The Schmidt Decomposition (Derivation)

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:

For more information on how to compute the SVD of a matrix, I found this video series extremely helpful:

[gautam]

Appreciate your work a lot. Thank you for uploading these videos!

[HopeGao-tj]

Great lecture, Diego! And thanks! Could you please comment on - 1. At the video time of approximately 14 minutes, how to make an analogy between C_AB|v_k = \lambda_k |u_k> to the standard eigenvalue analysis A \vec x = \lambda \vec x. 2. Do you have a video on how to perform MPS via Schmidt decomposition for a system with multiple qubits (a N-qubit system)? A question is on the steps when iterating over each qubit for Schmidt decomposition, I believe that one needs to update the bipartite subsystem, i.e., re-splitting the full system again for the qubit that is currently being iterated over. Is there an optimal approach to split and how does it impact the result? And, how to interpret the final MPS that is performed from the change of the subsystems for each qubit in the N-quite system? Thanks!

[TangJackson]

❤Your video is very helpful for me to understand the details. Thanks a lot.