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{{Short description|Theorem in quantum mechanics}}
In [[Physics:Mathematical physics|mathematical physics]], '''Gleason's theorem''' shows that the rule one uses to calculate [[Probability|probabilities]] in quantum physics, the [[Physics:Born rule|Born rule]], can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of [[Physics:Quantum contextuality|non-contextuality]]. [[Biography:Andrew M. Gleason|Andrew M. Gleason]] first proved the theorem in 1957,<ref name="gleason1957"/> answering a question posed by [[Biography:George Mackey|George W. Mackey]], an accomplishment that was historically significant for the role it played in showing that wide classes of [[Hidden-variable theory|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 [[Physics:Quantum logic|quantum logic]] and its attempt to find a minimal set of mathematical [[Axiom|axiom]]s for quantum theory.

== Statement of the theorem ==
{{Quantum mechanics|advanced}}

===Conceptual background===
In quantum mechanics, each physical system is associated with a [[Hilbert space]]. For the purposes of this overview, the Hilbert space is assumed to be finite-dimensional. In the approach codified by [[Biography:John von Neumann|John von Neumann]], a measurement upon a physical system is represented by a [[Self-adjoint operator|self-adjoint operator]] on that Hilbert space sometimes termed an "observable". The eigenvectors of such an operator form an [[Orthonormal basis|orthonormal basis]] for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis. A density operator is a positive-semidefinite operator on the Hilbert space whose trace is equal to 1. In the language of [[Biography:Carl Friedrich von Weizsäcker|von Weizsäcker]], a density operator is a "catalogue of probabilities": for each measurement that can be defined, the probability distribution over the outcomes of that measurement can be computed from the density operator.<ref>{{Cite journal| last1=Drieschner |first1=M.|last2=Görnitz|first2=Th.|last3=von Weizsäcker|first3=C. F.| date=1988-03-01 |title=Reconstruction of abstract quantum theory|journal=[[Physics:International Journal of Theoretical Physics|International Journal of Theoretical Physics]] |  language=en|volume=27|issue=3|pages=289–306|doi=10.1007/bf00668895|issn=0020-7748|bibcode=1988IJTP...27..289D|s2cid=122866239}}</ref> The procedure for doing so is the [[Physics:Born rule|Born rule]], which states that
<math display="block">P(x_i) = \operatorname{Tr}(\Pi_i \rho),</math>
where <math>\rho</math> is the density operator, and <math>\Pi_i</math> is the [[Projection (linear algebra)|projection operator]] onto the basis vector corresponding to the measurement outcome <math>x_i</math>.

The Born rule associates a probability with each unit vector in the Hilbert space, in such a way that these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis. Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like a choice of basis for that vector to be embedded in. Gleason's theorem establishes the converse: all assignments of probabilities to unit vectors (or, equivalently, to the operators that project onto them) that satisfy these conditions take the form of applying the Born rule to some density operator. Gleason's theorem holds if the dimension of the Hilbert space is 3 or greater; counterexamples exist for dimension 2.

===Deriving the state space and the Born rule===
The probability of any outcome of a measurement upon a quantum system must be a real number between 0 and 1 inclusive, and in order to be consistent, for any individual measurement the probabilities of the different possible outcomes must add up to 1.  Gleason's theorem shows that any function that assigns probabilities to measurement outcomes, as identified by projection operators, must be expressible in terms of a density operator and the Born rule. This gives not only the rule for calculating probabilities, but also determines the set of possible quantum states.

Let <math>f</math> be a function from projection operators to the [[Unit interval|unit interval]] with the property that, if a set <math>\{ \Pi_i\}</math> of projection operators sum to the [[Identity matrix|identity matrix]] (that is, if they correspond to an orthonormal basis), then
<math display="block">\sum_i f(\Pi_i) = 1.</math>

Such a function expresses an assignment of probability values to the outcomes of measurements, an assignment that is "noncontextual" in the sense that the probability for an outcome does not depend upon which measurement that outcome is embedded within, but only upon the mathematical representation of that specific outcome, i.e., its projection operator.<ref name="barnum2000">{{Cite journal|last1=Barnum|first1=H.|last2=Caves|first2=C. M.|last3=Finkelstein|first3=J.|last4=Fuchs|first4=C. A.|last5=Schack|first5=R.|date=2000-05-08|title=Quantum probability from decision theory?|url=http://rspa.royalsocietypublishing.org/content/456/1997/1175|journal=Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences|language=en|volume=456|issue=1997|pages=1175–1182|arxiv=quant-ph/9907024|doi=10.1098/rspa.2000.0557|issn=1364-5021|bibcode=2000RSPSA.456.1175B|citeseerx=10.1.1.769.8732|s2cid=11563591}}</ref><ref name="pitowsky2003">{{Cite journal|last=Pitowsky|first=Itamar|date=2003|title=Betting on the outcomes of measurements: a Bayesian theory of quantum probability|journal=Studies in History and Philosophy of Modern Physics|volume=34|issue=3|pages=395–414|arxiv=quant-ph/0208121|doi=10.1016/S1355-2198(03)00035-2|bibcode=2003SHPMP..34..395P}}</ref>{{rp|§1.3}}<ref name="pitowsky2006">{{Cite book|title=Physical Theory and its Interpretation: Essays in Honor of Jeffrey Bub|last=Pitowsky|first=Itamar|publisher=[[Company:Springer Science+Business Media|Springer]]|year=2006|isbn=9781402048760|editor-last=Demopoulos|editor-first=William|pages=213|chapter=Quantum mechanics as a theory of probability|arxiv=quant-ph/0510095|editor-last2=Pitowsky|editor-first2=Itamar|bibcode=2005quant.ph.10095P |oclc=917845122}}</ref>{{rp|§2.1}}<ref>{{Cite journal |last1=Kunjwal |first1=Ravi|last2=Spekkens |first2=Rob W. |date=2015-09-09 |title=From the Kochen–Specker theorem to noncontextuality inequalities without assuming determinism |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=115 |issue=11 |pages=110403 |arxiv=1506.04150 |doi=10.1103/PhysRevLett.115.110403 |pmid=26406812 |bibcode=2015PhRvL.115k0403K| s2cid=10308680 }}</ref> Gleason's theorem states that for any such function <math>f</math>, there exists a positive-semidefinite operator <math>\rho</math> with unit trace such that
<math display="block">f(\Pi_i) = \operatorname{Tr}(\Pi_i \rho).</math>

Both the Born rule and the fact that "catalogues of probability" are positive-semidefinite operators of unit trace follow from the assumptions that measurements are represented by orthonormal bases, and that probability assignments are "noncontextual". In order for Gleason's theorem to be applicable, the space on which measurements are defined must be a real or complex Hilbert space, or a quaternionic [[Module (mathematics)|module]].{{efn|For additional discussion on this point, see Piron,<ref>{{Cite journal |last=Piron |first=C. |date=1972-10-01 |title=Survey of general quantum physics |journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |language=en |volume=2 |issue=4 |pages=287–314 |doi=10.1007/bf00708413 |issn=0015-9018 |bibcode=1972FoPh....2..287P|s2cid=123364715 }}</ref>{{rp|§6}} Drisch,<ref>{{Cite journal|last=Drisch|first=Thomas|date=1979-04-01|title=Generalization of Gleason's theorem|journal=[[Physics:International Journal of Theoretical Physics|International Journal of Theoretical Physics]]|language=en|volume=18|issue=4|pages=239–243| doi=10.1007/bf00671760|issn=0020-7748| bibcode=1979IJTP...18..239D|s2cid=121825926}}</ref> Horwitz and Biedenharn,<ref>{{Cite journal |last1=Horwitz |first1=L. P. |last2=Biedenharn |first2=L. C. |title=Quaternion quantum mechanics: Second quantization and gauge fields |year=1984 |journal=[[Physics:Annals of Physics|Annals of Physics]] |volume=157 |issue=2 |pages=432–488 |doi=10.1016/0003-4916(84)90068-x |bibcode=1984AnPhy.157..432H}}</ref> Razon and Horwitz,<ref>{{Cite journal|last1=Razon|first1=Aharon|last2=Horwitz|first2=L. P.|date=1991-08-01|title=Projection operators and states in the tensor product of quaternion Hilbert modules|journal=[[Acta Applicandae Mathematicae]]| language=en|volume=24|issue=2|pages=179–194|doi=10.1007/bf00046891| s2cid=119666741| issn=0167-8019}}</ref> Varadarajan,<ref>{{Cite book |last=Varadarajan |first=Veeravalli S. |author-link=Veeravalli S. Varadarajan |title=Geometry of Quantum Theory |edition=2nd |publisher=[[Company:Springer Science+Business Media|Springer Science+Business Media]] |year=2007 |isbn=978-0-387-96124-8 |oclc=764647569}}</ref>{{rp|83}} Cassinelli and Lahti,<ref name="cassinelli2017"/>{{rp|§2}} and Moretti and Oppio.<ref>{{Cite journal| last1=Moretti|first1=Valter|last2=Oppio|first2=Marco|date=2018-10-16|title=The Correct Formulation of Gleason's Theorem in Quaternionic Hilbert Spaces|journal=[[Annales Henri Poincaré]]|volume=19| issue=11|pages=3321–3355| doi=10.1007/s00023-018-0729-8| arxiv=1803.06882| bibcode=2018AnHP...19.3321M|s2cid=53630146}}</ref>}} (Gleason's argument is inapplicable if, for example, one tries to construct an [[Physics:P-adic quantum mechanics|analogue of quantum mechanics using ''p''-adic numbers]].)

== History and outline of Gleason's proof ==
[[File:GleasonAndrewMattei_Berlin1959.jpg|thumb|Gleason in 1959]]
In 1932, [[Biography:John von Neumann|John von Neumann]] also managed to derive the Born rule in his textbook ''[[Organization:Mathematical Foundations of Quantum Mechanics|Mathematische Grundlagen der Quantenmechanik]]'' [''Mathematical Foundations of Quantum Mechanics'']. However, the assumptions on which von Neumann built his proof were rather strong and eventually regarded to not be well-motivated.<ref name="bell1966">{{cite journal |first=John |last=Bell|title=On the Problem of Hidden Variables in Quantum Mechanics |journal=[[Physics:Reviews of Modern Physics|Reviews of Modern Physics]] |date=1966 |volume=38 |issue=3 |page=447 |doi=10.1103/RevModPhys.38.447|bibcode=1966RvMP...38..447B |osti=1444158 }}</ref> Specifically, von Neumann assumed that the probability function must be linear on all observables, commuting or non-commuting. His proof was derided by [[Biography:John Stewart Bell|John Bell]] as "not merely false but foolish!".<ref>{{cite journal |first=Jeffrey |last=Bub|author-link=Jeffrey Bub |title=Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal |journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |date=2010 |volume=40 |issue=9–10 |pages=1333–1340 |doi=10.1007/s10701-010-9480-9|arxiv=1006.0499|bibcode=2010FoPh...40.1333B |s2cid=118595119 }}</ref><ref name="mermin2018"/> Gleason, on the other hand, did not assume linearity, but merely additivity for commuting projectors together with noncontextuality, assumptions seen as better motivated and more physically meaningful.<ref name="mermin2018">{{cite journal|title=Homer nodded: von Neumann's surprising oversight |year=2018 |arxiv=1805.10311 |last1=Mermin |first1=N. David |last2=Schack |first2=Rüdiger |author-link1=N. David Mermin |doi=10.1007/s10701-018-0197-5 |journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |volume=48 |number=9 |pages=1007–1020 |bibcode=2018FoPh...48.1007M|s2cid=118951033 }}</ref><ref>{{cite journal|last=Peres |first=Asher |author-link=Asher Peres |date=1992 |doi=10.1016/0375-9601(92)91005-C |title=An experimental test for Gleason's theorem |journal=Physics Letters A |volume=163 |issue=4 |pages=243–245|bibcode=1992PhLA..163..243P }}</ref>

By the late 1940s, George Mackey had grown interested in the mathematical foundations of quantum physics, wondering in particular whether the Born rule was the only possible rule for calculating probabilities in a theory that represented measurements as orthonormal bases on a Hilbert space.<ref>{{Cite journal |last=Mackey |first=George W. |title=Quantum Mechanics and Hilbert Space |journal=[[The American Mathematical Monthly]] |year=1957 |volume=64 |number=8P2 |pages=45–57 |doi=10.1080/00029890.1957.11989120 |jstor=2308516}}</ref><ref name="chernoff2009">{{Cite journal|last=Chernoff |first=Paul R. |author-link=Paul Chernoff |title=Andy Gleason and Quantum Mechanics |journal=Notices of the AMS |volume=56 |number=10 |pages=1253–1259 |url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}</ref> Mackey discussed this problem with Irving Segal at the [[Organization:University of Chicago|University of Chicago]], who in turn raised it with Richard Kadison, then a graduate student. Kadison showed that for 2-dimensional Hilbert spaces there exists a probability measure that does not correspond to quantum states and the Born rule. Gleason's result implies that this only happens in dimension 2.<ref name="chernoff2009"/>

Gleason's original proof proceeds in three stages.<ref name="hp2004">{{Cite journal |last1=Hrushovski |first1=Ehud |last2=Pitowsky |first2=Itamar |date=2004-06-01 |title=Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |volume=35 |issue=2 |pages=177–194 |doi=10.1016/j.shpsb.2003.10.002 |arxiv=quant-ph/0307139 |bibcode=2004SHPMP..35..177H|s2cid=15265001 }}</ref>{{rp|§2}} In Gleason's terminology, a ''frame function'' is a real-valued function <math>f</math> on the unit sphere of a Hilbert space such that
<math display="block">\sum_i f(x_i) = 1</math>
whenever the vectors <math>x_i</math> comprise an orthonormal basis. A noncontextual probability assignment as defined in the previous section is equivalent to a frame function.{{efn|Gleason allows for the possibility that a frame function is normalized to a constant other than 1, but focusing on the case of "unit weight" as done here does not result in any [[Without loss of generality|loss of generality]].}} Any such measure that can be written in the standard way, that is, by applying the Born rule to a quantum state, is termed a ''regular'' frame function. Gleason derives a sequence of [[Lemma (mathematics)|lemmas]] concerning when a frame function is necessarily regular, culminating in the final theorem. First, he establishes that every [[Continuous function|continuous]] frame function on the Hilbert space <math>\mathbb{R}^3</math> is regular. This step makes use of the theory of [[Physics:Spherical harmonics|spherical harmonics]]. Then, he proves that frame functions on <math>\mathbb{R}^3</math> have to be continuous, which establishes the theorem for the special case of <math>\mathbb{R}^3</math>. This step is regarded as the most difficult of the proof.<ref name="cooke1985">{{Cite journal|last1=Cooke |first1=Roger |last2=Keane |first2=Michael |last3=Moran |first3=William |title=An elementary proof of Gleason's theorem |journal=[[Organization:Mathematical Proceedings of the Cambridge Philosophical Society|Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=98 |number=1 |year=1985 |pages=117–128 |doi=10.1017/S0305004100063313|bibcode=1985MPCPS..98..117C |s2cid=124627182 }}</ref><ref name="pitowsky1998">{{cite journal|last=Pitowsky |first=Itamar |title=Infinite and finite Gleason's theorems and the logic of indeterminacy |doi=10.1063/1.532334 |journal=[[Physics:Journal of Mathematical Physics|Journal of Mathematical Physics]] |volume=39 |number=1 |year=1998 |pages=218–228 |bibcode=1998JMP....39..218P|doi-access=free }}</ref> Finally, he shows that the general problem can be reduced to this special case. Gleason credits one lemma used in this last stage of the proof to his doctoral student [[Biography:Richard Palais|Richard Palais]].<ref name="gleason1957">{{cite journal|first=Andrew M.|year = 1957|title = Measures on the closed subspaces of a Hilbert space|url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050|journal = [[Organization:Indiana University Mathematics Journal|Indiana University Mathematics Journal]]|volume = 6|issue=4|pages = 885–893|doi=10.1512/iumj.1957.6.56050|mr=0096113|last = Gleason|doi-access = free}}</ref>{{rp|fn 3}}

[[Biography:Robin Lyth Hudson|Robin Lyth Hudson]] described Gleason's theorem as "celebrated and notoriously difficult".<ref>{{Cite journal|last=Hudson |first=Robin Lyth |title=Geometry of quantum theory |journal=The Mathematical Gazette |year=1986 |volume=70 |number=454 |pages=332–333 |doi=10.2307/3616230|jstor=3616230 }}</ref> Cooke, Keane and Moran later produced a proof that is longer than Gleason's but requires fewer prerequisites.<ref name="cooke1985"/>

== Implications ==
Gleason's theorem highlights a number of fundamental issues in quantum measurement theory. As Fuchs argues, the theorem "is an extremely powerful result", because "it indicates the extent to which the Born probability rule and even the state-space structure of density operators are ''dependent'' upon the theory's other postulates". In consequence, quantum theory is "a tighter package than one might have first thought".<ref name="fuchs2011">{{cite book|first=Christopher A.|last=Fuchs|title=Coming of Age with Quantum Information: Notes on a Paulian Idea|year = 2011|publisher = {{wipe|Cambridge University Press}}|isbn = 978-0-521-19926-1|location = Cambridge |oclc=535491156}}</ref>{{rp|94–95}} Various approaches to rederiving the quantum formalism from alternative axioms have, accordingly, employed Gleason's theorem as a key step, bridging the gap between the structure of Hilbert space and the Born rule.{{efn|This is discussed by Barnum et al.,<ref name="barnum2000"/> Cassinelli and Lahti,<ref name="cassinelli2017">{{Cite journal |last1=Cassinelli |first1=G. |last2=Lahti |first2=P. |date=2017-11-13 |title=Quantum mechanics: why complex Hilbert space? |journal=Philosophical Transactions of the Royal Society A |language=en |volume=375 |issue=2106 |pages=20160393 |doi=10.1098/rsta.2016.0393 |issn=1364-503X |pmid=28971945 |bibcode=2017RSPTA.37560393C|doi-access=free }}</ref>{{rp|§2}} Stairs,<ref>{{cite journal|last=Stairs |first=Allen |title=Quantum Logic and Quantum Reconstruction |journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |volume=45 |year=2015 |issue=10 |pages=1351–1361 |doi=10.1007/s10701-015-9879-4 |arxiv=1501.05492 |bibcode=2015FoPh...45.1351S|s2cid=126435 }}</ref> and Wilce.<ref name="wilce2017"/>{{rp|§1.4}}}}

===Hidden variables===
Moreover, the theorem is historically significant for the role it played in ruling out the possibility of certain classes of [[Hidden-variable theory|hidden variables]] in quantum mechanics. A hidden-variable theory that is [[Determinism|deterministic]] implies that the probability of a given outcome is ''always'' either 0 or 1. For example, a [[Physics:Stern–Gerlach experiment|Stern–Gerlach measurement]] on a [[Physics:Spin|spin-1]] atom will report that the atom's angular momentum along the chosen axis is one of three possible values, which can be designated <math>-</math>, <math>0</math> and <math>+</math>. In a deterministic hidden-variable theory, there exists an underlying physical property that fixes the result found in the measurement. Conditional on the value of the underlying physical property, any given outcome (for example, a result of <math>+</math>) must be either impossible or guaranteed. But Gleason's theorem implies that there can be no such deterministic probability measure. The mapping <math>u \to \langle \rho u, u \rangle</math> is continuous on the [[Unit sphere|unit sphere]] of the Hilbert space for any density operator <math>\rho</math>. Since this unit sphere is connected, no continuous probability measure on it can be deterministic.<ref name="wilce2017">{{cite SEP|last=Wilce |first=A. |date=2021-08-10 |url-id=qt-quantlog |title=Quantum Logic and Probability Theory |edition=Spring 2017}}</ref>{{rp|§1.3}} Gleason's theorem therefore suggests that quantum theory represents a deep and fundamental departure from the [[Physics:Classical mechanics|classical]] intuition that uncertainty is due to ignorance about hidden degrees of freedom.<ref name="mermin1993">{{Cite journal|last=Mermin|first=N. David|date=1993-07-01|title=Hidden variables and the two theorems of John Bell|journal=[[Physics:Reviews of Modern Physics|Reviews of Modern Physics]]|volume=65|issue=3|pages=803–815|doi=10.1103/RevModPhys.65.803|bibcode=1993RvMP...65..803M|arxiv=1802.10119|s2cid=119546199}}</ref> More specifically, Gleason's theorem rules out hidden-variable models that are "noncontextual". Any hidden-variable model for quantum mechanics must, in order to avoid the implications of Gleason's theorem, involve hidden variables that are not properties belonging to the measured system alone but also dependent upon the external context in which the measurement is made. This type of dependence is often seen as contrived or undesirable; in some settings, it is inconsistent with [[Physics:Special relativity|special relativity]].<ref name="mermin1993"/><ref>{{Cite journal|last=Shimony |first=Abner |author-link=Abner Shimony |title=Contextual Hidden Variable Theories and Bell's Inequalities |journal=British Journal for the Philosophy of Science |year=1984 |volume=35 |number=1 |pages=25–45 |doi=10.1093/bjps/35.1.25}}</ref>

[[File:Bloch sphere.svg|thumb|In the Bloch sphere representation of a [[Qubit|qubit]], each point on the unit sphere stands for a pure state. All other density matrices correspond to points in the interior.]]
To construct a counterexample for 2-dimensional Hilbert space, known as a [[Qubit|qubit]], let the hidden variable be a unit vector <math>\vec{\lambda}</math> in 3-dimensional Euclidean space. Using the [[Physics:Bloch sphere|Bloch sphere]], each possible measurement on a qubit can be represented as a pair of [[Antipodal point|antipodal point]]s on the unit sphere. Defining the probability of a measurement outcome to be 1 if the point representing that outcome lies in the same hemisphere as <math>\vec{\lambda}</math> and 0 otherwise yields an assignment of probabilities to measurement outcomes that obeys Gleason's assumptions. However, this probability assignment does not correspond to any valid density operator. By introducing a probability distribution over the possible values of <math>\vec{\lambda}</math>, a hidden-variable model for a qubit that reproduces the predictions of quantum theory can be constructed.<ref name="mermin1993"/><ref name=":0">{{cite journal|first1=Nicholas |last1=Harrigan |first2=Robert W. |last2=Spekkens |title=Einstein, incompleteness, and the epistemic view of quantum states |journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |volume=40 |pages=125–157 |year=2010 |issue=2 |doi=10.1007/s10701-009-9347-0 |arxiv=0706.2661|bibcode=2010FoPh...40..125H |s2cid=32755624 }}</ref>

Gleason's theorem motivated later work by [[Biography:John Stewart Bell|John Bell]], [[Biography:Ernst Specker|Ernst Specker]] and Simon Kochen that led to the result often called the [[Kochen–Specker theorem]], which likewise shows that noncontextual hidden-variable models are incompatible with quantum mechanics. As noted above, Gleason's theorem shows that there is no probability measure over the rays of a Hilbert space that only takes the values 0 and 1 (as long as the dimension of that space exceeds 2). The Kochen–Specker theorem refines this statement by constructing a specific finite subset of rays on which no such probability measure can be defined.<ref name="mermin1993"/><ref>{{Cite journal|last=Peres|first=Asher|date=1991|title=Two simple proofs of the Kochen-Specker theorem|url=http://stacks.iop.org/0305-4470/24/i=4/a=003|journal=Journal of Physics A: Mathematical and General|language=en|volume=24|issue=4|pages=L175–L178|doi=10.1088/0305-4470/24/4/003|issn=0305-4470|bibcode=1991JPhA...24L.175P}}</ref> The fact that such a finite subset of rays must exist follows from Gleason's theorem by way of a [[Compactness theorem|logical compactness]] argument, but this method does not construct the desired set explicitly.<ref name="hp2004"/>{{rp|§1}} In the related no-hidden-variables result known as [[Bell's theorem]], the assumption that the hidden-variable theory is noncontextual instead is replaced by the assumption that it is [[Local hidden-variable theory|local]]. The same sets of rays used in Kochen–Specker constructions can also be employed to derive Bell-type proofs.<ref name="mermin1993"/><ref>{{cite journal|last=Stairs |first=Allen |title=Quantum Logic, Realism, and Value Definiteness |journal=Philosophy of Science |year=1983 |volume=50 |number=4 |pages=578–602 |doi=10.1086/289140|s2cid=122885859 }}</ref><ref>{{cite journal|last1=Heywood |first1=Peter |last2=Redhead |first2=Michael L. G. |title=Nonlocality and the Kochen–Specker paradox |journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |year=1983 |volume=13 |issue=5 |pages=481–499 |doi=10.1007/BF00729511|bibcode=1983FoPh...13..481H |s2cid=120340929 }}</ref>

Pitowsky uses Gleason's theorem to argue that quantum mechanics represents a new theory of probability, one in which the structure of the space of possible events is modified from the classical, Boolean algebra thereof. He regards this as analogous to the way that special relativity modifies the [[Physics:Kinematics|kinematics]] of [[Physics:Newtonian mechanics|Newtonian mechanics]].<ref name="pitowsky2003"/><ref name="pitowsky2006"/>

The Gleason and Kochen–Specker theorems have been cited in support of various philosophies, including [[Philosophy:Perspectivism|perspectivism]], [[Philosophy:Constructive empiricism|constructive empiricism]] and agential realism.<ref>{{cite journal
 |  author = Edwards, David
 |  title = The Mathematical Foundations of Quantum Mechanics
 |  journal = Synthese
 |  volume = 42
 |  year = 1979
 |  pages = 1–70
 | doi=10.1007/BF00413704
 | s2cid = 46969028
 }}</ref><ref>{{Cite book |last=van Fraassen |first=Bas |title=Quantum Mechanics: An Empiricist View |year=1991 |publisher=Clarendon Press |isbn=9780198239802 |oclc=1005285550 |url-access=registration |url=https://archive.org/details/quantummechanics0000vanf }}</ref><ref>{{cite book |last=Barad |first=Karen |author-link=Karen Barad |title = Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning |publisher = Duke University Press |year = 2007 |isbn = 9780822339175 |oclc = 894219980}}</ref>

===Quantum logic===
Gleason's theorem finds application in quantum logic, which makes heavy use of [[Lattice (order)|lattice theory]].  Quantum logic treats the outcome of a [[Physics:Measurement in quantum mechanics|quantum measurement]] as a logical proposition and studies the relationships and structures formed by these logical propositions. They are organized into a lattice, in which the distributive law, valid in classical logic, is weakened, to reflect the fact that in quantum physics, not all pairs of quantities can be [[Physics:Uncertainty principle|measured simultaneously]].<ref>{{cite book|title = Gleason's Theorem and Its Applications|publisher = [[Company:Wolters Kluwer|Kluwer Acad. Publ.]]|year = 1992|isbn = 978-0-7923-1990-0|series = Mathematics and its Applications, Vol. 60|location = Dordrecht|pages = 348|author = Dvurecenskij, Anatolij |oclc=751579618}}</ref> The ''representation theorem'' in quantum logic shows that such a lattice is [[Isomorphism|isomorphic]] to the lattice of [[Space (mathematics)|subspaces]] of a [[Vector space|vector space]] with a scalar product.<ref name="pitowsky2006"/>{{rp|§2}}  Using [[Solèr's theorem]], the (skew) [[Field (mathematics)|field]] ''K'' over which the vector space is defined can be proven, with additional hypotheses, to be either the [[Real number|real number]]s, [[Complex number|complex number]]s, or the [[Quaternion|quaternion]]s, as is needed for Gleason's theorem to hold.<ref name="cassinelli2017"/>{{rp|§3}}<ref>{{Cite web|url=https://golem.ph.utexas.edu/category/2010/12/solers_theorem.html|title=Solèr's Theorem|last=Baez|first=John C.|date=2010-12-01|website=[[NLab|The n-Category Café]]|access-date=2017-04-24}}</ref><ref>{{Cite journal|last1=Moretti|first1=Valter|last2=Oppio|first2=Marco|date=2019-06-01|title=Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one|journal=[[Physics:Reviews in Mathematical Physics|Reviews in Mathematical Physics]]|volume=31|issue=4|pages=1950013–502|doi=10.1142/S0129055X19500132|arxiv=1709.09246|bibcode=2019RvMaP..3150013M|s2cid=119733863}}</ref>

By invoking Gleason's theorem, the form of a probability function on lattice elements can be restricted. Assuming that the mapping from lattice elements to probabilities is noncontextual, Gleason's theorem establishes that it must be expressible with the Born rule.

== Generalizations ==
Gleason originally proved the theorem assuming that the measurements applied to the system are of the von Neumann type, i.e., that each possible measurement corresponds to an [[Orthonormal basis|orthonormal basis]] of the Hilbert space. Later, [[Biography:Paul Busch (physicist)|Busch]]<ref>{{Cite journal|last=Busch|first=Paul|date=2003| title=Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem|journal=[[Physics:Physical Review Letters|Physical Review Letters]]|volume=91|issue=12|pages=120403|arxiv=quant-ph/9909073|bibcode=2003PhRvL..91l0403B|doi=10.1103/PhysRevLett.91.120403|pmid=14525351|s2cid=2168715}}</ref> and independently [[Biography:Carlton M. Caves|Caves]] ''et al.''<ref name="fuchs2011" />{{rp|116}}<ref name="caves2004">{{Cite journal|last1=Caves|first1=Carlton M.|author-link=Carlton M. Caves|last2=Fuchs|first2=Christopher A.|last3=Manne|first3=Kiran K.|last4=Renes|first4=Joseph M.|date=2004|title=Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements|journal=[[Philosophy:Foundations of Physics|Foundations of Physics]]|volume=34|issue=2|pages=193–209|arxiv=quant-ph/0306179|bibcode=2004FoPh...34..193C|doi=10.1023/B:FOOP.0000019581.00318.a5|s2cid=18132256}}</ref> proved an analogous result for a more general class of measurements, known as [[POVM|positive-operator-valued measures]] (POVMs). The set of all POVMs includes the set of von Neumann measurements, and so the assumptions of this theorem are significantly stronger than Gleason's. This made the proof of this result simpler than Gleason's, and the conclusions stronger. Unlike the original theorem of Gleason, the generalized version using POVMs also applies to the case of a single qubit.<ref>{{cite journal|first=Robert W. |last=Spekkens|author-link=Robert Spekkens |date=2014|title=The Status of Determinism in Proofs of the Impossibility of a Noncontextual Model of Quantum Theory|journal=[[Philosophy:Foundations of Physics|Foundations of Physics]] |volume=44|issue=11|pages=1125–1155|arxiv=1312.3667|doi=10.1007/s10701-014-9833-x|bibcode=2014FoPh...44.1125S |s2cid=118469528}}</ref><ref>{{cite journal|last1=Wright |first1=Victoria J. |last2=Weigert |first2=Stephan |title=A Gleason-type theorem for qubits based on mixtures of projective measurements |arxiv=1808.08091 |journal=[[Physics:Journal of Physics A|Journal of Physics A]] |volume=52 |pages=055301 |doi=10.1088/1751-8121/aaf93d |year=2019|issue=5 |bibcode=2019JPhA...52e5301W |s2cid=119309162 }}</ref> Assuming noncontextuality for POVMs is, however, controversial, as POVMs are not fundamental, and some authors defend that noncontextuality should be assumed only for the underlying von Neumann measurements.<ref>{{cite journal |first1=Andrzej |last1=Grudka |first2=Paweł |last2=Kurzyński |title=Is There Contextuality for a Single Qubit? |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=2008 |volume=100 |issue=16 |page=160401 |doi=10.1103/PhysRevLett.100.160401|pmid=18518167 |arxiv=0705.0181|bibcode=2008PhRvL.100p0401G |s2cid=13251108 }}</ref> Gleason's theorem, in its original version, does not hold if the Hilbert space is defined over the [[Rational number|rational number]]s, i.e., if the components of vectors in the Hilbert space are restricted to be rational numbers, or complex numbers with rational parts. However, when the set of allowed measurements is the set of all POVMs, the theorem holds.<ref name="caves2004"/>{{rp|§3.D}}

The original proof by Gleason was not [[Constructive proof|constructive]]: one of the ideas on which it depends is the fact that every continuous function defined on a [[Compact space|compact space]] attains its minimum. Because one cannot in all cases explicitly show where the minimum occurs, a proof that relies upon this principle will not be a constructive proof. However, the theorem can be reformulated in such a way that a constructive proof can be found.<ref name="hp2004"/><ref>{{Cite journal|last1=Richman|first1=Fred|last2=Bridges|first2=Douglas|date=1999-03-10|title=A Constructive Proof of Gleason's Theorem|journal=[[Journal of Functional Analysis]]|volume=162|issue=2|pages=287–312|doi=10.1006/jfan.1998.3372|doi-access=free}}</ref>

Gleason's theorem can be extended to some cases where the observables of the theory form a [[Von Neumann algebra|von Neumann algebra]]. Specifically, an analogue of Gleason's result can be shown to hold if the algebra of observables has no [[Direct sum|direct sum]]mand that is representable as the algebra of 2×2 matrices over a commutative von Neumann algebra (i.e., no direct summand of type ''I''<sub>2</sub>). In essence, the only barrier to proving the theorem is the fact that Gleason's original result does not hold when the Hilbert space is that of a qubit.<ref>{{Cite book|url=https://books.google.com/books?id=jU7jzm4PylUC|title=Quantum Measure Theory|last=Hamhalter|first=Jan|date=2003-10-31|publisher=[[Company:Springer Science+Business Media|Springer Science & Business Media]]|isbn=9781402017148|language=en|mr=2015280|zbl=1038.81003 |oclc=928681664}}</ref>

== Notes ==
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== References ==
{{Reflist}}

{{Quantum computing}}

{{DEFAULTSORT:Gleasons theorem}}
[[Category:Quantum measurement]]
[[Category:Probability theorems]]

{{Sourceattribution|Gleason's theorem}}