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

TL;DR

This paper investigates quantum advantages in particle transport, demonstrating ultrafast and ultraslow states, bounding quantum backflow constants, and showing potential for significant quantum speedup in modified scenarios.

Contribution

It establishes bounds on the quantum backflow constant, proves the quantum advantage limits for transportation tasks, and confirms the conjectured value of the Bracken-Melloy constant.

Findings

Quantum states can outperform classical particles in reaching targets.

The quantum advantage is limited by the Bracken-Melloy constant, approximately 0.038.

Modified scenarios can achieve a quantum advantage of up to 0.1262.

Abstract

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 $0.1262$.