Editing Bell's theorem (section)

===Background===
{{main|EPR paradox|History of quantum mechanics}}

The question of whether quantum mechanics can be "completed" by hidden variables dates to the early years of quantum theory. In his [[Mathematical Foundations of Quantum Mechanics|1932 textbook on quantum mechanics]], the Hungarian-born polymath [[John von Neumann]] presented what he claimed to be a proof that there could be no "hidden parameters". The validity and definitiveness of von Neumann's proof were questioned by [[Hans Reichenbach]], in more detail by [[Grete Hermann]], and possibly in conversation though not in print by Albert Einstein.{{refn|group=note|See Reichenbach<ref>{{cite book|first=Hans |last=Reichenbach |author-link=Hans Reichenbach |title=Philosophic Foundations of Quantum Mechanics |year=1944 |publisher=University of California Press |page=14 |oclc=872622725}}</ref> and Jammer,<ref name="jammer1974">{{cite book|last=Jammer|first=Max|title=The Philosophy of Quantum Mechanics|publisher=John Wiley and Sons|year=1974|isbn=0-471-43958-4|author-link=Max Jammer}}</ref>{{Rp|276}} Mermin and Schack,<ref>{{cite journal|title=Homer nodded: von Neumann's surprising oversight |journal=[[Foundations of Physics]] |volume=48 |issue=9 |pages=1007–1020 |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 |bibcode=2018FoPh...48.1007M |s2cid=118951033 }}</ref> and for Einstein's remarks, Clauser and Shimony<ref>{{cite journal | last1 = Clauser | first1 = J. F. | last2 = Shimony | first2 = A. | title = Bell's theorem: Experimental tests and implications | url = http://www.physics.oregonstate.edu/~ostroveo/COURSES/ph651/Supplements_Phys651/RPP1978_Bell.pdf | journal = Reports on Progress in Physics | volume = 41 | issue = 12 | pages = 1881–1927 | year = 1978 | doi = 10.1088/0034-4885/41/12/002 | bibcode = 1978RPPh...41.1881C | citeseerx = 10.1.1.482.4728 | s2cid = 250885175 | access-date = 2017-10-28 | archive-date = 2017-09-23 | archive-url = https://web.archive.org/web/20170923004338/http://physics.oregonstate.edu/~ostroveo/COURSES/ph651/Supplements_Phys651/RPP1978_Bell.pdf | url-status = live }}</ref> and Wick.<ref name=":1"/>{{Rp|286}}}} ([[Simon B. Kochen|Simon Kochen]] and [[Ernst Specker]] rejected von Neumann's key assumption as early as 1961, but did not publish a criticism of it until 1967.<ref>{{Cite book |author-first1=John |author-last1=Conway |author-link1=John Horton Conway |author-first2=Simon |author-last2=Kochen |author-link2=Simon B. Kochen |chapter=The Geometry of the Quantum Paradoxes |pages=257–269 |title=Quantum [Un]speakables: From Bell to Quantum Information |date=2002 |publisher=Springer |editor-first1=Reinhold A. |editor-last1=Bertlmann |editor-link1=Reinhold Bertlmann |editor-first2=Anton |editor-last2=Zeilinger |editor-link2=Anton Zeilinger |isbn=3-540-42756-2 |location=Berlin |oclc=49404213}}</ref>)

Einstein argued persistently that quantum mechanics could not be a complete theory. His preferred argument relied on a principle of locality:
:Consider a mechanical system constituted of two partial systems ''A'' and ''B'' which have interaction with each other only during limited time. Let the ψ function before their interaction be given. Then the [[Schrödinger equation]] will furnish the ψ function after their interaction has taken place. Let us now determine the physical condition of the partial system ''A'' as completely as possible by measurements. Then the quantum mechanics allows us to determine the ψ function of the partial system ''B'' from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon ''which'' of the determining magnitudes specifying the condition of ''A'' has been measured (for instance coordinates ''or'' momenta). Since there can be only ''one'' physical condition of ''B'' after the interaction and which can reasonably not be considered as dependent on the particular measurement we perform on the system ''A'' separated from ''B'' it may be concluded that the ψ function is not unambiguously coordinated with the physical condition. This coordination of several ψ functions with the same physical condition of system ''B'' shows again that the ψ function cannot be interpreted as a (complete) description of a physical condition of a unit system.<ref>{{cite journal|first=Albert |last=Einstein |author-link=Albert Einstein |title=Physics and reality |journal=Journal of the Franklin Institute |volume=221 |number=3 |date=March 1936 |pages=349–382 |doi=10.1016/S0016-0032(36)91047-5 |bibcode=1936FrInJ.221..349E}}</ref>

The EPR thought experiment is similar, also considering two separated systems ''A'' and ''B'' described by a joint wave function. However, the EPR paper adds the idea later known as the EPR criterion of reality, according to which the ability to predict with probability 1 the outcome of a measurement upon ''B'' implies the existence of an "element of reality" within ''B''.<ref>{{cite journal|first1=Nicholas |last1=Harrigan |first2=Robert W. |last2=Spekkens |title=Einstein, incompleteness, and the epistemic view of quantum states |journal=[[Foundations of Physics]] |volume=40 |issue=2 |pages=125 |year=2010 |doi=10.1007/s10701-009-9347-0 |arxiv=0706.2661|bibcode=2010FoPh...40..125H |s2cid=32755624 }}</ref>

In 1951, [[David Bohm]] proposed a variant of the EPR thought experiment in which the measurements have discrete ranges of possible outcomes, unlike the position and momentum measurements considered by EPR.<ref>{{cite book|last=Bohm |first=David |author-link=David Bohm |year=1989 |orig-date=1951 |title=Quantum Theory |publisher=Prentice-Hall |edition=Dover reprint |isbn=978-0-486-65969-5 |oclc=1103789975 |pages=614–623}}</ref> The year before, [[Chien-Shiung Wu]] and Irving Shaknov had successfully measured polarizations of photons produced in entangled pairs, thereby making the Bohm version of the EPR thought experiment practically feasible.<ref>{{cite journal |last1=Wu |first1=C.-S. |author-link=Chien-Shiung Wu |last2=Shaknov |first2=I. |year=1950 |title=The Angular Correlation of Scattered Annihilation Radiation |journal=[[Physical Review]] |volume=77 |issue=1 |pages=136 |bibcode=1950PhRv...77..136W |doi=10.1103/PhysRev.77.136}}</ref>

By the late 1940s, the mathematician [[George Mackey]] had grown interested in the foundations of quantum physics, and in 1957 he drew up a list of postulates that he took to be a precise definition of quantum mechanics.<ref>{{Cite journal |last=Mackey |first=George W. |author-link=George Mackey |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> Mackey conjectured that one of the postulates was redundant, and shortly thereafter, [[Andrew M. Gleason]] proved that it was indeed deducible from the other postulates.<ref name="gleason1957">{{cite journal|first=Andrew M.|author-link=Andrew M. Gleason|year = 1957|title = Measures on the closed subspaces of a Hilbert space|url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050|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><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> [[Gleason's theorem]] provided an argument that a broad class of hidden-variable theories are incompatible with quantum mechanics.{{refn|group=note|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 Stern–Gerlach measurement on a [[Spin (physics)|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, because it proves that any probability measure must take the form of a mapping <math>u \to \langle \rho u, u \rangle</math> for some density operator <math>\rho</math>. This mapping is continuous on the [[unit sphere]] of the Hilbert space, and since this unit sphere is [[Connected (topology)|connected]], no continuous probability measure on it can be deterministic.<ref name="wilce2017">{{cite book|last=Wilce |first=A. |year=2017 |chapter-url=https://plato.stanford.edu/entries/qt-quantlog/ |chapter=Quantum Logic and Probability Theory |title=Stanford Encyclopedia of Philosophy |title-link=Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University }}</ref>{{rp|§1.3}}}} 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 [[special relativity]].<ref name = "ND Mermin 1993-07">{{cite journal | last = Mermin |first = N. David |author-link=N. David Mermin |title = Hidden Variables and the Two Theorems of John Bell | journal = [[Reviews of Modern Physics]] | volume = 65 |pages = 803–815 | number = 3| date = July 1993  | url = http://cqi.inf.usi.ch/qic/Mermin1993.pdf |arxiv=1802.10119 |doi = 10.1103/RevModPhys.65.803 |bibcode = 1993RvMP...65..803M |s2cid = 119546199 }}</ref><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> 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="ND Mermin 1993-07" /><ref>{{Cite journal|last=Peres|first=Asher|author-link=Asher Peres|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>

[[Tsung-Dao Lee]] came close to deriving Bell's theorem in 1960. He considered events where two [[kaon]]s were produced traveling in opposite directions, and came to the conclusion that hidden variables could not explain the correlations that could be obtained in such situations. However, complications arose due to the fact that kaons decay, and he did not go so far as to deduce a Bell-type inequality.<ref name="jammer1974"/>{{Rp|308}}