Deriving the Kijowski Arrival-Time POVM from the Schr\"odinger Current: Minimal Positivity and Uniqueness

Abstract:Quantum backflow refers here to the appearance of a negative Schrödinger current for a state whose momentum support is entirely positive. We ask for the smallest modification of the free Schrödinger current that makes it nonnegative for every such state, while preserving the current of each individual momentum component. We show that the required minimal modification changes the free-particle momentum kernel according to \[ K_{\rm Sch}(p,p')=\frac{p+p'}{2m} \;\longrightarrow\; K_{\min}(p,p')=\frac{\sqrt{pp'}}{m}. \] The resulting current is positive and normalized and therefore defines an arrival-time POVM. Extending the directional no-backflow requirement to states containing both momentum signs forces the cross-sector kernel to vanish, \[ K_{\min}^{+-}=K_{\min}^{-+}=0, \] so that the full current is the sum of two independent directional contributions. The resulting POVM is exactly the Kijowski time-of-arrival POVM, providing a current-based physical motivation for both its directional kernels and their separation. Within the diagonal-preserving pairwise-minimal construction considered here, the result is unique. The construction itself does not impose a first-arrival condition.

arXiv:2609.08458 [quant-ph]

(or arXiv:2609.08458v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.08458 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.