.

Abstract

Transmitting energy and information are two essential aspects of nature. Recent findings suggest they are closely related, while a quantitative equivalence between them is still unknown. This thus motivates us to ask: Can information transmission tasks equal certain energy transmission tasks? We answer this question positively by bounding various one-shot classical capacities via different energy transmission tasks. Such bounds provide the physical implication that, in the one-shot regime, transmitting $n$ bits of classical information is equivalent to $n\times k_BT\ln2$ transmitted energy. Unexpectedly, these bounds further uncover a dynamical version of Landauer's principle, showing the strong link between "transmitting" (rather than "erasing") information and energy. Finally, in the asymptotic regime, our findings further provide thermodynamic meanings for Holevo-Schumacher-Westmoreland Theorem and a series of strong converse properties as well as no-go theorems.

Dynamical Landauer principle: Thermodynamic criteria of transmitting classical information

TL;DR

The paper establishes a quantitative link between transmitting information and transmitting energy in quantum channels by developing a one-shot framework that bounds classical communication capacities with energy-transmission tasks. It introduces entropic bounds based on and , and defines energy-transmission figures and that bound the one-shot capacity via . A dynamical version of Landauer's principle is derived, showing that transmitting bits necessitates at least of work-like energy in the one-shot scenario, with asymptotic results recovering the HSW theorem and revealing no-go results for leveraging informational non-equilibrium to boost asymptotic communication. The framework further interprets thermodynamic aspects of channel capacities and establishes rigorous bounds linking information processing and energy exchange, offering a thermodynamic perspective on classical communication limits.

Abstract

Transmitting energy and information are two essential aspects of nature. Recent findings suggest they are closely related, while a quantitative equivalence between them is still unknown. This thus motivates us to ask: Can information transmission tasks equal certain energy transmission tasks? We answer this question positively by bounding various one-shot classical capacities via different energy transmission tasks. Such bounds provide the physical implication that, in the one-shot regime, transmitting bits of classical information is equivalent to transmitted energy. Unexpectedly, these bounds further uncover a dynamical version of Landauer's principle, showing the strong link between "transmitting" (rather than "erasing") information and energy. Finally, in the asymptotic regime, our findings further provide thermodynamic meanings for Holevo-Schumacher-Westmoreland Theorem and a series of strong converse properties as well as no-go theorems.

Paper Structure

This paper contains 18 sections, 13 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Classical Communication via Quantum Channels
  4. Thermodynamics and Informational Non-equilibrium
  5. Entropic Bounds on Classical Capacities
  6. Bounding One-Shot $\Theta$-Assisted Classical Capacities
  7. Proof of Theorem \ref{['AppThm:CC']}
  8. Energy Transmission via Channels
  9. Å berg's $\epsilon$-Deterministic Work Extraction
  10. Work Extraction from Correlation
  11. Energy Transmission Tasks
  12. Thermodynamic Bounds on Classical Capacities
  13. Bounding One-Shot $\Theta$-Assisted Classical Capacities by Energy Transmission Tasks
  14. Dynamical Landauer's Principle
  15. Asymptotic Limit and Holevo-Schumacher-Westmoreland Theorem
  16. ...and 3 more sections

Key Result

Theorem 1

(Bounding One-Shot $\Theta$-Assisted Classical Capacity)For a set of superchannels $\Theta$, a channel $\mathcal{N}$, and errors $0<\delta\le\omega<\epsilon\le{\color{black}1/2}$, we have that where $S'$ is some finite-dimensional auxiliary system which is not necessarily of the same size with $S_{\rm in|\Pi}$, $\eta_{S_{\rm in|\Pi}S'}$ is a separable state in $S_{\rm in|\Pi}S'$ that can be writt

Figures (3)

  • Figure 1: Summary of this work. This paper is structured as follows. Section \ref{['App:Proof-MainResult']} contains preliminary notions, including the framework of classical communication tasks. Section \ref{['Sec:Entropic bound on CC']} provides entropic bounds on one-shot classical capacities (Theorems \ref{['AppThm:MainResult']} and \ref{['AppThm:CC']}). Section \ref{['Sec:energy transmission']} details the energy transmission tasks. Section \ref{['Sec:TCTCI']} contains results bridging information and energy transmission. Section \ref{['App:Proof']} provides thermodynamic bounds on one-shot classical capacities (Theorem \ref{['Result:TCTCI']}). Section \ref{['Sec:Landauer']} uncovers the dynamical version of Landauer's principle (Corollaries \ref{['coro:weak dynamical Landauer']} and \ref{['coro:strong dynamical Landauer']}). Section \ref{['App:AsymptoticLimit']} provides the thermodynamic meaning of the HSW Theorem (Proposition \ref{['AppResult:HSWThermo']}). Section \ref{['Sec:no-go']} reports strong converse properties and no-go results (Corollaries \ref{['coro:strong converse property chi']} and \ref{['coro:no-go']}). Section \ref{['Sec:Conclusion']} concludes the paper.
  • Figure 2: Two equivalent formulations of classical communication.(a) The task corresponding to Definition \ref{['AppDef:CCapacity']}. The sender encodes the classical index $m$ into the state $\rho_m$. After sending it via $\Pi(\mathcal{N})$ for some $\Pi\in\Theta$, the receiver decodes it by the measurement $\{E_{m'}\}_{m'=0}^{M-1}$. The communication is successful if $m'=m$. (b) The task corresponding to Fact \ref{['AppLemma:AlternativeCCapacity']} (which is equivalent to Definition \ref{['AppDef:CCapacity']}). The sender encodes $m$ into a pre-fixed computational basis $|m\rangle$. Then, they send it via a classical version of $\mathcal{N}$ assisted by $\Theta$ as given in Definition \ref{['Def:classical_ver']}; namely, the classical-to-classical channel $\Pi_M(\mathcal{N})$. Finally, the receiver decodes the information by applying the projective measurement $\{|m'\rangle\langle m'|\}_{m'=0}^{M-1}$.
  • Figure 3: Tasks for extracting and transmitting work-like energy.(a) The task corresponding to Eq. \ref{['AppEq:App:Proof-Com']}, which aims to extract work from the correlation of a given state $\rho_{AB}$. This can be done by first quenching the local Hamiltonians to make the state locally thermal (i.e., making both $\rho_A$ and $\rho_B$ the thermal states of the local systems $A$ and $B$) and then extracting work from the bipartite state. (b) The task corresponding to Eq. \ref{['Eq: one-shot transmitted energy']}, which aims to measure the work-like energy that can only due to transmission via the channel $\mathcal{N}$ (with the assistance of $\Pi_M\in\Theta_M$). As argued in the text, the extracted work from the bipartite output's correlation can only result from transmission by $\Pi_M(\mathcal{N})$.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • proof
  • proof
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Definition 4
  • Theorem 3
  • ...and 17 more