In this video, we explain how to calculate the Schmidt Decomposition of a bipartite pure state using the singular value decomposition (SVD) of the probability-amplitude matrix. Later in the series we will discuss how the Schmidt Decomposition can be performed using the reduced density matrices of the two subsystems.

NOTE:
In this video I use the term canonical to refer to the most common way of representing a quantum state, which is by using a standard basis such as the computational basis. In some references, the term canonical is used to denote the "simplest" representation of a state (with the least amount of summation terms), which corresponds to its Schmidt Decomposition. Sorry if this caused confusion.

CORRECTION:
Around 13:45, the last non-zero diagonal element of the matrix should be λ_d-1, not λ_d because I index the λₖ coefficients starting from k = 0.
Around 24:30, I misspoke and said that setting full_matrices=False will give us matrices of the same size. What I meant to say is that the matrices U and V will have the same number of columns (i.e., same number of Schmidt vectors equal to d). The number of rows of U and V will be given by the size of their spaces N and M, respectively.

This video is part of a series on quantum entanglement:
   • Quantum Entanglement  

For more information on how to compute the SVD of a matrix, I found this video series extremely helpful:
   • Singular Value Decomposition  [Data-D...