# Prove that the Bures metric satisfies a contractive property and has unitary invariance

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?

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.
• 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? Apr 30, 2022 at 22:02