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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 $v\in H$ can be decomposed into $v_1\in H_1$ and $v_2\in H_2$ 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.

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  • $\begingroup$ I think you're looking for the Schmidt Decompositon. en.wikipedia.org/wiki/Schmidt_decomposition $\endgroup$ – Sam Palmer Jun 12 at 20:16
  • $\begingroup$ you are asking about how to decompose a vector in a basis. That is the first thing discussed in any linear algebra course $\endgroup$ – glS Jun 13 at 5:48

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