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In this paper, the authors assert that the Bures metric satisfies a contractive property and has unitary invariance. These terms aren't defined or proved in the paper, nor is any reference given for a definition or proof. Can anyone provide a concrete definition of these terms (and a proof that the Bures metric has these properties) or a place where I can find such details?

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Contractivity refers to the fact that, under the action of any CPTP map $\mathcal{E}$, a given metric satisfies $M(\rho,\sigma) \geq M(\mathcal{E}(\rho), \mathcal{E}(\sigma))$. Unitary invariance means that the above is an equality when $\mathcal{E}$ is a unitary channel (and is actually a consequence of the standard contractivity).

The fact that the Bures metric satisfies the above follows from the fact that the fidelity does (with $\leq$ rather than $\geq$), which is proved e.g. in Nielsen and Chuang, Theorem 9.6.

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  • $\begingroup$ Hmmm, in the paper they seem to use the contractive property to argue that $d(A^m, B^m) \leq md(A,B)$ for channels $A,B$, where $X^m$ denotes the channel $X$ applied $m$ times. Maybe I'm just being dumb but I don't see how this follows then from the definition you gave. Do you know what they might be using here? $\endgroup$
    – SescoMath
    Apr 30, 2022 at 22:02
  • $\begingroup$ actually I think they are using a triangle inequality... $\endgroup$
    – SescoMath
    May 1, 2022 at 4:11

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