This is both linear algebra and physics problem, and I decided to post in physics because I want a "physics-framed" answer.

Suppose you have a system with two objects (subsystems) in it described by the state:

|ψ> = Ʃ

_{i}Ʃ

_{j}c

_{ij}|i>|j>

where |i> and |j> are orthonormal bases for the two different subsystems.

I apply a local unitary transformation to each subsystem, evolving the subsystem forward in time: U acts on |i> system, and V acts on |j> system.

According to the Singular Value Decomposition, this can be written as

U[\otimes]V|ψ> = Ʃ

_{i}√λ

_{i}|i>|i>

Notice, by choosing the special bases correctly using the SVD, we get a single sum over one index instead of a double sum over two indices.

My best understanding of this is that the SVD takes a system in one orthonormal basis and transforms it into another ("future state") orthonormal basis (transformed in size by the singular values, √λ). We have somehow coupled the two bases for the two different subsystems.

Since they are both orthonormal bases, we now only need one index to keep track of the wave-functions of the two sub-systems, whereas before we required taking a tensor product of every possible combination of the bases of the different subspaces.

I am not confident on this. What is the benefit of expressing with a single sum over a double sum (aside from easier math)? What does this represent physically?

Thanks in advance for your time...