Quantum Pigeonhole Research Articles

Experimental implementation of Hardy-like quantum pigeonhole paradoxes

We present the general Hardy-like quantum pigeonhole paradoxes for \textit{n}-particle states, and find that each of such paradoxes can be simply associated to an un-colorable solution of a specific vertex-coloring problem induced from the projected-coloring graph (a kind of unconventional graph). Besides, as a special kind of Hardy's paradox, several kinds of Hardy-like quantum pigeonhole paradoxes can even give rise to higher success probability in demonstrating the conflict between quantum mechanics and local or noncontextual realism than the previous Hardy's paradoxes. Moreover, not only multi-qubit states, but high-dimensional states can exhibit the paradoxes. In contrast to only one type of contradiction presented in the original quantum pigeonhole paradox, two kinds of three-qubit projected-coloring graph states as the minimal illustration are discussed in our work, and an optical experiment to verify such stronger paradox is performed. This quantum paradox provides innovative thoughts and methods in exploring new types of stronger multi-party quantum nonlocality and may have potential applications in multi-party untrusted communications and device-independent random number generation.

Hardy-like quantum pigeonhole paradox and the projected-coloring graph state

A Hardy-like version of the quantum pigeonhole paradox is proposed, which can also be considered as a special kind of Hardy's paradox. Besides an example induced from the minimal system, a general construction of this paradox from an $n$-qubit quantum state is also discussed. Moreover, by introducing the projected-coloring graph and the projected-coloring graph state, a pictorial representation of the Hardy-like quantum pigeonhole paradox can be presented. This Hardy-like version of quantum pigeonhole paradox can be implemented more directly in the experiment than the original one, since it does not require some sophisticated techniques such as weak measurements. In addition, from the angle of Hardy's paradox, some Hardy-like quantum pigeonhole paradoxes can even set a new record for the success probability of demonstrating Bell nonlocality.

Apparent quantum paradoxes as simple interference: Quantum violation of the pigeonhole principle and exchange of properties between quantum particles

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 [Y. Aharonov et al., PNAS 113, 532 (2016)]. An experimental verification of this effect was recently reported [M.-C. Chen et al., PNAS 116, 1549 (2019)]. In another recent experimental work, it was argued that two entities were observed to exchange properties without meeting each other [Z.-H. Liu et al., 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.

Footprints of quantum pigeons

This paper proposes a quantum setup violating the pigeonhole principle. It describes how the counterfactual reasoning of the paradox may be operationally grounded in the analysis of the tiny footprints left in the environment by the pigeons. Identifying the drawbacks of recent experiments of the quantum pigeonhole effect, the authors argue that a definitive experimental violation of the pigeonhole principle is still needed and propose an implementation using modern quantum computing hardware.

Experimental demonstration of quantum pigeonhole paradox

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.

Non-Local Parity Measurements and the Quantum Pigeonhole Effect.

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 quantum pigeonhole principle as a violation of the principle of bivalence

In the paper, it is argued that the phenomenon known as the quantum pigeonhole principle (namely, three quantum particles are put in two boxes, yet no two particles are in the same box) can be explained not as a violation of Dirichlet's box principle in the case of quantum particles but as a nonvalidness of a bivalent logic for describing not-yet verified propositions relating to quantum mechanical experiments.

Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect

Previous experimental tests of quantum contextuality based on the Bell-Kochen-Specker (BKS) theorem have demonstrated that not all observables among a given set can be assigned noncontextual eigenvalue predictions, but have never identified which specific observables must fail such assignment. We now remedy this shortcoming by showing that BKS contextuality can be confined to particular observables by pre- and postselection, resulting in anomalous weak values that we measure using modern neutron interferometry. We construct a confined contextuality witness from weak values, which we measure experimentally to obtain a $5\sigma$ average violation of the noncontextual bound, with one contributing term violating an independent bound by more than $99\sigma$. This weakly measured confined BKS contextuality also confirms the quantum pigeonhole effect, wherein eigenvalue assignments to contextual observables apparently violate the classical pigeonhole principle.

Contextuality, Pigeonholes, Cheshire Cats, Mean Kings, and Weak Values

The Kochen–Specker (KS) theorem shows that noncontextual hidden variable models of reality that allow random choice are inconsistent with quantum mechanics. Such noncontextual models predict certain outcomes for specific experiments that are not observed in practice, and this is how the theorem is proved. A realist hidden-variable model suggested by the Aharonov–Bergmann–Lebowitz reformulation of quantum mechanics is introduced to explain why those outcomes are never observed. Just as the KS theorem requires them due to noncontextuality, this model requires independent truth-value assignments for each observable, but now allows that the entire set of assignments depends on both the pre-selected and post-selected quantum states in a time-symmetric manner. Using sets that prove the KS theorem, along with pre- and post-selected states, we find that particular projectors within the set cannot be assigned logically consistent truth-values, and furthermore that the weak values of these projectors have a corresponding signature. The contextual behavior has effectively been confined to a particular context by the pre- and post-selection. This inconsistency is never observed because it is never in the pre-selected or post-selected quantum state, but its signature can be experimentally revealed through weak measurements. We also show that for specific cases where the logical inconsistency is in a ‘classical basis,’ this gives rise to the quantum pigeonhole effect. As a related issue, we show using weak values that any KS set can be used to ensure that the Mean King always wins the game.

Quantum Weak Values and Logic: An Uneasy Couple

Quantum mechanical weak values of projection operators have been used to answer which-way questions, e.g. to trace which arms in a multiple Mach–Zehnder setup a particle may have traversed from a given initial to a prescribed final state. I show that this procedure might lead to logical inconsistencies in the sense that different methods used to answer composite questions, like “Has the particle traversed the way X or the way Y?”, may result in different answers depending on which methods are used to find the answer. I illustrate the problem by considering some examples: the “quantum pigeonhole” framework of Aharonov et al., the three-box problem, and Hardy’s paradox. To prepare the ground for my main conclusion on the incompatibility in certain cases of weak values and logic, I study the corresponding situation for strong/projective measurements. In this case, no logical inconsistencies occur provided one is always careful in specifying exactly to which ensemble or sample space one refers. My results cast doubts on the utility of quantum weak values in treating cases like the examples mentioned.

Quantum violation of the pigeonhole principle and the nature of quantum correlations

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.

Statistical Mechanics for Weak Measurements and Quantum Inseparability

In weak measurement thought experiment, an ensemble consists of M quantum particles and N states. We observe that separability of the particles is lost, and hence we have fuzzy occupation numbers for the particles in the ensemble. Without sharply measuring each particle state, quantum interferences add extra possible configurations of the ensemble, this explains the Quantum Pigeonhole Principle. This principle adds more entropy to the system; hence the particles seem to have a new kind of correlations emergent from particles not having a single, well-defined state. We formulated the Quantum Pigeonhole Principle in the language of abstract Hilbert spaces, then generalized it to systems consisting of mixed states. This insight into the fundamentals of quantum statistical mechanics could help us understand the interpretation of quantum mechanics more deeply, and possibly have implication on quantum computing and information theory.

NMR investigation of the quantum pigeonhole effect

NMR quantum simulators have been used for studying various quantum phenomena. Here, using a four-qubit NMR quantum simulator, we investigate the recently postulated quantum pigeonhole effect. In this phenomenon, a set of three particles in a two-path interferometer often appears to be in such a superposition that no two particles can be assigned a single path, thus exhibiting the nonclassical behavior. In our experiments, quantum pigeons are emulated by three nuclear qubits whose states are probed jointly and noninvasively by an ancillary spin. The experimental results are in good agreement with quantum theoretical predictions.