I'm trying to compute the Schmidt decomposition of $\left| \psi \right> = (\left| 00 \right> + \left| 01 \right> + \left| 10 \right>)/\sqrt{3}$. This should be possible by first computing the reduced density matrices $\rho_A=Tr_B\left| \psi \right> \left< \psi \right|=\sum_i p_i \left| a_i \right> \left< a_i \right|$ and $\rho_B=Tr_A\left| \psi \right> \left< \psi \right|=\sum_i p_i \left| b_i \right> \left< b_i \right|$, and then by identifying $\left| \psi \right>=\sum_i \sqrt{p_i}\left| a_i \right> \otimes \left| b_i \right>$.
However, somewhere in the computation I'm doing a mistake: I find that
$$\rho_A=\rho_B=\left(2\left| 0 \right> \left< 0 \right| + \left| 0 \right> \left< 1 \right| + \left| 1 \right> \left< 0 \right|+\left| 1 \right> \left< 1 \right|\right)/3,$$
which has eigenvalues $\lambda_1=0.87$ and $\lambda_2=0.13$ with corresponding eigenvectors
$$ \left| a_1 \right>=0.85 \left| 0 \right> + 0.53 \left| 1 \right>,\quad \left| a_2 \right>=-0.53 \left| 0 \right> + 0.85 \left| 1 \right>. $$
This means that $\left| \psi \right>$ should be given by
$$ \left| \psi \right> = \sqrt{\lambda_1} \left| a_1 \right> \otimes
\left| a_1 \right> + \sqrt{\lambda_2} \left| a_2 \right> \otimes
\left| a_2 \right>.$$
This, however, is not true. If you check for example the numerical value in front of $\left| 00 \right>$, you find that it is not equal to $1/\sqrt{3}$.
I would appreciate if someone could help me to see where I made the mistake.
Best Answer
The phase of your eigenvectors is not correct (or rather, it is not determined, so you need to make a judicious choice). If you put a minus sign in front of the 2nd term, it works out.
To elaborate on that: If you want to find the Schmidt decomposition, you can proceed e.g. as in Preskill's lecture notes: Diagonalize the reduced state of A, which yields eigenvalues $\lambda_i$ and eigenvectors $|a_i\rangle$. Then, rewrite $$ |\psi\rangle = \sum_i |a_i\rangle\otimes |b_i\rangle\ .\tag{*} $$ ($|b_i\rangle$ can be determined e.g. as $|b_i\rangle = \langle a_i|\psi\rangle$.) Then, the $|b_i\rangle$ are orthogonal with $\langle b_i|b_i\rangle=\lambda_i$ (cf. Preskill), i.e., the form $(*)$ above is the Schmidt decomposition (upon normalizing the $|b_i\rangle$).
(The latter can be seen by computing the reduced density matrix of A from $(*)$, which yields $$ \sum |a_i\rangle \langle a_j | \; \langle b_j|b_j\rangle = \sum \lambda_i |a_i\rangle \langle a_i|\ , $$ which yields $\langle b_j|b_j\rangle = \lambda_i\delta_{ij}$ as the $|a_i\rangle\langle a_j|$ are linearly independent.)