Complementary Observables in Quantum Mechanics

Jukka Kiukas, Pekka Lahti, Juha-Pekka Pellonpaa, Kari Ylinen

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We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity—some of which new—in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position–momentum, position–energy, number–phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul’s work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.
Original languageEnglish
Pages (from-to)506-531
Number of pages26
JournalFoundations of Physics
Issue number6
Early online date03 May 2019
Publication statusPublished - 15 Jun 2019


  • quantum observables
  • positive operator measures
  • effects
  • complementarity
  • joint measurability
  • position - momentum pair
  • position - energy pair
  • momentum - energy pair
  • time - energy pair
  • phase - number pair
  • Quantum observables
  • Complementarity
  • Positive operator measures
  • Joint measurability
  • Effects
  • Position–momentum pair
  • Phase–number pair
  • Time–energy pair
  • Momentum–energy pair
  • Position–energy pair


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