Bell state measurement

The Bell state $\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$ is prepared by $H$ on qubit 0 followed by $\mathrm{CX}_{0\to1}$: starting from $|00\rangle$, $H\otimes I$ gives $\frac{1}{\sqrt2}(|00\rangle+|10\rangle)$, and CX then copies the control's value onto the target, $\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$ — exactly the statevector this record's exact-state check compares against. Measuring both qubits in the computational basis yields $00$ or $11$ each with probability $1/2$ and never $01$ or $10$: the two outcomes are perfectly correlated even though neither qubit's individual outcome is determined before measurement, the operational signature of entanglement (this state violates a Bell/CHSH inequality; it cannot be reproduced by any local hidden-variable model tossing independent coins). Because it is the smallest circuit exhibiting genuine quantum correlation, the Bell state anchors teleportation, superdense coding, and entanglement-verification protocols throughout this catalog.