Small Incremental Entangling conjecture

Let $A$ and $B$ be the systems controlled by Alice and Bob, respectively, and let $a$ and $b$ be arbitrary local ancillas. For a pure state $|\Psi\rangle$ on $aABb$ evolving under a non-local Hamiltonian $H_{AB}$, let $\Gamma(\Psi,H)$ be the entangling rate, and set

$$d=\min\{\dim(A),\dim(B)\}.$$

Bravyi's Small Incremental Entangling conjecture. There is a universal constant $c$ such that, for all ancilla dimensions, all states $|\Psi\rangle$, and all such Hamiltonians,

$$\Gamma(\Psi,H)\leq c\|H\|\ln d.$$

The conjecture asks for a dimension-independent logarithmic bound on the rate at which a non-local Hamiltonian can generate entanglement. The paper presents it as the entangling analogue underlying the Small Incremental Mixing problem; its resolution status is not specified in the supplied text.