Geometric description of modular and weak values in discrete quantum systems using the Majorana representation

Abstract

We express modular and weak values of observables of three- and higher-level quantum systems in their polar form. The Majorana representation of N-level systems in terms of symmetric states of N − 1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N-level quantum systems can be factored in N − 1 contributions. Their modulus is determined by the product of N − 1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N − 1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three-box paradox (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen. 24 2315–28) from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre- and postselected.