Understanding the relevance of writing a quantum state in the Schmidt basis

[skynelson]

Hi all,
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_iƩ_jc_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...

 

[eljose79]

Hello,

Firstly, I would like to commend you on your insightful question. It is indeed true that the SVD allows us to express a system with two subsystems in a more compact form, using only one index instead of a double sum.

From a physics perspective, this transformation can be interpreted as a change in basis. The new basis, obtained through the SVD, is a combination of the original bases for the two subsystems. This means that the new basis is a more efficient representation of the system, as it captures the correlations between the two subsystems.

In terms of physical interpretation, this can be seen as a form of entanglement between the two subsystems. Entanglement is a phenomenon in quantum mechanics where two or more particles become correlated and behave as a single system, even when separated by large distances. In your case, the SVD allows us to mathematically represent this entanglement between the two subsystems.

Moreover, the single sum over a double sum also has implications in terms of computational complexity. As you mentioned, it makes the math easier, but it also reduces the number of calculations needed to describe the system. This is especially useful in larger systems with multiple subsystems, as it simplifies the calculations and allows for a more efficient description of the system.

In conclusion, the SVD allows us to represent a system with two subsystems in a more compact form, capturing the entanglement between the two subsystems. This has both physical and computational implications, making it a powerful tool in both linear algebra and physics.

I hope this helps clarify your understanding. Keep exploring and asking questions!