1

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?

glS
  • 27,510
  • 7
  • 37
  • 125
Sergio Escobar
  • 811
  • 4
  • 9

1 Answers1

1

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.