Checkpoint: Formalism Workout

This checkpoint pulls together every tool from the module — kets, inner products, completeness, projectors, basis expansions, and tensor products — to construct and analyze the most famous entangled state in quantum mechanics, the Bell state $|\Phi^+\rangle$.

The target state

We want to build

a two-qubit state living in $\mathbb{C}^2 \otimes \mathbb{C}^2 = \mathbb{C}^4$. It is normalized, since $\langle\Phi^+|\Phi^+\rangle = \tfrac{1}{2}(\langle 00|00\rangle + \langle 11|11\rangle) = \tfrac{1}{2}(1 + 1) = 1$, using orthonormality of the product basis.

Why it is entangled

Apply the factorization test from the tensor-product lesson. The amplitudes are $c_{00} = c_{11} = 1/\sqrt{2}$ and $c_{01} = c_{10} = 0$, so

The condition for a product state fails, so $|\Phi^+\rangle$ is entangled: there is no pair of single-qubit states $|\psi\rangle, |\phi\rangle$ with $|\Phi^+\rangle = |\psi\rangle\otimes|\phi\rangle$.

Constructing it from formalism

Build it in two steps, each a manipulation you have already practiced.

Step 1 — superpose the control. Apply a Hadamard to qubit 0, leaving qubit 1 in $|0\rangle$:

This is still a product state — H acts on one factor only.

Step 2 — correlate with CNOT. The CNOT gate with control 0, target 1 acts on the product basis by flipping the target when the control is $|1\rangle$: it fixes $|00\rangle$ and sends $|10\rangle \to |11\rangle$. Applying it,

The entangling step is the CNOT: it converts a product state into a non-factorizable one.

Checking with projectors and completeness

The probability of measuring both qubits and finding $00$ is the expectation of the projector $P_{00} = |00\rangle\langle 00|$:

Likewise the outcome $11$ has probability $1/2$, while $01$ and $10$ each have probability $0$. The four projectors $\{P_{00}, P_{01}, P_{10}, P_{11}\}$ satisfy the completeness relation $\sum P = I$, and the probabilities $\tfrac12 + 0 + 0 + \tfrac12 = 1$ — total probability is conserved, exactly as Parseval's identity guarantees. The perfect correlation (only $00$ and $11$ ever occur) is the experimental signature of entanglement.

Try it

Build the Bell state $|\Phi^+\rangle = \tfrac{1}{\sqrt2}(|00\rangle + |11\rangle)$ on two qubits using a Hadamard followed by a CNOT, and return the circuit. The grader checks the full statevector: amplitudes $1/\sqrt2$ on $|00\rangle$ and $|11\rangle$, zero elsewhere.

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