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Suppose I have the state $\frac{1}{2}(|0\rangle + |1\rangle)$, how can I refer to ket 0 and ket 1 in general within a state? Would "term 1" and "term 2" work?

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If you have $|\psi \rangle = \alpha |0\rangle + \beta|1\rangle$ then you can refer to $|0\rangle$ and $|1\rangle$ as the basis states. This is generalize to $n$ qubit system as well.

And yes, you can just just refer to them as the $|0\rangle$ or "ket 0" state, and $|1\rangle$ or "ket 1" state.

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Generally, a quantum state $|\psi\rangle$ is composed of basis states $|0\rangle$ and $|1\rangle$. This come from linear algebra where any vector can be written as a sum of basis states multplied by an appropriate number. Since quantum states can be described by vectors from $\mathbb{C}^{2^n}$ (where $n$ is a number of qubits in the state), term basis states is more than appropriate.

You can also call these states ZERO and ONE in this particular case (a basis composed of $|0\rangle$ and $|1\rangle$ is called computational basis).

Just note that your state is not normalized, it should be $\frac{1}{\sqrt{2}}(|0\rangle +|1\rangle)$. Maybe, you were confused by notion that $|\psi\rangle$ collapses to either 0 or 1 with 50 % probability after measurement. However, coefficients in a quantum state are complex numbers called (quantum) amplitudes. A probability is square of amplitude absolute value.

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