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arXiv:2306.17743

Quantum paradoxical knowledge

Benjamin W. Schumacher, Michael D. Westmoreland

Published Jun 30, 2023• 0 citations
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Abstract

We generalize the quantum "pigeonhole paradox" to quantum paradoxes involving arbitrary types of particle relations, including orderings, functions and graphs.
DOI
10.48550/arxiv.2306.17743

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The general Hardy-like quantum pigeonhole paradoxes for n-qubit states were presented by Tang [W. Tang, Phys. Rev. A 105, 032457 (2022)]. It has been shown that each of such paradoxes can be associated with an uncolorable solution of a specific vertex-coloring problem induced from the projected-coloring graph (a kind of unconventional graph). The Hardy-like quantum pigeonhole paradoxes can even give rise to a higher suc- cess probability in demonstrating the conflict between quantum mechanics and local or noncontextual realism than the previous Hardy's paradoxes. Moreover, multiqubit states and high-dimensional states can exhibit paradoxes. In contrast to only one type of contradiction presented in the original quantum pigeonhole paradox, we discussed two kinds of three-qubit projected-coloring graph states as the minimal illustration in our work. An optical experiment to verify such a stronger paradox is performed. This quantum paradox provides innovative thoughts and methods for exploring new types of stronger multiparty quantum nonlocality. It may have potential applications in untrusted multiparty communications and device-independent random number generation.
Coherence as a resource for NNN-box and quantum pigeonhole paradoxes
Pre- and post-selection (PPS) paradoxes are striking demonstrations of quantum nonclassicality. Logical PPS paradoxes, where inferences made with the Aharonov-Bergmann-Lebowitz (ABL) rule are exactly 0 or 1, are linked to contextuality. Non-logical paradoxes lack this strong signature. In this work, we analyse more general, non-logical PPS scenarios involving mixed pre- and post-selected states. We show that two such scenarios, the NNN-box and quantum pigeonhole paradoxes, require coherence of both pre- and post-selected states in the basis of the intermediate measurement. This is done by showing each paradox holds if and only if there is weak-value anomaly for a single, paradox-specific operator, together with the fact that weak-value anomaly requires coherence. This clarifies the role of different notions of nonclassicality in these scenarios, highlighting the required quantitative departures from (strictly) classical explanations provided by incoherent sub-theories of quantum theory.
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 nnn-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 the quantum pigeonhole paradox can be implemented more directly in 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 may be able to set a new record for the success probability of demonstrating Bell nonlocality.