# Schmidt decomposition theorem proof

1. Jan 12, 2010

### Fredrik

Staff Emeritus
I would be interested in seeing a correct statement and proof of the Schmidt decomposition theorem that (I think) says that if $x\otimes y\in H_1\otimes H_2$, there exists a choice of bases for these Hilbert spaces, such that

$$x\otimes y=\sum_{n=1}^\infty \sqrt{p_n}\ a_n\otimes b_n$$

with $\sum_n\sqrt{p_n}=1$. I'm not even sure that this is what the theorem says. Maybe it applies to all members of the tensor product space, and not just members of the form $x\otimes y$. Wikipedia has a proof for the finite-dimensional case (here), but I'm more interested in the infinite-dimensional case. (I haven't made the effort to try to understand the finite-dimensional case yet). I found this, but I don't see how to relate it to what I wrote above. Maybe it's a completely different theorem, but there are some similarities with Wikipedia's approach (they both talk about eigenvalues of operators of the form (T*T)1/2) that suggest that this is in fact the right theorem, and that I just need to figure out how to use it. (The pages that can't be read at Google Books can be read at Amazon).

Last edited: Jan 12, 2010
2. Jan 14, 2010

### atyy

3. Jan 16, 2010

### Fredrik

Staff Emeritus