Escape from the Quantum Pigeon Conundrum

Abstract

It has recently been argued in that quantum mechanics violates the Pigeon Counting Principle (PCP): if one distributes three pigeons among two boxes there must be at least two pigeons in one of the boxes. The article describes how this Quantum Pigeon Conundrum (QPC) can be observed in an experiment using a Mach-Zehnder interferometer with three weakly coupled electrons. We provide a detailed theoretical analysis of the QPC and new experimental results obtained with the IBM Yorktown quantum processor. Although these results and previous experiments confirm certain predicted amplitudes, we argue that the PCP is not violated in quantum mechanics unless one makes untenable hidden variable assumptions. Simply put, the physical Hilbert space is spanned by states each having at least two particles in the same box. This leads to the quantum version of the PCP, an operator identity that forbids assigning values of zero to all three two-particle projection operators. The statement "there are no two particles in the same box" prior to the projection onto the final state is therefore no more viable than the contention that each particle in the Einstein-Podolsky-Rosen state has specific values for both position and momentum prior to any measurement. Our conclusion is that PCP and QPC provide a novel and elegant pedagogical framework for understanding the (non-)viability of local hidden variables but do not constitute a radically new quantum effect as suggested by Aharonov et al.