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arXiv:2609.08458

Deriving the Kijowski Arrival-Time POVM from the Schrödinger Current: Minimal Positivity and Uniqueness

Avi Marchewka

quant-phSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
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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.

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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.
Detector-based first-arrival models and operator-based arrival-time observables provide two conceptually distinct approaches to the quantum time-of-arrival problem. Here we connect these approaches by using the positive unidirectional first-arrival current of Marchewka and Schuss (MS) as the basis for constructing a family of positive arrival-time operators and, after normalization, an arrival-time POVM. The first-arrival character is inherited from the dynamics with a Dirichlet boundary, while the detector response is introduced at the amplitude level through a non-negative spectral function λ(k)λ(k)λ(k). For a one-dimensional free particle, resolution of the identity uniquely selects λ(k)=π4kλ(k)=\fracπ{4k}λ(k)=4kπ​, yielding an arrival-time POVM. The normalized unidirectional current can therefore be interpreted as a generalized Born-rule probability density rather than as a hazard rate, as in the original MS formulation. We further show that the normalized one-sided MS amplitude coincides, up to an irrelevant phase, with the corresponding momentum-sector amplitude in Kijowski's arrival-time distribution. Consequently, the two directional contributions in Kijowski's distribution can be reproduced by two first-arrival problems defined on opposite sides of the boundary. This establishes an equivalence between the resulting arrival-time statistics while emphasizing the different physical interpretations: MS describes a first-arrival process with coherent momentum components, whereas Kijowski's POVM treats the two momentum sectors as separate directional contributions. Thus, the normalized unidirectional current provides a POVM description of first arrival at a point, combining detector-based dynamics with a positive-operator measurement structure.
In its standard formulation, quantum backflow is a classically impossible phenomenon in which a free quantum particle in a positive-momentum state exhibits a negative probability current. Recently, Miller et al. [Quantum 5, 379 (2021)] have put forward a new, "experiment-friendly" formulation of quantum backflow that aims at extending the notion of quantum backflow to situations in which the particle's state may have both positive and negative momenta. Here, we investigate how the experiment-friendly formulation of quantum backflow compares to the standard one when applied to a free particle in a positive-momentum state. We show that the two formulations are not always compatible. We further identify a parametric regime in which the two formulations appear to be in qualitative agreement with one another.
We prove that the Bohmian arrival time of the one-dimensional Schrödinger evolution violates the quadratic form structure on which Kijowski's axiomatic treatment of arrival times is based. Within Kijowski's framework, for a free right moving wave packet Ψ, the various notions of arrival time (at a fixed point x on the real line) all yield the same average arrival time . We derive the inequality relating the average Bohmian arrival time to that of Kijowksi. We prove that if and only if Ψ leads to position probability backflow through x .
We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit–there are however problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the timeof-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski. 1
In this paper we study the construction of probability densities for time of arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time of arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of the Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit—there are, however, problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.
We contrast the average arrival time at x according to the Bohmian mechanics of one dimensional free Schrödinger evolution with the standard quantum mechanical one. For positive momentum wave functions the first cannot be larger than the second one. Equality holds if and only if the wave function does not lead to position probability backflow through x. This position probability backflow has the least upper bound of approximately 0.04. We describe a numerical method to determine this backflow constant, introduced by Bracken and Melloy, more precisely and we illustrate the approximate wave function of maximal backflow.