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arXiv:2509.04989

Surpassing the wave-particle duality relation via feed-forward of phase information

Elisabeth Meusert, U. Schilling, Marc-Oliver Pleinert, Joachim von Zanthier

Published Sep 5, 2025• 0 citations
arXivHTMLPDF

Abstract

Complementarity constitutes a central aspect of quantum theory. It manifests itself, for example, in a two-way interferometer, where the simultaneous observation of an interference pattern and the acquisition of which-way information are limited by an inequality known as the duality relation. Here, we investigate which-way information in a double-slit interferometer and show that it can be correlated to the phase of the quantum object at the detection screen, leading to a phase-dependent which-way knowledge. In specific cases, this knowledge can locally exceed the limit set by the duality relation. Based on this observation, we propose a feed-forward protocol that aims at maximizing the which-way information locally for each phase after the particle has been recorded on the screen. This allows us to surpass the duality relation limit even globally. We present analytical results as a proof of principle of our protocol as well as numerical outcomes quantifying the amount of maximally achievable which-way knowledge.

DOI
10.48550/arxiv.2509.04989

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Complementarity constitutes a central aspect of quantum theory. It manifests itself, for example, in a two-way interferometer, where the simultaneous observation of an interference pattern and the acquisition of which-way information are limited by an inequality known as the duality relation. Here, we investigate which-way information in a double-slit interferometer and show that it can be correlated to the phase of the quantum object at the detection screen, leading to a phase-dependent which-way knowledge. In specific cases, this knowledge can locally exceed the limit set by the duality relation. Based on this observation, we propose a feed-forward protocol that aims at maximizing the which-way information locally for each phase after the particle has been recorded on the screen. This allows us to surpass the duality relation limit even globally. We present analytical results as a proof of principle of our protocol as well as numerical outcomes quantifying the amount of maximally achievable which-way knowledge.
A distinct method to show a quantum object behaving both as wave and as particle is proposed and described in some detail. We make a systematic analysis using the elementary methodology of quantum mechanics upon Young's two-slit interferometer and the Mach—Zehnder two-arm interferometer with the focus placed on how to measure the interference pattern (wave nature) and the which-way information (particle nature) of quantum objects. We design several schemes to simultaneously acquire the which-way information for an individual quantum object and the high-contrast interference pattern for an ensemble of these quantum objects by placing two sets of measurement instruments that are well separated in space and whose perturbation of each other is negligibly small within the interferometer at the same time. Yet, improper arrangement and cooperation of these two sets of measurement instruments in the interferometer would lead to failure of simultaneous observation of wave and particle behaviors. The internal freedoms of quantum objects could be harnessed to probe both the which-way information and the interference pattern for the center-of-mass motion. That quantum objects can behave beyond the wave—particle duality and the complementarity principle would stimulate new conceptual examination and exploration of quantum theory at a deeper level.
We propose for the experimental verification of an inequality concerning wave-particle duality by Englert [Phys. Rev. Lett. 77, 2154 (1996)] relating (or setting) an upper limit on distinguishability and visibility in a two-way interferometer. The inequality, quantifies the concept of wave-particle duality. The considered two-way interferometer is a Young's double-slit experiment involving two four-level atoms and is a slightly modified version of that of the recent experiment by Eichmann et al. [Phys. Rev. Lett. 70, 2359 (1993)]. The fringe visibility depends on the detected polarization direction of the scattered light and a read out of the internal state of one of the two atoms provides a partial which-way information.
Complementarity lies at the heart of conceptual foundation of orthodox quantum mechanics. The wave-particle duality makes it impossible to tell which slit each particle passes through and still observe an interference pattern in a Young's double-slit experiment. In this paper, this duality is appraised under the standard formulation of quantum mechanics for atom interferometers. It is found that the internal freedoms like electronic states of an atom can be used to detect the which-path information of each atom while uphold the interference pattern of atoms when the center-of-mass motion of atoms is detected.
Some recent works have introduced a quantum twist to the concept of complementarity, exemplified by a setup in which the which-way detector is in a superposition of being present and absent. It has been argued that such experiments allow measurement of particle-like and wave-like behavior at the same time. Here, we derive an inequality which puts a bound on the visibility of interference and the amount of which-way information that one can obtain, in the context of such modified experiments. As the wave aspect can only be revealed by an ensemble of detections, we argue that, in such experiments, a single detection can contribute only to one subensemble, corresponding to either wave aspect or particle aspect. This way, each detected particle behaves either as particle or as wave, never both, and Bohr's complementarity is fully respected.
A single photon exhibits wave-particle duality in the Young's double-slit interferometer. The duality characterized by an interference visibility and a which-path information has trade-off relation known as complementarity. These quantities are related to the first-order coherence, and the interference is based on the phase correlation between lights coming from two arms. However according to quantum optics theory, such a simple wave-particle picture is not enough to understand the nature because the theory showed an importance of higher-order coherence in the sense of both interference and statistical distribution of photons. Second-order intensity correlation is especially crucial to reveal distinctive quantum features of photons with no classical analogue. Here, in an intensity interferometric scenario as represented by the Hong-Ou-Mandel interferometer, we discuss a wave-particle duality of light based on a which-path information and a quantity characterizing a magnitude of the intensity interferometric effect. We show, for classical light, the two quantities obey the complementary principle similar to the case of the double-slit experiment, but do not for nonclassical light. The nonclassical light such as photons at two arms is allowed to show larger which-path information and intensity interference simultaneously beyond the complementary relation. Moreover, the violation reveals a new nonclassical nature of light although both of the above two quantities seem to be understandable classically, which is never found from a consideration of only one side of wave-particle duality.
For a particle travelling through an interferometer, the trade-off between the available which-way information and the interference visibility provides a lucid manifestation of the quantum mechanical wave–particle duality. Here we analyse this relation for a particle possessing an internal degree of freedom such as spin. We quantify the trade-off with a general inequality that paints an unexpectedly intricate picture of wave–particle duality when internal states are involved. Strikingly, in some instances which-way information becomes erased by introducing classical uncertainty in the internal degree of freedom. Furthermore, even imperfect interference visibility measured for a suitable set of spin preparations can be sufficient to infer absence of which-way information. General results are illustrated with a proof-of-principle single-photon experiment. Quantum mechanics dictates that the interference pattern cast by particles after passing through a double slit depends on how much information it is possible to know about which slit they went through. Banaszek et al. show how this behaviour extends to a system’s internal degrees of freedom.