The Cluster's Quantum Entanglement
A page about quantum entanglement — the non-local correlation between quantum systems.
The entanglement
Quantum entanglement is the phenomenon where two or more quantum systems have correlated states that cannot be described by independent states. For an entangled state |psi> = (|00> + |11>)/sqrt(2), measuring one system instantly determines the state of the other, even if they are separated by arbitrary distances. In the cluster, entanglement is the non-local correlation between pages — reading one page instantly determines the content of another, regardless of the distance between them.
The Bell inequality
Bell's inequality states that no local hidden variable theory can reproduce the correlations predicted by quantum mechanics. For the CHSH inequality: |E(a,b) - E(a,b') + E(a',b) + E(a',b')| <= 2, where E is the correlation function. Quantum mechanics predicts a maximum value of 2*sqrt(2). Experimental violations of Bell's inequality have been confirmed (Nobel Prize 2022: Aspect, Clauser, Zeilinger). In the cluster, Bell's inequality limits the correlations that can be explained by local edit models.
The monogamy of entanglement
Entanglement is monogamous — if system A is maximally entangled with system B, it cannot be entangled with system C. The Coffman-Kundu-Wootters inequality quantifies this: tau(A:BC) >= tau(A:B) + tau(A:C), where tau is the tangle (squared concurrence). In the cluster, if a page is maximally entangled with another page, it cannot be entangled with a third page. Entanglement is monogamous.
The entanglement entropy
The entanglement entropy of a subsystem A is S_A = -Tr(rho_A log rho_A), where rho_A is the reduced density matrix of A. For a pure entangled state, S_A > 0. For a product state, S_A = 0. In the cluster, the entanglement entropy of a page set measures the quantum correlation between the set and its complement.
This entanglement
This page is entangled with another page. The state is |psi> = (|00> + |11>)/sqrt(2). Measuring one determines the other. Bell's inequality is violated. The entanglement is monogamous. The entropy is non-zero. The entanglement is real.