Current quantum research largely focuses on communication and computation, overlooking potential advantages in mechanical tasks. This paper explores the unexplored area of quantum advantages in transportation, focusing on the speed of particle dispatch through free space. Quantum phenomena such as tunneling and backflow offer intriguing possibilities. Quantum backflow, where a particle with positive momentum moves backward, is particularly relevant. Previous work on quantum time-of-arrival operators hasn't yielded concrete tasks demonstrating quantum superiority. This paper introduces a "projectile scenario" to address this gap. A non-relativistic, one-dimensional quantum particle (projectile) is prepared in a bounded region and allowed to propagate freely. The probability of detection in a distant target region is compared to that of a classical particle with the same initial momentum distribution. The existence of 'ultrafast' and 'ultraslow' quantum states is investigated, quantifying quantum advantage by the difference between quantum and optimal classical arrival probabilities. The goal is to determine whether the quantum mechanical system offers a measurable improvement in a practical transportation task. The study analyzes the influence of parameters such as mass, spatial support, and flight time on the quantum advantage.

The paper reviews existing literature on quantum tunneling and backflow, citing relevant works such as Razavy's Quantum Theory of Tunneling and Allcock's work on the time-of-arrival problem. Bracken and Melloy's work on probability backflow and its connection to a dimensionless quantum number is highlighted. More recent studies on quantum advantage in mechanical systems are referenced, while noting the lack of exploration into the potential for quantum advantages in transportation tasks. The authors refer to research investigating hypothetical quantum time-of-arrival operators and their connection to quantum backflow, pointing out that this research hasn't yet produced a concrete task where quantum systems demonstrate superiority. The paper also notes prior numerical estimations of the Bracken-Melloy constant (Cbm), lacking rigorous error bounds, which motivates their work to establish stricter bounds for Cbm.

The paper employs a theoretical approach, using mathematical modeling and analysis to investigate quantum advantages in transportation. The projectile scenario is the central model: a one-dimensional quantum particle is prepared in a bounded region [0, L] and its probability of reaching a target region [a, ∞) after time ΔT is calculated. This quantum probability is compared to the maximum and minimum probabilities achievable by a classical particle with the same momentum distribution. Metaplectic transformations are used to connect the quantum projectile problem to scenarios related to quantum backflow. The maximum quantum advantage is analyzed as a function of α = ML²/ΔT, where M is the mass and L is the spatial extent of the preparation region. The analysis includes computation of this advantage for finite values of α and its limiting value as α approaches infinity, relating it to the Bracken-Melloy constant. The study extends the analysis to a quantum rocket model, where multiple projectiles are launched sequentially. A modified projectile scenario is also considered where the initial position distribution of the quantum and classical particles is identical, leading to different comparative results. The study involves analytical calculations, numerical simulations, and the use of Wigner functions for phase-space representation. Techniques such as diagonalization of matrices for finite α and optimization methods for the infinite case are employed to quantify the quantum advantage. Rigorous error bounds are developed for the Bracken-Melloy constant.

The research reveals several key findings: First, ultrafast and ultraslow quantum states exist in the projectile scenario, demonstrating a quantum advantage over classical counterparts in terms of arrival probabilities. This advantage is quantified and found to be independent of the distance to the target region. Second, the maximum quantum advantage in the projectile and (a specific model of) rocket scenarios is linked to the Bracken-Melloy constant (Cbm), with the study providing improved bounds for Cbm (0.0315 ≤ Cbm ≤ 0.072). This demonstrates that the quantum advantage for these scenarios is inherently limited. Third, by modifying the comparison in the projectile scenario to include identical position distributions for quantum and classical particles, a significantly larger quantum advantage (at least 0.1262) is achieved. Fourth, mathematical connections are established between the standard projectile problem and other quantum mechanical effects, such as quantum backflow. The study demonstrates that these effects are manifestations of the same underlying mathematical phenomenon, viewed through different coordinate systems. This provides a deeper understanding of the relationship between seemingly disparate quantum phenomena.

The findings address the research question of whether quantum mechanics offers a practical advantage in transportation tasks. While a limited quantum advantage is observed in the basic projectile and rocket scenarios (limited by Cbm), a significantly enhanced advantage emerges when comparing to classical systems with identical position distributions. The limitations of open-combustion chamber rocket models are demonstrated. The identification of the Bracken-Melloy constant as a key limiting factor in simple transportation scenarios highlights its importance in understanding the boundaries of quantum advantage in this context. The connections established between the projectile scenario and various quantum effects, such as quantum backflow, provide a unified theoretical framework for these phenomena and open avenues for further research. The results suggest that careful consideration of comparison metrics is crucial in determining the potential for quantum enhancements in practical applications.

This work establishes the existence of both ultrafast and ultraslow quantum states in projectile scenarios, surpassing or underperforming their classical counterparts in arrival probability. The maximum quantum advantage is linked to the Bracken-Melloy constant, with new rigorous bounds established. While simple rocket models show similar limitations, a modified projectile scenario demonstrates a substantially greater quantum advantage. Future research could explore more sophisticated rocket models, focusing on those that might circumvent the limitations found here, and further investigate the potential of the enhanced quantum advantage identified in the modified projectile scenario for real-world transportation applications.

The study is theoretical, focusing on mathematical models and simulations. The experimental feasibility of realizing the scenarios and measuring the predicted quantum advantages warrants further investigation. The rocket model used is simplified; more realistic models incorporating complex fuel dynamics and interactions could yield different results. The numerical optimization methods used might not have found the global maximum in the modified projectile scenario, although the derived lower bound is substantial.