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arXiv:2103.15070 · Proceedings of the National Academy of Sciences

Experimental demonstration of quantum pigeonhole paradox

Ming-Cheng Chen, Chang Liu, Yi-Han Luo, He-Liang Huang, Bi-Ying Wang, Xi‐Lin Wang, Li Li, Nai-Le Liu, Chao‐Yang Lu, Jian-Wei Pan

Published Jan 11, 2019• 20 citations
arXivHTMLPDF

Abstract

We experimentally demonstrate that when three single photons transmit through two polarization channels, in a well-defined pre- and postselected ensemble, there are no two photons in the same polarization channel by weak-strength measurement, a counterintuitive quantum counting effect called the quantum pigeonhole paradox. We further show that this effect breaks down in second-order measurement. These results indicate the existence of the quantum pigeonhole paradox and its operating regime.

DOI
10.1073/pnas.1815462116

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"In our paper in PNAS (1), we describe a quantum violation of the pigeonhole principle. We describe a situation (involving pre- and postselection) in which we put three particles in two boxes and we never find two particles in the same box. We presented both a 'strong measurement' analysis and a 'weak measurement' one. In his comment, Svensson (2) makes two points. The first refers to our strong measurement experiment and the second to the weak measurement one."

Proceedings of the National Academy of Sciences•May 20, 2016•3 citations
We experimentally demonstrate that when three single photons transmit through two polarization channels, in a well-defined pre- and postselected ensemble, there are no two photons in the same polarization channel by weak-strength measurement, a counterintuitive quantum counting effect called the quantum pigeonhole paradox. We further show that this effect breaks down in second-order measurement. These results indicate the existence of the quantum pigeonhole paradox and its operating regime.
The quantum pigeonhole effect (QPE) appears to contradict the classical pigeonhole principle by allowing three quantum particles distributed between two boxes to exhibit no pairwise coincidence. We show that this effect does not signal a breakdown of classical counting, but instead arises from quantum contextuality. By deriving Bell-type inequalities directly from the pigeonhole principle and reformulating the weak-measurement protocol within a bipartite density-operator framework, we demonstrate that the QPE is a form of Bell's theorem without inequalities. The apparent paradox reflects the impossibility of non-contextual eigenvalue assignments rather than a violation of classical combinatorial logic.
When three pigeons are in two pigeonholes, at least two of them should be in the same pigeonhole, which is called pigeonhole principle. Recently Aharonov el al have shown that the principle can be violated in a pre-postselected quantum system. This violation sheds light on a new aspect of quantum correlations, as both of the pre and postselected states are separable states. To prove this kind of quantum correlations, a measurement apparatus must be arranged in entanglement to perform a joint measurement on the quantum system. In this paper we discuss how to design the state of entangled meters to achieve such a measurement. We also propose an experimental setup composed of linear optics for direct observation of violation of pigeonhole principle.
We implement a variant of the quantum pigeonhole paradox thought experiment to study whether classical counting principles survive in the quantum domain. We observe strong measurements significantly violate the pigeonhole principle (that among three pigeons in two holes, at least one pair must be in the same hole) and the sum rule (that the number of pigeon pairs in the same hole is the sum of the number of pairs across each of the holes) in an ensemble that is pre- and postselected into particular separable states. To investigate whether measurement disturbance is a viable explanation for these counterintuitive phenomena, we employ a we employ variable-strength nonlocal measurements. As we decrease the measurement strength, we find the violation of the sum rule decreases, yet the pigeonhole principle remains violated. In the weak limit, the sum rule is restored due to the cancellation between two weak values with equal and opposite imaginary parts. We observe the same kind of cancellation at higher measurement strengths, thus raising the question: do strong measurements have imaginary parts?
We report an experimental realization of the quantum paradox of the separation of a single photon from one of its properties (the so-called "quantum Cheshire cat"). We use a modified Sagnac interferometer with displaced paths to produce appropriately pre- and postselected states of heralded single photons. Weak measurements of photon presence and circular polarization are performed in each arm of the interferometer by introducing weak absorbers and small polarization rotations and analyzing changes in the postselected signal. The absorber is found to have an appreciable effect only in one arm of the interferometer, while the polarization rotation significantly affects the signal only when performed in the other arm. We carry out both sequential and simultaneous weak measurements and find good agreement between measured and predicted weak values. In the language of Aharonov et al. and in the sense of the ensemble averages described by weak values, the experiment establishes the separation of a particle from one its properties during the passage through the interferometer.
It was recently argued that the pigeonhole principle, which states that if three pigeons are put into two pigeonholes then at least one pigeonhole must contain more than one pigeon, is violated in quantum systems [Proc. Natl. Acad. Sci. USA 113, 532 (2016)]. An experimental verification of this effect was recently reported [Proc. Natl. Acad. Sci. USA 116, 1549 (2019)]. In another recent experimental work, it was argued that two entities were observed to exchange properties without meeting each other [Nat. Commun. 11, 3006 (2020)]. Here we describe all these proposals and experiments as simple quantum interference effects where no such dramatic conclusions appear. Besides demystifying some of the conclusions of the cited works, we also present physical insights for some interesting behaviors present in these treatments. For instance, we associate the anomalous particles behaviors in the quantum pigeonhole effect to a quantum interference of force.