Composite quantum systems: Kronecker and Hadamard/Schur products

[Dr_Nate]

TL;DR Summary

Can I use the Hadamard product between the spatial and spin kets?

In QM textbooks, authors will often jam two kets next to each other and say nothing about the binary operation between them. Other times, it may be called a tensor product, Kronecker product, direct product, or, in Griffith's case, a simple product. I ask the following question in this forum because I am looking for math experts who hopefully are aware of quantum mechanics.

I would like to expand the kets in the following equation to show unambiguously the binary operations between the spatial and spin states:
|Ψ⟩=|A↑,B↓⟩−|B↓,A↑⟩.

Let's concentrate on this term: |A↑,B↓⟩. I think many would say that it would be expanded like this:
(|A⟩⊗|↑⟩)⊗(|B⟩⊗|↓⟩),
where the parentheses are necessary.

Now, isn't the binary operation inside the parentheses slightly different from the central binary operation that links the two sets of parentheses? I believe we can call the central binary operation the Kronecker product and we know it is non-commutative. However, the binary operation inside the parentheses is commutative; after all, who cares if we put the spin to the left of the spatial state?

Could we then use the Hadamard/Schur product because it is commutative?
(|A⟩⊙|↑⟩)⊗(|B⟩⊙|↓⟩)

 

[jambaugh]

The tensor product inside the parentheses are also non-commuting. When two spaces are distinct it is a choice of convention which factor we decide will go on the left and which on the right in applications.

I believe that the seeming commutativity your are referencing is merely one playing ambiguous with that choice. Specifically we can "sort" the factors for convenience with an implicit isomorphism map (applied to all relevant operators as well as the representation vectors) e.g. writing the two spinor factors together.

This is clearer in component form with index notation as it is the order of the indices on the product form that matters, not the order of the factors.
$$(u\otimes v)^k = u^i v^j {[\otimes]_{ij}}^k$$

 

[Peter Morgan]

The Hadamard/Schur product is a component-by-component product of matrices, $M_{ij}=A_{ij}B_{ij}$ whereas you seem to be using it as a product of vectors (specifically kets, $|A\rangle$, et cetera)?
As a component-by-component product, the Hadamard/Schur product is basis dependent, so it is not likely to be useful unless the mathematical context is very sure about an appropriate basis (that does happen in my work because the Hadamard/Schur product preserves positive-definiteness, but that's in a very different context than your question gives.)