How do you decompose an arbitrary quantum state into its corresponding projection subspaces such that their direct sum is the quantum state?

If $H$ is finite-dimensional (e.g. $H$ is the state space of a qubit), the procedure amounts to expanding the quantum state in an appropriately chosen basis.

Choose an orthonormal basis $|u_1rangle, dots |u_krangle$ of $H_1$ and extend it to an orthonormal basis $|u_1rangle, dots, |u_krangle, |u_{k+1}rangle, dots, |u_nrangle$ of $H = H_1oplus H_2$. Expand $|vrangle$ in the basis

$$ |vrangle = underbrace{a_1 |u_1rangle + dots a_k |u_krangle}_{|v_1ranglein H_1} + underbrace{a_{k+1} |u_{k+1}rangle + dots + a_n |u_nrangle}_{|v_2ranglein H_2} $$

to see that $|vrangle = |v_1rangle + |v_2rangle$ is the desired decomposition.

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If $H$ is infinite-dimensional (e.g. $H$ is the state space of a quantum harmonic oscillator), we can use projectors instead of a basis.

Let $P_k$ denote the orthogonal projector onto $H_k$ for $k=1,2$. Note that $H_1$ and $H_2$ are eigenspaces of $P_1$ associated with eigenvalues $1$ and $0$ respectively and of $P_2$ associated with eigenvalues $0$ and $1$ respectively. Consequently, $H$ is the eigenspace of $P_1+P_2$ associated with eigenvalue $1$. Therefore, $P_1+P_2 = I$.

Now, let $|v_1rangle = P_1|vrangle$ and $|v_2rangle = P_2|vrangle$. Then,

$$ |vrangle = I |vrangle = P_1|vrangle + P_2|vrangle = |v_1rangle + |v_2rangle $$

is the desired decomposition.