Schmidt decomposition and entanglement
Schmidt decomposition theorem
For any state of a composite system $\ket{\Phi} \in \mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$, there exists a decomposition $$\ket{\Phi} = \sum_j d_j \ket{\phi^{(1)}_j}\otimes\ket{\phi^{(2)}_j},$$ where $\ket{\phi^{(i)}_j}$ is a basis of $\mathcal{H}^{(i)}$ respecitvely The coefficents $d_j$ are called Schmidt coefficents and are real and non-negative Note that the choice of the bases depends on $\ket{\Phi}$, so given fixed bases, the decomposition wont work for ever $\ket{\Phi}$.
Derivation of the Schmidt decomposition theorem
Derivation work by diagonalizing the coefficent matrix of the general decomposition $$\ket{\Phi} = \sum_{ij}\alpha_{ij} \ket{\phi_i^{(1)}}\otimes\ket{\phi_j^{(2)}},$$ so with $\alpha = uvd$ we find $$\ket{\Phi} = \sum_{ijk} u_{ij}d_{j}v_{jk} \ket{\phi_i^{(1)}}\otimes\ket{\phi_j^{(2)}},$$ and since $u$ and $v$ are unitary matrices, they only rotate the basis vectors $\ket{\phi_i^{(1)}}$ and $\ket{\phi_j^{(2)}}$, leading to the above form of the decomposition theorem.
The number of non-zero Schmitd coefficents is called Schmidt number. It is unique for a state. We can use it to define entanglement
Definition: entanglement
A composite state $\ket{\Phi} \in \mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$ is called entangled if its Schmidt number is larger than one. Then it cant be written as a product of two states $\ket{\Phi} = \ket{\phi^{(1)}}\otimes\ket{\phi^{(2)}}$.
Definition: maximally entanglement
An entangled composite state $\ket{\Phi} \in \mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$ is maximally entangled if the value of all non zero Schmidt numbers is equal. Then it cant be written as a product of two states $\ket{\Phi} = \ket{\phi^{(1)}}\otimes\ket{\phi^{(2)}}$.