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TheBorn rule is a postulate ofquantum mechanics that gives theprobability that ameasurement of a quantum system will yield a given result. In one commonly used application, it states that theprobability density for finding a particle at a given position is proportional to the square of the amplitude of the system'swavefunction at that position. It was formulated and published by German physicistMax Born in July 1926.^[1]

The Born rule states that anobservable, measured in a system with normalizedwave function${\displaystyle |\psi ⟩}$ (seeBra–ket notation), corresponds to aself-adjoint operator${\displaystyle A}$ whosespectrum is discrete if:

• the measured result will be one of theeigenvalues${\displaystyle \lambda }$ of${\displaystyle A}$, and
• the probability of measuring a given eigenvalue${\displaystyle {\lambda }_i}$ will equal${\displaystyle ⟨\psi |P_i|\psi ⟩}$, where${\displaystyle P_i}$ is the projection onto theeigenspace of${\displaystyle A}$ corresponding to${\displaystyle {\lambda }_i}$.

(In the case where the eigenspace of${\displaystyle A}$ corresponding to${\displaystyle {\lambda }_i}$ is one-dimensional and spanned by the normalized eigenvector${\displaystyle |{\lambda }_i⟩}$,${\displaystyle P_i}$ is equal to${\displaystyle |{\lambda }_i⟩⟨{\lambda }_i|}$, so the probability${\displaystyle ⟨\psi |P_i|\psi ⟩}$ is equal to${\displaystyle ⟨\psi |{\lambda }_i⟩⟨{\lambda }_i|\psi ⟩}$. Since thecomplex number${\displaystyle ⟨{\lambda }_i|\psi ⟩}$ is known as theprobability amplitude that the state vector${\displaystyle |\psi ⟩}$ assigns to the eigenvector${\displaystyle |{\lambda }_i⟩}$, it is common to describe the Born rule as saying that probability is equal to the amplitude-squared (really the amplitude times its owncomplex conjugate). Equivalently, the probability can be written as${\displaystyle |⟨{\lambda }_i|\psi ⟩{|}^2}$.)

In the case where the spectrum of${\displaystyle A}$ is not wholly discrete, thespectral theorem proves the existence of a certainprojection-valued measure (PVM)${\displaystyle Q}$, the spectral measure of${\displaystyle A}$. In this case:

• the probability that the result of the measurement lies in a measurable set${\displaystyle M}$ is given by${\displaystyle ⟨\psi |Q(M)|\psi ⟩}$.

For example, a single structureless particle can be described by a wave function${\displaystyle \psi }$ that depends upon position coordinates${\displaystyle (x,y,z)}$ and a time coordinate${\displaystyle t}$. The Born rule implies that the probability density function${\displaystyle p}$ for the result of a measurement of the particle's position at time${\displaystyle t_0}$ is:$${\displaystyle p(x,y,z,t_0)=|\psi (x,y,z,t_0){|}^2.}$$The Born rule can also be employed to calculate probabilities (for measurements with discrete sets of outcomes) or probability densities (for continuous-valued measurements) for other observables, like momentum, energy, and angular momentum.

In some applications, this treatment of the Born rule is generalized usingpositive-operator-valued measures (POVM). A POVM is ameasure whose values arepositive semi-definite operators on aHilbert space. POVMs are a generalization of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurements described by self-adjoint observables. In rough analogy, a POVM is to a PVM what amixed state is to apure state. Mixed states are needed to specify the state of a subsystem of a larger system (seepurification 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 inquantum field theory.^[2] They are extensively used in the field ofquantum information.

In the simplest case of a POVM with a finite number of elements acting on a finite-dimensionalHilbert space, a POVM is a set ofpositive semi-definitematrices${\displaystyle \{F_i\}}$ on a Hilbert space${\displaystyle \mathcal{H}}$ that sum to theidentity matrix,:^[3]: 90$${\displaystyle \sum _{i=1}^n{F}_i=I.}$$

The POVM element${\displaystyle F_i}$ is associated with the measurement outcome${\displaystyle i}$, such that the probability of obtaining it when making a measurement on the quantum state${\displaystyle \rho }$ is given by:

$${\displaystyle p(i)=\mathrm{tr}(\rho F_i),}$$

where${\displaystyle \mathrm{tr}}$ is thetrace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state${\displaystyle |\psi ⟩}$ this formula reduces to:$${\displaystyle p(i)=\mathrm{tr}(|\psi ⟩⟨\psi |F_i)=⟨\psi |F_i|\psi ⟩.}$$

The Born rule, together with theunitarity of the time evolution operator${\displaystyle e^{-i\hat{H}t}}$ (or, equivalently, theHamiltonian${\displaystyle \hat{H}}$ beingHermitian), implies theunitarity of the theory: a wave function that is time-evolved by a unitary operator will remain properly normalized. (In the more general case where one considers the time evolution of adensity matrix, proper normalization is ensured by requiring that the time evolution is atrace-preserving, completely positive map.)

The Born rule was formulated by Born in a 1926 paper.^[4] In this paper, Born solves theSchrödinger equation for a scattering problem and, inspired byAlbert Einstein andEinstein's probabilistic rule for thephotoelectric effect,^[5] concludes, in a footnote, that the Born rule gives the only possible interpretation of the solution. (The main body of the article says that the amplitude "gives the probability" [bestimmt die Wahrscheinlichkeit], while the footnote added in proof says that the probability is proportional to the square of its magnitude.) In 1954, together withWalther Bothe, Born was awarded theNobel Prize in Physics for this and other work.^[5]John von Neumann discussed the application ofspectral theory to Born's rule in his 1932 book.^[6]

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 ofnon-contextuality.Andrew M. Gleason first proved the theorem in 1957,^[7] prompted by a question posed byGeorge W. Mackey.^[8][9] This theorem was historically significant for the role it played in showing that wide classes ofhidden-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 themany-worlds interpretation. These include the decision-theory approach pioneered byDavid Deutsch^[11] and later developed byHilary Greaves^[12] and David Wallace;^[13] and an "envariance" approach byWojciech H. Zurek.^[14] These proofs have, however, been criticized as circular.^[15] In 2018, an approach based on self-locating uncertainty was suggested by Charles Sebens andSean M. Carroll;^[16] this has also been criticized.^[17] In 2019, Lluís Masanes, Thomas Galley, and Markus Müller proposed a derivation based on postulates including the possibility of state estimation.^[18][19] In 2021,Simon Saunders produced a branch counting derivation of the Born rule. The crucial feature of this approach is to define the branches so that they all have the same magnitude or2-norm. The ratios of the numbers of branches thus defined give the probabilities of the various outcomes of a measurement, in accordance with the Born rule.^[20]

It has also been claimed thatpilot-wave theory can be used to statistically derive the Born rule, though this remains controversial.^[21]

Within theQBist interpretation of quantum theory, the Born rule is seen as an extension of the normative principle ofcoherence, which ensures self-consistency of probability assessments across a whole set of such assessments. It can be shown that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum-like system but refuses to use the Born rule when placing their bets is vulnerable to aDutch book.^[22]

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