Quantum advantages for transportation tasks: projectiles, rockets and quantum backflow

Consider a scenario where a quantum particle is initially prepared in some bounded region of space and left to propagate freely. After some time, we verify if the particle has reached some distant target region. We find that there exist "ultrafast" ("ultraslow") quantum states, whose probability of arrival is greater (smaller) than that of any classical particle prepared in the same region with the same momentum distribution. For both projectiles and rockets, we prove that the quantum advantage, quantified by the difference between the quantum and optimal classical arrival probabilities, is limited by the Bracken-Melloy constant $c_{bm}$, originally introduced to study the phenomenon of quantum backflow. In this regard, we substantiate the $29$-year-old conjecture that $c_{bm}\approx 0.038$ by proving the bounds $0.0315\leq c_{bm}\leq 0.072$. Finally, we show that, in a modified projectile scenario where the initial position distribution of the particle is also fixed, the quantum advantage can reach $0.1262$.