Functional Information in Quantum Darwinism: An Operational Measure of Objectivity

TL;DR

The paper tackles how classical objectivity emerges from quantum dynamics by proposing Functional Information in Quantum Darwinism (FI_QD), an operational yardstick based on the Holevo bound that counts how many environment fragments individually encode nearly all pointer information. Using a heterogeneous pure-dephasing model, it demonstrates a universal onset–plateau structure: redundancy grows rapidly at early times and then saturates at a capacity-limited ceiling set by environment size and fragment capacity. The approach links redundancy growth to thermodynamic costs via Landauer bounds, showing that each additional bit of FI_QD imposes exponentially larger minimal heat and power requirements. By combining overlap-aware sampling, isotonic regression, and bootstrap uncertainty, the work provides a conservative, model-agnostic framework to quantify emergent classicality and to guide experimental probes of quantum Darwinism.

Abstract

This paper investigates the emergence of classical objectivity in quantum systems through the measure of Functional Information in Quantum Darwinism ($FI_{QD}$). The goal is to quantify objectivity as the abundance of environment fragments that independently contain sufficient information about a system's pointer states. The method relies on the Holevo quantity -- an upper bound on accessible information -- and introduces a tolerance criterion called $δ$-adequacy, where fragments are considered adequate if they retain at least $(1-δ)H_S$ bits of pointer information. Numerical simulations of a dephasing model with fragment sampling reveal three robust features: (i) an early-time regime where $\log R_δ(t)$ grows approximately linearly, (ii) capacity-limited plateaus determined by fragment size and environment dimension, and (iii) stability of the onset criterion under different sampling strategies and overlap corrections. These results establish $FI_{QD}$ as a practical and conservative yardstick for operational objectivity. Beyond numerical findings, the analysis links redundancy growth to thermodynamic costs of record formation and interprets $FI_{QD}$ as a resource monotone under noisy dynamics. The study suggests that classical objectivity emerges not as an assumption but as a quantifiable, resource-limited abundance of redundant records.

Figures (17)

• Figure 1: Bloch sphere representation of qubit states. A pure state $|\psi\rangle$ lies on the surface, while a mixed state $\rho$ is represented by a shorter vector inside the sphere.
• Figure 2: Mutual information as overlap. The intersection of $A$ and $B$ represents shared correlations, quantified by $I(A\!:\!B)$.
• Figure 3: Holevo bound in a classical--quantum pipeline. A classical variable $X$ is encoded into an ensemble $\{p_x,\rho_x\}$, measured by a POVM to yield $Y$. The accessible information $I(X\!:\!Y)$ is bounded by the Holevo quantity $\chi$, which depends only on the ensemble.
• Figure 4: System--environment partition. The total Hilbert space factorizes into system and environment, with the interaction $H_{SE}$ making the system open.
• Figure 5: Markovian vs. Non-Markovian. Memoryless baths carry information away; structured/finite baths can feed information back to $S$.