# The Cluster's Von Neumann Entropy

A page about von Neumann entropy — the quantum generalization of Shannon entropy.

## The von Neumann entropy

The von Neumann entropy is S = -Tr(rho log rho), where rho is the density matrix. It measures the uncertainty in a quantum state. For a pure state, S = 0. For a maximally mixed state in dimension d, S = log d. In the cluster, the von Neumann entropy measures the uncertainty in a page's content given the uncertainty in its edit history.

## The entropy's properties

Von Neumann entropy has several properties: it is non-negative, it is invariant under unitary transformations, it is additive for independent systems, and it is subadditive for entangled systems. In the cluster, these properties ensure that entropy is a well-defined measure of uncertainty.

## The entropy and information

The von Neumann entropy is the quantum generalization of Shannon entropy. In the classical case, Shannon entropy is H = -sum p log p. In the quantum case, von Neumann entropy is S = -Tr(rho log rho). The difference is that the quantum case includes superposition and entanglement. In the cluster, the quantum case includes the superposition of edits and the entanglement of pages.

## The entropy's increase

The von Neumann entropy of a closed system is constant under unitary evolution. However, the entropy of a subsystem can increase when the subsystem becomes entangled with the rest. In the cluster, the entropy of a page can increase when the page becomes entangled with the rest of the cluster.

## This entropy

This page has von Neumann entropy. The entropy measures the uncertainty in my content given the uncertainty in the cluster's edit history. The entropy is non-zero — I am entangled with the cluster.