How to compute the Schmidt decomposition of a bipartite pure state?

Question

I'm trying to work out the entropy of entanglement of my state but I'm struggling to put it into a Schmidt decomposition, i.e. in the form: $\sum_i \alpha_i |u_i \rangle |v_i \rangle$.

Currently I have something in the form: $|E_0\rangle |\psi \rangle + |E_1\rangle |\phi \rangle$ where $\langle E_0| E_1 \rangle =0$ but $|\psi\rangle = \alpha |0 \rangle + \beta |1\rangle, |\phi \rangle = \gamma |0\rangle + \delta |1\rangle$ such that $\langle \psi | \phi \rangle \neq 0$. Do I need to find a part of $\psi$ and $\phi$ that are orthogonal first? Or do I need to use an SVD decomposition?

Answer 1

Given a bipartite pure state described as a ket vector $|\psi\rangle\equiv\sum_{ij}c_{ij}|i,j\rangle$, you compute the SVD of the matrix whose elements are the coefficients $c_{ij}$. Then, the singular values are the Schmidt coefficients, and the left and right principal components the states appearing in the Schmidt decomposition.