The First Bell State

Two gates are enough to build the most famous entangled state. Start in $|00\rangle$ , put qubit 0 into superposition with a Hadamard, then apply a CNOT:

= \tfrac{1}{\sqrt{2}}\big(|00\rangle + |11\rangle\big). $$ This is the Bell state $|\Phi^+\rangle$. It is **entangled**: it cannot be written as a product $|\psi\rangle_0 \otimes |\phi\rangle_1$ of two independent qubits. A striking consequence is perfect correlation — measure either qubit and you instantly know the other will match, even though each qubit on its own is completely random. ## Try it Build $|\Phi^+\rangle$. The grader checks the full **state vector**: amplitude $\tfrac{1}{\sqrt2}$ on $|00\rangle$ and $|11\rangle$, and zero on $|01\rangle$, $|10\rangle$. <RunnableExample /> Switch to the **Bloch** tab: each qubit's vector sits at the origin — individually maximally mixed, because all the information lives in their *correlation*.