Footprints of quantum pigeons

We show that in the mathematical framework of the quantum theory, the classical pigeonhole principle can be violated more directly than previously suggested, i.e., in a setting closer to the traditional statement of the principle. We describe 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. After identifying the drawbacks of recent experiments of the quantum pigeonhole effect, we argue that a definitive experimental violation of the pigeonhole principle is still needed and propose such an implementation using modern quantum computing hardware: a superconducting circuit with transmon qubits.

Figure 1

Proposed setup for demonstrating the quantum pigeonhole paradox with superconducting transmons. Each pair $(j,k)$ of three transmons $j,k=1,2,3$ is coupled dispersively to parity readout resonators. The transmon states determine the dispersive shifts ${\chi }_{j,k}$ of the resonator frequencies. Pumping the readout resonators will produce signals $I_{j,k}\left(t\right)$. For task (iii), one resonator is pumped for a long duration to projectively measure a parity eigenvalue ${\sigma }_z^{\left(j\right)}{\sigma }_z^{\left(k\right)}=-1$. For task (iv), all three resonators are pumped for a short time to measure the parity weak values of −1.