Checkpoint: Add Two Spins

What this checkpoint tests

You have now seen the complete picture for adding two spin-$\tfrac12$ particles: the uncoupled and coupled bases, the total-spin operator $\mathbf{S}^2$, the singlet–triplet split, the Clebsch–Gordan coefficients, the triangle rule, and the exchange symmetry. This checkpoint asks you to build a coupled state explicitly and confirm it on the simulator.

The target state

The middle member of the triplet is the symmetric, maximally entangled combination

It has total spin $s = 1$ and projection $m = 0$. Contrast it with the singlet $|0,0\rangle = \tfrac{1}{\sqrt2}(|01\rangle - |10\rangle)$, which differs only by the sign of the second term — symmetric versus antisymmetric exchange.

A recipe to recall

A clean way to reach $|1,0\rangle$ from $|00\rangle$:

1. $H$ on qubit 0 to make $\tfrac{1}{\sqrt2}(|00\rangle + |10\rangle)$.
2. $X$ on qubit 1 to reach $\tfrac{1}{\sqrt2}(|01\rangle + |11\rangle)$.
3. $\mathrm{CX}(0,1)$ to convert the $|11\rangle$ term into $|10\rangle$, giving the triplet.

If you instead wanted the singlet, you would insert a $Z$ on qubit 0 after step 1 to flip the relative sign — a one-gate difference that encodes the whole symmetry distinction.

Try it

Build the triplet state $|1,0\rangle = \tfrac{1}{\sqrt2}(|01\rangle + |10\rangle)$ as a two-qubit statevector and press Check.

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