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Proceedings of the National Academy of Sciences

Reply to Svensson: Quantum violations of the pigeonhole principle

Yakir Aharonov, Fabrizio Colombo, Sandu Popescu, Irene Sabadini, Daniele Carlo Struppa, Jeff Tollaksen

Published May 20, 2016• 3 citations
Publisher

Abstract

"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."

DOI
10.1073/pnas.1602477113

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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."
The pigeonhole principle upholds the idea that by ascribing to three different particles either one of two properties, we necessarily end up in a situation when at least two of the particles have the same property. In quantum physics, this principle is violated in experiments involving postselection of the particles in appropriately-chosen states. Here, we give two explicit constructions using standard gates and measurements that illustrate this fact. Intriguingly, the procedures described are manifestly non-local, which demonstrates that the correlations needed to observe the violation of this principle can be created without direct interactions between particles.
The pigeonhole principle: "If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole," is an obvious yet fundamental principle of nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above "quantum pigeonhole principle" is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.
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?
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.