Complete-complement quantitative bounds

Define the complete leakage distance

The infimum is attained because the finite state space is compact.

We use two standard finite-dimensional fidelity facts in the sufficiency direction below:

1. root fidelity $f(\rho ,\sigma )={\Vert \sqrt{\rho }\sqrt{\sigma }\Vert }_1$ satisfies $f(\rho ,\sigma )\ge 1-d_{\mathrm{tr}}(\rho ,\sigma )$ FuchsVanDeGraaf;

2. purifications can be aligned, after enlarging the purifying space, with overlap $f(\rho ,\sigma )$ Uhlmann.

These are explicitly identified standard inputs, not consequences of the supplied modular manuscript.