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.