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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.