Editing Bell's theorem (section)

===Kochen–Specker theorem (1967)===
{{main|Kochen–Specker theorem}}
In quantum theory, orthonormal bases for a [[Hilbert space]] represent measurements that can be performed upon a system having that Hilbert space. Each vector in a basis represents a possible outcome of that measurement.{{refn|group=note|In more detail, as developed by [[Paul Dirac]],<ref>{{cite book|first=Paul Adrien Maurice |last=Dirac |author-link=Paul Dirac |title=The Principles of Quantum Mechanics |title-link=The Principles of Quantum Mechanics |publisher=Clarendon Press |location=Oxford |year=1930}}</ref> [[David Hilbert]],<ref>{{cite book|first=David |last=Hilbert |author-link=David Hilbert |title=Lectures on the Foundations of Physics 1915–1927: Relativity, Quantum Theory and Epistemology |publisher=Springer |doi=10.1007/b12915 |editor-first1=Tilman |editor-last1=Sauer |editor-first2=Ulrich |editor-last2=Majer |year=2009 |isbn=978-3-540-20606-4 |oclc=463777694}}</ref> [[John von Neumann]],<ref>{{cite book|first=John |last=von Neumann |author-link=John von Neumann |title=Mathematische Grundlagen der Quantenmechanik |publisher=Springer |location=Berlin |year=1932}} English translation: {{cite book|title=Mathematical Foundations of Quantum Mechanics |title-link=Mathematical Foundations of Quantum Mechanics |publisher=Princeton University Press |year=1955 |translator-first=Robert T. |translator-last=Beyer |translator-link=Robert T. Beyer}}</ref> and [[Hermann Weyl]],<ref>{{cite book|first=Hermann |last=Weyl |author-link=Hermann Weyl |title=The Theory of Groups and Quantum Mechanics |title-link=Gruppentheorie und Quantenmechanik |orig-year=1931 |publisher=Dover |year=1950 |isbn=978-0-486-60269-1 |translator-first=H. P. |translator-last=Robertson |translator-link=Howard P. Robertson}} Translated from the German {{cite book |title=Gruppentheorie und Quantenmechanik |year=1931 |edition=2nd |publisher={{ill|S. Hirzel Verlag|de}}}}</ref> the state of a quantum mechanical system is a vector <math>|\psi\rangle</math> belonging to a ([[Separable space|separable]]) Hilbert space <math>\mathcal H</math>. Physical quantities of interest — position, momentum, energy, spin — are represented by "observables", which are [[self-adjoint operator|self-adjoint]] linear [[Operator (physics)|operator]]s acting on the Hilbert space. When an observable is measured, the result will be one of its eigenvalues with probability given by the [[Born rule]]: in the simplest case the eigenvalue <math>\eta</math> is non-degenerate and the probability is given by <math>|\langle \eta|\psi\rangle|^2</math>, where <math>|\eta\rangle</math> is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by <math>\langle \psi|P_\eta\psi\rangle</math>, where <math>P_\eta</math> is the projector onto its associated eigenspace. For the purposes of this discussion, we can take the eigenvalues to be non-degenerate.}} Suppose that a hidden variable <math>\lambda</math> exists, so that knowing the value of <math>\lambda</math> would imply certainty about the outcome of any measurement. Given a value of <math>\lambda</math>, each measurement outcome – that is, each vector in the Hilbert space – is either ''impossible'' or ''guaranteed.'' A Kochen–Specker configuration is a finite set of vectors made of multiple interlocking bases, with the property that a vector in it will always be ''impossible'' when considered as belonging to one basis and ''guaranteed'' when taken as belonging to another. In other words, a Kochen–Specker configuration is an "uncolorable set" that demonstrates the inconsistency of assuming a hidden variable <math>\lambda</math> can be controlling the measurement outcomes.<ref>{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |year=1993 |publisher=[[Kluwer]] |isbn=0-7923-2549-4 |oclc=28854083}}</ref>{{Rp|196–201}}