Page Curve of average subsystem entropy

TL;DR

The paper analyzes the Page curve, describing how the average entropy of a subsystem in a bipartite pure state behaves as the subsystem size varies, rising to a maximum near half the total system and then decreasing. It develops a framework for averaging over quantum states using a unitarily invariant measure, connects subsystem entropy to entanglement via the Schmidt decomposition, and derives concrete expressions for average purity (Lubkin's formula) that explain why local entropy is typically high even though the global state remains pure. The work further discusses the relevance to black hole information paradox, showing how unitary evolution yields information preservation through correlations, and provides an entanglement-based route to thermodynamics, where reduced states resemble Gibbs states. Overall, the approach links randomness in quantum states, entanglement structure, and thermodynamic behavior, offering a coherent picture of how typical quantum subsystems behave and why high local entropy coexists with global purity.

Abstract

The Page curve is a curve of average subsystem entropy as a function of subsystem size. The curve starts from 0, rises, peaks near the maximal possible entropy when the subsystem makes up half of the total system, and then falls back to 0. We here describe subsystem entropy, averaging over quantum states, and why the curve rises and falls in that manner. We also discuss the connection between the curve and the black hole information paradox.

Figures (2)

• Figure 1: Page curve. The entropy of a subsystem, averaged over the uniform distribution over pure quantum states, rises with the subsystem size, peaks when the subsystem is half of the total subsystem (50/100 subsystems in this case) and then falls off in a symmetric manner down to 0. The Page Curve contrasts with the behaviour of the entropy curve of black hole radiation derived semiclassically by Hawking and others. The latter curve increases monotonically as the radiation increases, suggesting a black hole evaporation leaves behind high entropy radiation rather than a pure state radiation field.
• Figure 2: Non-uniform density on Bloch sphere. Toy picture of how non-uniform density of pure states changes under unitary evolution. The circle represents the Bloch sphere real vector representation of a qubit as seen from above the z-axis. The shading indicates density. The unitary acts as a rotation, altering the overall density. In the case depicted, the density changes under rotation and is thus not unitarily invariant. In contrast, a uniform density would be rotationally invariant.