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

Mirko Cormann, Yves Caudano

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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.

Original languageEnglish
Article number305302
Number of pages26
JournalJournal of Physics A: Mathematical and Theoretical
Volume50
Issue number30
DOIs
Publication statusPublished - 29 Jun 2017

Keywords

  • geometric phase
  • Majorana representation
  • modular value
  • three-box paradox
  • weak measurement
  • Weak value

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