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

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?

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.

• – glS
Commented Feb 12, 2023 at 11:13