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arXiv:2103.08194 · Physical Review A

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

Weidong Tang

Published Mar 31, 2022• 5 citations
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Abstract

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 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.

DOI
10.1103/physreva.105.032457

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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.
Hardy-like proofs for some quantum mechanical properties, such as Hardy-like nonlocality and Hardy-like contextuality, provide us two probabilistic no-go theorems by logical contradictions. Inspired by this, one may conjure that there are some other nonclassical properties which can also be revealed by Hardy-like arguments. In this work, we present an unconventional kind of quantum pigeonhole effects using Hardy-like arguments and show how to systematically construct them. Besides, we introduce a kind of ``projected-stabilizer state, say projected-coloring graph states, associated with a special type of coloring graphs called projected-coloring graph states. We build a one-to-one connection between a given projected-coloring graph (state) and a Hardy-like quantum pigeonhole In this sense a Hardy-like quantum pigeonhole paradox may also be referred to as a ``quantum coloring paradox.
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.
Here we present the most general framework for nnn-particle Hardy's paradoxes, which include Hardy's original one and Cereceda's extension as special cases. Remarkably, for any n≥3n\ge 3n≥3 we demonstrate that there always exist generalized paradoxes (with the success probability as high as 1/2n−11/2^{n-1}1/2n−1) that are stronger than the previous ones in showing the conflict of quantum mechanics with local realism. An experimental proposal to observe the stronger paradox is also presented for the case of three qubits. Furthermore, from these paradoxes we can construct the most general Hardy's inequalities, which enable us to detect Bell's nonlocality for more quantum states.
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.
Here, we present the most general framework for n-particle Hardy's paradoxes, which include Hardy's original one and Cereceda's extension as special cases. Remarkably, for any n≥3, we demonstrate that there always exist generalized paradoxes (with the success probability as high as 1/2^{n-1}) that are stronger than the previous ones in showing the conflict of quantum mechanics with local realism. An experimental proposal to observe the stronger paradox is also presented for the case of three qubits. Furthermore, from these paradoxes we can construct the most general Hardy's inequalities, which enable us to detect Bell's nonlocality for more quantum states.
Bell nonlocality is a special quantum nonlocality and has emerged as an important resource in quantum information processing tasks. Typically, there are two types of strategies for testing the Bell nonlocality: the Bell inequality method and the “all-versus-nothing proof", the later includes methods such as the GHZ argument and Hardy paradox. The objective of this work is to provide new methods for detecting Bell nonlocality by understanding Hardy-like paradoxes (HLPs) and Hardy-Bell inequalities (HBIs) as well as Hardy inequality. First, the weak HLP (WHLP) is established, and an HBI is obtained for tripartite correlation tensors (CTs) with two inputs and two outcomes. Based on an existing Hardy inequality for an n-qubit pure state, a Hardy inequality is proven for n-partite Bell local CTs. Second, the WHLP and HBI are proven for tripartite states in terms of conditional probabilities. Our WHLP is theoretically and practically easier to construct and more efficient in checking Bell nonlocality compared to the usual HLP.