Can Operators A and B Always Be Applied to a General State in Tensor Space?

[Kreizhn]

Hey,

My brain seems to have shut down. Let's say I'm working in the space $H_a \otimes H_b$ and I have an operator A and B on each of these spaces respectively. Furthermore, consider a general state in the tensor space $|\psi \rangle = | x_1 \rangle |y_1\rangle + |x_2 \rangle |y_2 \rangle$ (ignoring normalization). Now in my head, I keep on thinking that if I wanted to apply the operator $A \otimes B$ to $| \psi \rangle$ I couldn't simply go

$$A \otimes B | \psi \rangle = A| x_1 \rangle B|y_1\rangle + A|x_2 \rangle B|y_2 \rangle$$

However, I was playing around with some values today (assuming finite dimensions), and from what I tried it always seemed to work. Can we do this in general, or did I just get lucky in all my trials?

 

[javierR]

It's fine. First, AxB is defined so that (AxB)(|y> x |z>)=(A|y> x B|z>), and second, the operator (AxB) is linear on the tensor space...call $$(A\times B)=O$$ and $$|y> \times |z>=\Psi$$. Then $$O (\Psi_1+\Psi_2)=(O\Psi_{1}+O\Psi_{2})$$. Then use the first property for each of the terms. (the "x" is product)