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

Quantum Computing Asked by Sam Michael on June 20, 2021

I understand that every Hilbert space $H$ can be decomposed into two mutually orthogonal subspaces $H_1$ and $H_2$ whose direct sum is $H$.

Therefore, every vector $vin H$ can be decomposed into $v_1in H_1$ and $v_2in H_2$ such that direct sum of $v_1$ and $v_2$ is $v$.

I just want to see the mathematical procedure for an arbitrary quantum state.

One Answer

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.


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.

Answered by Adam Zalcman on June 20, 2021

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