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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?

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  • $\begingroup$ What do you mean by "in terms of the difference between these unitaries"? Will you only accept functions of $U-V$? $\endgroup$
    – Quantum Mechanic
    Sep 18 at 18:47
  • $\begingroup$ Or something like $(U^\dagger\rho U - V^\dagger\rho V)$ (i.e. a single application of the channel). Or potentially something similar. $\endgroup$
    – Hans Schmuber
    Sep 18 at 19:03
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    $\begingroup$ 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 $\endgroup$
    – Quantum Mechanic
    Sep 18 at 20:23
  • $\begingroup$ 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. $\endgroup$
    – Hans Schmuber
    Sep 18 at 21:25
  • $\begingroup$ @QuantumMechanic yeah, if $U$ and $V$ are simultaneously diagonalizable, then it would make things a lot easier. $\endgroup$
    – FDGod
    Sep 18 at 23:01

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