A geometric approach to modular and weak values

  • Mirko Cormann

Student thesis: Doc typesDoctor of Sciences


Weak and modular values are unbounded complex numbers that usually describe observations in weak measurements of pre- and postselected ensembles. In practice, only their real or/and imaginary part have been measured directly so far, or related to observations. Our work brings a wholly new perspective to the research of weak and modular values by studying their polar form. An interferometric measurement of the visibility and the phase in a quantum eraser experiment allows us to probe directly the polar form of weak and modular values: the interferometric visibility is related to the modulus; the phase provides the argument. Our proof-of-concept experiment relies on nonlocal correlations of two qubits (entangled photons), which act as the meter and probed systems, respectively. This system has a fundamental quantum nature; yet it remains relatively simple. The Majorana representation of N-level quantum systems in terms of symmetric states of N-1 qubits provides us with a description on the Bloch sphere. With this geometric approach, weak and modular values of N-level 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 and modular values in two- and three-level systems. This geometric approach opens also the way to describe weak measurements of high-level quantum systems by the manipulation of multi-qubit states. Furthermore, the three-box paradox (a so-called quantum paradox) is analyzed 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.
Date of Award7 Jul 2017
Original languageEnglish
Awarding Institution
  • University of Namur
SupervisorYves Caudano (Supervisor), Bertrand Hespel (Co-Supervisor), Philippe Lambin (President), John Martin (Jury) & Alexandre Matzkin (Jury)


  • quantum optics
  • weak measurement
  • weak value
  • geometric phase
  • Majorana representation

Attachment to an Research Institute in UNAMUR

  • naXys

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