Difference between the outer product

[Dragonfall]

Given $$\left| v\right>$$ and $$\left| u\right>$$ what is the difference between the outer product $$\left| v\right>\left< u\right|$$ and the tensor product $$\left| v\right>\otimes\left|u\right>$$? Is the latter a matrix representation of the former in some basis? Which basis would that be?

 

[Dragonfall]

And what's the difference between a tensor product and a kronecker product?

 

[Dragonfall]

I meant the tensor product $$\left< v\right|\otimes\left| u\right>$$.

 

[StatusX]

|u><v| is a way of writing the tensor product of a vector and a dual vector (ie, an element of the Hilbert space and an element of its dual, which is usually casually identified with the Hilbert space using the inner product). This is a linear operator on the Hilbert space, sending |w> to <v|w>|u>. In general, the tensor product of a vector space and its dual is the space of (finite rank) linear operators on the vector space.

On the other hand, |u>|v> is an element of the tensor product of the vector space with itself, usually used in physics for describing a composite of two identical systems. Again, since there is an isomorphism between the vector space and its dual, there is one between the space of composite states and the space of linear operators. This is interesting, but I've never seen this put to good use.

Finally, the Kronecker product is just a particular representation of the tensor product, convenient for dealing with tensor products of linear operators.