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

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.

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