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The **Born rule** (also called **Born's rule**) is a key 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 particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. 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 .

Given a wave function for a single structureless particle in position space, implies that the probability density function for a measurement of the 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).

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] }

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. While it has been claimed that the Born rule can be derived from the many-worlds interpretation, the existing proofs have been criticized as circular.^{ [11] } It has also been claimed that Pilot wave theory can be used to statistically derive the Born rule, though this remains controversial.^{ [12] } Kastner claims that the transactional interpretation is unique in giving a physical explanation for the Born rule.^{ [13] }

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.^{ [14] } 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.^{ [15] }

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.^{ [16] }

The **mathematical formulations of quantum mechanics** are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L^{2} space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

**Quantum mechanics** is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum mechanics, a **density matrix** is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent *mixed states*. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

**Quantum decoherence** is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In mathematics, **spectral theory** is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

In quantum physics, a **measurement** is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

**Quantum statistical mechanics** is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble is described by a density operator *S*, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space *H* describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.

In mathematics, particularly in functional analysis, a **projection-valued measure (PVM)** is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.

In quantum statistical mechanics, the **von Neumann entropy**, named after John von Neumann, is the extension of classical Gibbs entropy concepts to the field of quantum mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is

In functional analysis and quantum measurement theory, a **positive operator-valued measure** (**POVM**) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs.

In quantum mechanics, notably in quantum information theory, **fidelity** is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.

In mathematical physics, **Gleason's theorem** shows that the rule one uses to calculate probabilities in quantum physics, 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, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.

In quantum physics, **unitarity** is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A **unitarity bound** is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space.

In quantum mechanics, the **expectation value** is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the *most* probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.

A **symmetric, informationally complete, positive operator-valued measure** (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem.

In quantum physics, a **quantum state** is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called **pure quantum states**, while all other states are called **mixed quantum states**. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The **Koopman–von Neumann mechanics** is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.

In quantum mechanics, **weak measurements** are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little. From Busch's theorem the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp, fuzzy, dull, noisy, approximate, and gentle measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.

- ↑ 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.
- ↑ 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. - ↑ 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. - ↑ 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*.**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. - 1 2 Born, Max (11 December 1954). "The statistical interpretation of quantum mechanics" (PDF).
*www.nobelprize.org*. nobelprize.org. 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). - ↑ 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. - ↑ 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. - ↑ 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. - ↑ Chernoff, Paul R. (November 2009). "Andy Gleason and Quantum Mechanics" (PDF).
*Notices of the AMS*.**56**(10): 1253–1259. - ↑ 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. - ↑ 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

- ↑ Goldstein, Sheldon (2017). "Bohmian Mechanics". In Zalta, Edward N. (ed.).
*Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University. - ↑ Kastner, R. E. (2013).
*The Transactional Interpretation of Quantum Mechanics*. Cambridge University Press. p. 35. ISBN 978-0-521-76415-5. - ↑ Masanes, Lluís; Galley, Thomas; Müller, Markus (2019). "The measurement postulates of quantum mechanics are operationally redundant".
*Nature Communications*.**10**(1): 1361. doi:10.1038/s41467-019-09348-x. PMC 6434053 . PMID 30911009. - ↑ Ball, Philip (February 13, 2019). "Mysterious Quantum Rule Reconstructed From Scratch]".
*Quanta Magazine*. - ↑ 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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