# Error in repeated applications of a quantum channel?

Suppose I have two quantum channels. Assume they they consist of $$r\in \mathbb{Z}$$ applications of unitaries, $$U$$ and $$V$$ respectively. Let the error between the channels acting on some state $$\rho$$ be: $$\varepsilon(U,V) = U^{\dagger r} \rho U^r - V^{\dagger r} \rho V^r.$$

Ideally, I want an expression for $$\epsilon(U,V)$$ in terms of the difference between these unitaries and $$r$$. Ideally I want an expression for $$\varepsilon(U,V)$$ as defined above, not the norm of the expression.

My first attempt was to write that $$U^r - V^r = \sum_{k=1}^{r-1} U^{r-k}(U-V)V^k$$ and substitute into the expression for $$\varepsilon(U,V)$$. However, this introduces a lot of cross terms which make things messy, and one ends up with $$O(r^2)$$ many terms when it feels like there should only be $$O(r)$$. Is there a better way to do this?

• What do you mean by "in terms of the difference between these unitaries"? Will you only accept functions of $U-V$? Sep 18 at 18:47
• Or something like $(U^\dagger\rho U - V^\dagger\rho V)$ (i.e. a single application of the channel). Or potentially something similar. Sep 18 at 19:03
• Hm. I was thinking along the lines of $\epsilon(U,V)=U^{\dagger r}[\rho - W \rho W^\dagger ]U^r$ for $W=U^r V^{\dagger r}$, and finding $W$ in terms of $U V^\dagger$, but even that requires lots and lots of arbitrary commutators Sep 18 at 20:23
• To be honest, I think I'd be happy with anything which is linear in $r$ and some characterisation of the error between a single application of the separate unitaries. Sep 18 at 21:25
• @QuantumMechanic yeah, if $U$ and $V$ are simultaneously diagonalizable, then it would make things a lot easier. Sep 18 at 23:01