Thermodynamic Constraints on Information Transmission in Quantum Ensembles

TL;DR

The paper investigates how finite thermodynamic resources constrain the encoding, transmission, and retrieval of classical information in quantum ensembles. By formulating a framework with Gibbs states as free resources and unitary encodings, it derives fundamental rank-based limits, proving a no-go for encoding classical information into non-orthogonal pure-state ensembles and establishing a thermodynamic mixture theorem for linearly dependent ensembles. An optimal encoding/decoding protocol based on controlled unitaries is shown to maximize the distinguishability of the ensemble, with the success probability equaling the maximal correlation $C_{\max}$; the Holevo information is interpreted thermodynamically as $\chi(\Sigma)=\beta Q- D(\tilde{\rho}_S||\gamma_{\beta})$, yielding $\chi(\Sigma)\le \beta Q$ and linking information to heat. Together, these results provide a tight, second-law–like bound on information processing under realistic thermodynamic constraints and offer a principled guide for designing thermodynamically feasible quantum communication protocols.

Abstract

The processing of quantum information is limited by fundamental physical constraints on how information can be encoded, transmitted, and extracted. In particular, the non-orthogonality of quantum states limits their distinguishability, and thermodynamic constraints, including the energetic cost of state preparation and quantum operations, further restrict the viability of realistic information protocols. This work explores the impact of such constraints on the preparation, evolution, and readout of quantum information. We demonstrate that preparing the system for encoding and measurement affects the distinguishability and purity of the resulting ensemble of states. Furthermore, we analyze a noisy communication channel and propose an optimal protocol for encoding and decoding the transmitted information. For this realistic protocol, we show that the maximum probability of successfully retrieving the information is equal to the maximum correlation that can be achieved between the system and the register. The protocol uses only Gibbs states as free resources, ensuring minimal thermodynamic cost. Based on this, we provide a thermodynamic interpretation of the Holevo information, which quantifies the capacity of the transmitted information and establishes a fundamental limit on its retrieval in thermodynamically constrained scenarios.

Figures (2)

• Figure 1: Thermodynamic Coding/Decoding Protocol: Thermal states denoted by $\gamma_{\space\raisebox{-1.0pt}{\tiny{$\beta$}}}$ are freely accessible through interaction with a thermal bath at inverse temperature $\beta$ and Hamiltonian $H$. The sender prepares a register state given by $\rho_{\space\raisebox{-1pt}{\tiny{$R$}}}$ by applying unitary operations $U_R$ to a Gibbs state $\gamma_{\space\raisebox{-1.0pt}{\tiny{$\beta$}}}$. The information $X$, encoded in $\rho_{\space\raisebox{-1pt}{\tiny{$R$}}}$, is transmitted to the receiver via a unitary interaction $U$ among the register and a specified number of copies of the Gibbs state $\gamma_{\space\raisebox{-1.0pt}{\tiny{$\beta$}}}$. The receiver attempts to decode the information by analyzing the statistical properties of the received states $\rho_x$. The measurement process is inherently non-ideal, as the measurement pointer is also initialized in a Gibbs state $\gamma_{\space\raisebox{-1.0pt}{\tiny{$\beta$}}}$. Heat is generated in proportion to the amount of encoded and decoded information throughout the encoding and decoding process.
• Figure 2: Communication protocols: The sketch shows a quantum communication protocol that is repeated $N$ times. In this protocol, the sender prepares a register state by acting on a random unitary $U_{\space\raisebox{-1pt}{\tiny{$R$}}}$. Then, in each $i-th$ round, the sender encodes the information into the transmitted state through a controlled operation $U=\sum_{x_i}|{x_i}\rangle\!\langle{x_i}|\otimes U_{x_i}$, where $U_{x_i}$ indicates the action over the transmitted system. The receiver then measures its apparatus and records the result on a $n$-dimensional classical tape, $c_n$. Fig. $2a)$ represents a perfect communication protocol, in which the sender can prepare the register and the transmitted system at pure state $\left|\right.\!{0}\!\left.\right\rangle$. The receiver can perform a perfect measurement. Fig. $2b)$ represents the noisy version of the protocol. In this version, the sender prepares the register and the transmitted system, while the receiver prepares the measurement apparatus at a thermal state, $\gamma_{\space\raisebox{-1.0pt}{\tiny{$\beta$}}}$, in contact with a thermal bath at an inverse temperature, $\beta$.

Theorems & Definitions (11)

• Lemma 1: Ensemble Rank Relation
• Theorem 1: No-go pure states ensemble
• proof
• Theorem 2: Thermodynamic Mixture Theorem for Linearly Dependent States
• proof
• Remark 1: Orthogonal pure states
• Theorem 3: Optimal ensemble via Controlled Operations
• Lemma 1: Ensemble Rank Relation
• proof
• Theorem 3: Optimal ensemble via Controlled Operations