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Say we have two quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ with channel fidelities $\mathcal{F}_1$ and $\mathcal{F}_2$. Can we place any bounds on the fidelity of the channel $\mathcal{E}=\mathcal{E}_1\circ \mathcal{E}_2$ (let's call the fidelity of this channel $\mathcal{F}$)? Is it possible to show something like $\mathcal{F}\geq \mathcal{F}_1\mathcal{F}_2$ for a general case?

I define channel fidelity as: $\mathcal{F}(\mathcal{E})=\textrm{min}_\rho~Tr(\rho,\mathcal{E}(\rho)),$ similar to the definition in Nielsen and Chuang.

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    $\begingroup$ What do you call "minimum fidelity"? $\endgroup$
    – Norbert Schuch
    Commented Aug 2, 2021 at 19:13
  • $\begingroup$ The definition of channel fidelity is given by $\mathcal{F}(\mathcal{E})=\textrm{min}_{\rho}Tr(\rho,\mathcal{E}(\rho))$ (from Nielsen and Chuang). I am calling this minimum fidelity but I guess the better name is channel fidelity. $\endgroup$
    – Vikas
    Commented Aug 2, 2021 at 20:20
  • $\begingroup$ Can you include that definition in the question? $\endgroup$
    – Norbert Schuch
    Commented Aug 2, 2021 at 21:00
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    $\begingroup$ If you use the trace distance instead it is easy to see that there is a relation (triangle inequality). In turn, you can relate the trace distance to the fidelity. Not sure the resulting inequality is tight, though. $\endgroup$
    – Norbert Schuch
    Commented Aug 2, 2021 at 21:03

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