# Heisenberg Representation of Quantum Computers explain observable transformations

The Heisenberg Representation of Quantum Computers (Daniel Gottesman) http://arxiv.org/abs/quant-ph/9807006

Suppose we have a quantum computer in the state $$|\psi\rangle$$, and we apply the operator $$U$$. Then

$$U N |\psi\rangle = U N U^\dagger U|\psi\rangle$$

so after the operation, the operator $$U N U^\dagger$$ acts on states in just the way the operator $$N$$ did before the operation. Therefore, applying $$U$$ to the computer transforms an arbitrary operation $$N$$ by

$$N \rightarrow U N U^\dagger$$

For the first step, I understand that placing $$U^\dagger U$$ on the right hand side has no effect; it resolves to the identity matrix.

I follow what the text is saying about considering the the right hand side as applying $$U N U^\dagger$$ after $$U$$ has been applied to the state.

I don't quite follow the result for how the observable $$N$$ is changed when the operator $$U$$ is applied: $$N \rightarrow U N U^\dagger$$. I'd appreciate it if someone who "gets" it could put a little more explanatory text between the two steps.

The author shows that since $UN$ equivalent to $UNU^\dagger U$, that it is possible to change the order of the operations, if $N$ is replaced by $M=UNU^\dagger$. In that case, $UN$ = $MU$.
Note that $M$ is just $N$ in the $U$ basis; the choice of $U$ uniquely determines how $N$ changes to $M$ for this special commutation relation.