On Bounding Entangling Rates and Mixing Rates in Some Special Cases

We derive bounds on mixing rates in a special case when the probabilistic ensemble of states is under certain restrictions. In this case we can prove the small incremental mixing conjecture with a constant c ′ = 1, which is improved over the previous results of which the tightest is c ′ = 2. Furthermore, we demonstrate a bound Γ(Ψ, H ) ≤ 2∥ H ∥ log( d ) for the small incremental entangling conjecture if d is sufficiently large.