Born rule

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The Born rule (also called Born's rule) is a postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result.[1] In its simplest form, it states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state. It was formulated by German physicist Max Born in 1926.


The Born rule states that if an observable corresponding to a self-adjoint operator with discrete spectrum is measured in a system with normalized wave function (see Bra–ket notation), then:

  • the measured result will be one of the eigenvalues of , and
  • the probability of measuring a given eigenvalue will equal , where is the projection onto the eigenspace of corresponding to .
(In the case where the eigenspace of corresponding to is one-dimensional and spanned by the normalized eigenvector , is equal to , so the probability is equal to . Since the complex number is known as the probability amplitude that the state vector assigns to the eigenvector , it is common to describe the Born rule as saying that probability is equal to the amplitude-squared (really the amplitude times its own complex conjugate). Equivalently, the probability can be written as .)

In the case where the spectrum of is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure , the spectral measure of . In this case:

  • the probability that the result of the measurement lies in a measurable set is given by .

A wave function for a single structureless particle in space position implies that the probability density function for a measurement of the particles's position at time is:

In some applications, this treatment of the Born rule is generalized using positive-operator-valued measures. A POVM is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by self-adjoint observables. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics and can also be used in quantum field theory.[2] They are extensively used in the field of quantum information.

In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices on a Hilbert space that sum to the identity matrix,[3]: 90 :

The POVM element is associated with the measurement outcome , such that the probability of obtaining it when making a measurement on the quantum state is given by:

where is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state this formula reduces to:

The Born rule, together with the unitarity of the time evolution operator (or, equivalently, the Hamiltonian being Hermitian), implies the unitarity of the theory, which is considered required for consistency. For example, unitarity ensures that the probabilities of all possible outcomes sum to 1 (though it is not the only option to get this particular requirement[clarification needed]).


The Born rule was formulated by Born in a 1926 paper.[4] In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Albert Einstein and Einstein’s probabilistic rule for the photoelectric effect,[5] concludes, in a footnote, that the Born rule gives the only possible interpretation of the solution. In 1954, together with Walther Bothe, Born was awarded the Nobel Prize in Physics for this and other work.[5] John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.[6]

Derivation from more basic principles[edit]

Gleason's theorem shows that the Born rule can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957,[7] prompted by a question posed by George W. Mackey.[8][9] This theorem was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics.[10]

Several other researchers have also tried to derive the Born rule from more basic principles. A number of derivations have been proposed in the context of the many-worlds interpretation. These include the decision-theory approach pioneered by David Deutsch[11] and later developed by Hilary Greaves[12] and David Wallace;[13] and an "envariance" approach by Wojciech H. Zurek;[14] These proofs have, however, been criticized as circular.[15] More recently, an approach based on self-locating uncertainty has been suggested by Charles Sebens and Sean M. Carroll.[16]

It has also been claimed that pilot-wave theory can be used to statistically derive the Born rule, though this remains controversial.[17] Kastner claims that the transactional interpretation is unique in giving a physical explanation for the Born rule.[18]

In 2019, Lluis Masanes and Thomas Galley of the Perimeter Institute for Theoretical Physics, and Markus Müller of the Institute for Quantum Optics and Quantum Information presented a derivation of the Born rule.[19] While their result does not use the same initial assumptions as Gleason's theorem, it does presume a Hilbert-space structure and a type of context independence.[20]

Within the QBist interpretation of quantum theory, the Born rule is seen as a modification of the standard law of total probability, which takes into account the Hilbert space dimension of the physical system involved. Rather than trying to derive the Born rule, as many interpretations of quantum mechanics do, QBists take a formulation of the Born rule as primitive and aim to derive as much of quantum theory as possible from it.[21]


  1. ^ The time evolution of a quantum system is entirely deterministic according to the Schrödinger equation. It is through the Born Rule that probability enters into the theory.
  2. ^ Peres, Asher; Terno, Daniel R. (2004). "Quantum information and relativity theory". Reviews of Modern Physics. 76 (1): 93–123. arXiv:quant-ph/0212023. Bibcode:2004RvMP...76...93P. doi:10.1103/RevModPhys.76.93. S2CID 7481797.
  3. ^ Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information (1st ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-63503-5. OCLC 634735192.
  4. ^ Born, Max (1926). "I.2". In Wheeler, J. A.; Zurek, W. H. (eds.). Zur Quantenmechanik der Stoßvorgänge [On the quantum mechanics of collisions]. Zeitschrift für Physik. Vol. 37. Princeton University Press (published 1983). pp. 863–867. Bibcode:1926ZPhy...37..863B. doi:10.1007/BF01397477. ISBN 978-0-691-08316-2. S2CID 119896026.
  5. ^ a b Born, Max (11 December 1954). "The statistical interpretation of quantum mechanics" (PDF). Retrieved 7 November 2018. Again an idea of Einstein's gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the psi-function: |psi|2 ought to represent the probability density for electrons (or other particles).
  6. ^ Neumann (von), John (1932). Mathematische Grundlagen der Quantenmechanik [Mathematical Foundations of Quantum Mechanics]. Translated by Beyer, Robert T. Princeton University Press (published 1996). ISBN 978-0691028934.
  7. ^ Gleason, Andrew M. (1957). "Measures on the closed subspaces of a Hilbert space". Indiana University Mathematics Journal. 6 (4): 885–893. doi:10.1512/iumj.1957.6.56050. MR 0096113.
  8. ^ Mackey, George W. (1957). "Quantum Mechanics and Hilbert Space". The American Mathematical Monthly. 64 (8P2): 45–57. doi:10.1080/00029890.1957.11989120. JSTOR 2308516.
  9. ^ Chernoff, Paul R. (November 2009). "Andy Gleason and Quantum Mechanics" (PDF). Notices of the AMS. 56 (10): 1253–1259.
  10. ^ Mermin, N. David (1993-07-01). "Hidden variables and the two theorems of John Bell". Reviews of Modern Physics. 65 (3): 803–815. arXiv:1802.10119. Bibcode:1993RvMP...65..803M. doi:10.1103/RevModPhys.65.803. S2CID 119546199.
  11. ^ Deutsch, David (8 August 1999). "Quantum Theory of Probability and Decisions". Proceedings of the Royal Society A. 455 (1988): 3129–3137. arXiv:quant-ph/9906015. doi:10.1098/rspa.1999.0443. S2CID 5217034. Retrieved December 5, 2022.
  12. ^ Greaves, Hilary (21 December 2006). "Probability in the Everett Interpretation". Philosophy Compass. 2 (1): 109–128. doi:10.1111/j.1747-9991.2006.00054.x. Retrieved 6 December 2022.
  13. ^ Wallace, David (2009). "A Formal Proof of the Born Rule from Decision-Theoretic Assumptions". arXiv:0906.2718 [quant-ph].
  14. ^ Zurek, Wojciech H. (25 May 2005). "Probabilities from entanglement, Born's rule from envariance". Physical Review A. 71: 052105. arXiv:quant-ph/0405161. doi:10.1103/PhysRevA.71.052105. Retrieved 6 December 2022.
  15. ^ Landsman, N. P. (2008). "The Born rule and its interpretation" (PDF). In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.). Compendium of Quantum Physics. Springer. ISBN 978-3-540-70622-9. The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle
  16. ^ Sebens, Charles T.; Carroll, Sean M. (March 2018). "Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics". The British Journal for the Philosophy of Science. 69 (1): 25–74. doi:10.1093/bjps/axw004.
  17. ^ Goldstein, Sheldon (2017). "Bohmian Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
  18. ^ Kastner, R. E. (2013). The Transactional Interpretation of Quantum Mechanics. Cambridge University Press. p. 35. ISBN 978-0-521-76415-5.
  19. ^ Masanes, Lluís; Galley, Thomas; Müller, Markus (2019). "The measurement postulates of quantum mechanics are operationally redundant". Nature Communications. 10 (1): 1361. arXiv:1811.11060. Bibcode:2019NatCo..10.1361M. doi:10.1038/s41467-019-09348-x. PMC 6434053. PMID 30911009.
  20. ^ Ball, Philip (February 13, 2019). "Mysterious Quantum Rule Reconstructed From Scratch". Quanta Magazine. Archived from the original on 2019-02-13.
  21. ^ Healey, Richard (2016). "Quantum-Bayesian and Pragmatist Views of Quantum Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.

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