Wigner–Araki–Yanase theorem

The Wigner–Araki–Yanase theorem, also known as the WAY theorem, is a result in quantum physics establishing that the presence of a conservation law limits the accuracy with which observables that fail to commute with the conserved quantity can be measured.[1][2][3] It is named for the physicists Eugene Wigner,[4] Huzihiro Araki and Mutsuo Yanase.[5][6]

The theorem can be illustrated with a particle coupled to a measuring apparatus.[7]:421 If the position operator of the particle is and its momentum operator is , and if the position and momentum of the apparatus are and respectively, assuming that the total momentum is conserved implies that, in a suitably quantified sense, the particle's position itself cannot be measured. The measurable quantity is its position relative to the measuring apparatus, represented by the operator . The Wigner–Araki–Yanase theorem generalizes this to the case of two arbitrary observables and for the system and an observable for the apparatus, satisfying the condition that is conserved.[8][9]

References
  1. Baez, John C. (1994-05-10). "Week 33". This Week's Finds in Mathematical Physics. Retrieved 2020-02-10.
  2. Ahmadi, Mehdi; Jennings, David; Rudolph, Terry (2013-01-28). "The Wigner–Araki–Yanase theorem and the quantum resource theory of asymmetry". New Journal of Physics. 15 (1): 013057. doi:10.1088/1367-2630/15/1/013057. ISSN 1367-2630.
  3. Loveridge, L.; Busch, P. (2011). "'Measurement of quantum mechanical operators' revisited". The European Physical Journal D. 62 (2): 297–307. arXiv:1012.4362. Bibcode:2011EPJD...62..297L. doi:10.1140/epjd/e2011-10714-3. ISSN 1434-6060.
  4. Wigner, E. P. (1995), Mehra, Jagdish (ed.), "Die Messung quantenmechanischer Operatoren", Philosophical Reflections and Syntheses, Springer Berlin Heidelberg, pp. 147–154, doi:10.1007/978-3-642-78374-6_10, ISBN 978-3-540-63372-3. For an English translation, see Busch, P. "Translation of "Die Messung quantenmechanischer Operatoren" by E.P. Wigner". arXiv:1012.4372.
  5. Araki, Huzihiro; Yanase, Mutsuo M. (1960-10-15). "Measurement of Quantum Mechanical Operators". Physical Review. 120 (2): 622–626. doi:10.1103/PhysRev.120.622. ISSN 0031-899X.
  6. Yanase, Mutsuo M. (1961-07-15). "Optimal Measuring Apparatus". Physical Review. 123 (2): 666–668. doi:10.1103/PhysRev.123.666. ISSN 0031-899X.
  7. Peres, Asher (1995). Quantum Theory: Concepts and Methods. Kluwer Academic Publishers. ISBN 0-7923-2549-4.
  8. Ghirardi, G. C.; Miglietta, F.; Rimini, A.; Weber, T. (1981-07-15). "Limitations on quantum measurements. I. Determination of the minimal amount of nonideality and identification of the optimal measuring apparatuses". Physical Review D. 24 (2): 347–352. doi:10.1103/PhysRevD.24.347. ISSN 0556-2821.
  9. Ghirardi, G. C.; Miglietta, F.; Rimini, A.; Weber, T. (1981-07-15). "Limitations on quantum measurements. II. Analysis of a model example". Physical Review D. 24 (2): 353–358. doi:10.1103/PhysRevD.24.353. ISSN 0556-2821.
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