The three-box paradox revisited using the Majorana representation of the quantum states

In the quantum three-box paradox, seemingly contradictory conclusions are drawn about the intermediate location of a particle between its initial (pre-selected) state and its final (post-selected) state. We use the Majorana representation of the equivalent 3-level quantum system to reformulate the paradox in terms of the symmetric states of a pair of qubits. This provides us with a geometric description of the paradox on the Bloch sphere. In particular, we find that the phase of the weak values associated with the sensing of the particle location in the weak measurement formalism is related to solid angles defining geometric phases. In the Majorana representation of this paradox, the pre- and post-selected pairs of particles have initial and final states that are separable but their intermediate states are found to be maximally entangled. We describe an experiment which we are currently setting up to investigate this formulation of the paradox using weak measurements of pairs of entangled photons produced by spontaneous parametric down-conversion. Reference: M. Cormann and Y. Caudano, J. Phys. A: Math. Theor. 50 (2017) 305302.