# Composite Hilbert Spaces and Operators

## Main Question or Discussion Point

So, say I have a composite hilbert space $$H = H_A \otimes H_B$$, can I write any operator in H as $$U_A \otimes U_B$$?

Thanks

## Answers and Replies

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dextercioby
Homework Helper
Not really. IT should be something like $$U_{A}\otimes\hat{1}_{B}+\hat{1}_{A}\otimes U_{B}$$

Haelfix
You also need some assumptions on the properties of the particular hilbert spaces in question (Ha and Hb).

Like are they seperable or not.

You can find some operators that are simple tensor products of local operators, however, in general any operator acting on the composite space can be decomposed as a linear combination of products of dyad operators. To see this use the completeness relation (i.e. that the sum of projection operators equals the identity for both spaces):

$$I_A \otimes I_B = \sum_j \sum_k \vert e_j \rangle \langle e_j \vert \otimes \vert f_k \rangle \langle f_k \vert$$

Then for any operator O on the composite space we get

$$O = I . O . I = \sum_j \sum_k \sum_m \sum_n \langle e_j , f_k \vert O \vert e_m, f_n \rangle \vert e_j \rangle \langle e_m \vert \otimes \vert f_k \rangle \langle f_n \vert$$

Last edited:
vanesch
Staff Emeritus
Gold Member
So, say I have a composite hilbert space $$H = H_A \otimes H_B$$, can I write any operator in H as $$U_A \otimes U_B$$?
As others pointed out, the operators of the form $$U_A \otimes U_B$$ span the algebra of operators, but not all operators in that algebra are of that form themselves.
Exactly like the vectors of $$H = H_A \otimes H_B$$ which are not all of the form $$|\psi_A> \otimes |\psi_B >$$, but the entire space is nevertheless spanned by those vectors.

reilly