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 ${\displaystyle q}$ and its momentum operator is ${\displaystyle p}$, and if the position and momentum of the apparatus are ${\displaystyle Q}$ and ${\displaystyle P}$ respectively, assuming that the total momentum ${\displaystyle p+P}$ 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 ${\displaystyle q-Q}$. The Wigner–Araki–Yanase theorem generalizes this to the case of two arbitrary observables ${\displaystyle A}$ and ${\displaystyle B}$ for the system and an observable ${\displaystyle C}$ for the apparatus, satisfying the condition that ${\displaystyle B+C}$ is conserved.[8][9]

Mikko Tukiainen gave a generalized version of the WAY theorem, which makes no use of conservation laws, but uses quantum incompatibility instead.[10]

Yui Kuramochi and Hiroyasu Tajima proved a generalized form of the theorem for possibly unbounded and continuous conserved observables.[11]

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. arXiv:1209.0921. 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. S2CID 17995482.
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. (2010). "Translation of "Die Messung quantenmechanischer Operatoren" by E.P. Wigner". arXiv:1012.4372 [quant-ph].
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.
10. Tukiainen, Mikko (20 January 2017). "Wigner-Araki-Yanase theorem beyond conservation laws". Physical Review A. 95 (1): 012127. arXiv:1611.05905. doi:10.1103/PhysRevA.95.012127.
11. Kuramochi, Yui; Tajima, Hiroyasu (2023-11-21). "Wigner-Araki-Yanase Theorem for Continuous and Unbounded Conserved Observables". Phys. Rev. Lett. 131: 210201. arXiv:2208.13494. doi:10.1103/PhysRevLett.131.210201. Retrieved 29 November 2023.