# How to apply the Schmidt Decomposition to a Bell state?

I am trying to understand the Schmidt Decomposition, currently in my QC class. We had a tutorial where we were told if $$|\psi\rangle$$ is a pure state of a composite system A then there exists $$|i_A\rangle$$ and $$|i_B\rangle$$ which are orthonomral states for systems A and B respectively.

$$|\psi\rangle = \sum_i \lambda_i |i_A\rangle|i_B\rangle$$

I more or less understand that we write $$|\psi\rangle$$ as:

$$|\psi\rangle = \sum_{j,k} c_{jk}|a_j\rangle|b_k\rangle$$

Where $$|a_j\rangle_j$$ and $$|b_k\rangle_k$$ are orthonormal bases for A and B and $$C=(c_{jk})$$ a complex matrix. Then proceed by using singular value decomposition on C. My question is how should I move on to apply the Schmidt decomposition on a numerical example such as the $$|\beta_{00}\rangle$$. Some pointers would be lovely as I struggle to understand, and would prefer it if there were no direct answers as I want to struggle through it myself as much as possible. Thanks!

Instead, why not try an example such as $$|\psi\rangle=\frac{3}{5\sqrt{2}}|00\rangle+\frac{3}{5\sqrt{2}}|01\rangle+\frac{4}{5\sqrt{2}}|10\rangle-\frac{4}{5\sqrt{2}}|11\rangle$$ So, as you say, work out what the matrix $$C$$ is, and perform a singular value decomposition on it. From there, try to extract $$|i_A\rangle, |i_B\rangle$$ and $$\lambda_i$$ values. (is this all you were after? since you don't want too many hints, I'm not sure what else to say.)
• Just to make sure: $\sum_{i=0,1}= \frac{1}{\sqrt(2)}|i\rangle|i\rangle$ would be the decomposition of $|\beta_{00}\rangle$ right? Ah thanks so much everything is proper now I was stuck at such a weird place. – Yiğit Aras Tunalı Jun 16 '20 at 13:44