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

TL;DR

This paper presents a geometric framework using the Majorana representation to analyze modular and weak values in N-level quantum systems, revealing their structure on the Bloch sphere and explaining phase discontinuities.

Contribution

It introduces a novel geometric approach to express modular and weak values in terms of Majorana representation, linking them to solid angles and projection ratios.

Findings

Modular and weak values factor into N-1 contributions.

The modulus is a product of projection probability ratios.

The argument relates to sums of solid angles on the Bloch sphere.

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 [1] 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.