# Schmidt decomposition of bipartite states

When performing the Schmidt decomposition of a bipartite state $$\left|\psi\right>_{AB}=\sum_{ij}c_{ij}\left|v_i\right>_{A}\left|w_j\right>_{B}$$ is computing the eigenvectors of the reduced density matrices $$\rho_{A}=CC^{\dagger}$$ and $$\rho_{B}=C^{\dagger}C$$ to the eigenvalues $$\sigma(CC^{\dagger})=\sigma(C^{\dagger}C)=\{\lambda_{i}\}_{i}$$ (singular values of $$C$$), denoted by $$\{\left|a_i\right>_{A}\}$$, $$\{\left|b_i\right>_{B}\}$$, in order to decompose the state like $$\left|\psi\right>_{AB}=\sum_{i}\lambda_{i}\left|a_i\right>_{A}\left|b_i\right>_{B}$$ a general procedure which always works? Do I have to try in a different manner when having multiple singular values? The state I'm tryng to decompose is $$\left|\psi\right>_{AB}=\frac{1}{2}(\left|00\right>+\left|01\right>+\left|10\right>-\left|11\right>)$$, for which this method doesn't seem to work.

$$\left|\psi\right>_{AB}=\frac{1}{2}(\left|00\right>+\left|01\right>+\left|10\right>-\left|11\right>)$$ leads to the coefficients matrix

$$C= \left[ {\begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} \\ \end{array} } \right]$$,

which in turn results in the reduced density matrices

$$\rho_{A}=CC^{\dagger}= \left[ {\begin{array}{cc} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \\ \end{array} } \right]=C^{\dagger}C=\rho_{B}$$.

Thus, the reduced density matrices share the eigenvectors $$\left|0\right>$$ and $$\left|1\right>$$ for the repetead eigenvalue $$\frac{1}{2}$$. I thought then the bipartite state could have been decomposed like $$\left|\psi\right>_{AB}=\sum_{i}\lambda_{i}\left|a_i\right>_{A}\left|b_i\right>_{B}$$, using the singular values and the eigenvectors of $$\rho_{A}$$ and $$\rho_{B}$$. However, $$\frac{1}{\sqrt{2}}\left|0\right>\otimes\left|0\right>+\frac{1}{\sqrt{2}}\left|1\right>\otimes\left|1\right>$$ does not yield the state I wanted to decompose.

• it's not clear what you are trying to do. To find the Schmidt decomposition you need only find the SVD of $C$ (which in this case is also symmetric, making the task easier). You cannot reconstruct the state from the reduced density matrices, like you seem to be trying to do – glS Jan 9 at 1:00