Editing Einstein–Podolsky–Rosen paradox (section)

=== Bohm's variant ===
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>Bohm, D. (1951). ''Quantum Theory'', Prentice-Hall, Englewood Cliffs, page 29, and Chapter 5 section 3, and Chapter 22 Section 19.</ref><ref>{{cite journal |author1=D. Bohm |author2=Y. Aharonov |title=Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky |journal=Physical Review |date=1957 |volume=108 |issue=4 |page=1070 |doi=10.1103/PhysRev.108.1070|bibcode=1957PhRv..108.1070B }}</ref><ref>{{Cite journal|last1=Reid|first1=M. D.|last2=Drummond|first2=P. D.|last3=Bowen|first3=W. P.|last4=Cavalcanti|first4=E. G.|last5=Lam|first5=P. K.|last6=Bachor|first6=H. A.|last7=Andersen|first7=U. L.|last8=Leuchs|first8=G.|date=2009-12-10|title=Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications|journal=Reviews of Modern Physics|volume=81|issue=4|pages=1727–1751|doi=10.1103/RevModPhys.81.1727|bibcode=2009RvMP...81.1727R|arxiv=0806.0270|s2cid=53407634}}</ref> The EPR–Bohm thought experiment can be explained using electron–[[positron]] pairs. Suppose we have a source that emits electron–positron pairs, with the electron sent to destination ''A'', where there is an observer named [[Alice and Bob|Alice]], and the positron sent to destination ''B'', where there is an observer named [[Alice and Bob|Bob]]. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a [[spin singlet]]. The particles are thus said to be [[quantum entanglement|entangled]]. This can be viewed as a [[quantum superposition]] of two states, which we call state I and state II. In state I, the electron has [[Spin (physics)|spin]] pointing upward along the ''z''-axis (+''z'') and the positron has spin pointing downward along the ''z''-axis (−''z''). In state II, the electron has spin −''z'' and the positron has spin +''z''. Because it is in a superposition of states, it is impossible without measuring to know the definite state of spin of either particle in the spin singlet.<ref name=Griffiths2004>{{cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics |edition=2nd | publisher=Prentice Hall |year=2004 |isbn= 978-0-13-111892-8}}</ref>{{rp|421–422}}

[[File:EPR illustration.svg|thumb|500px|center|The EPR thought experiment, performed with electron–positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to Bob (right), who can perform spin measurements.]]
Alice now measures the spin along the ''z''-axis. She can obtain one of two possible outcomes: +''z'' or −''z''. Suppose she gets +''z''. Informally speaking, the quantum state of the system [[wavefunction collapse|collapses]] into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the ''z''-axis, there is 100% probability that he will obtain −''z''. Similarly, if Alice gets −''z'', Bob will get +''z''. There is nothing special about choosing the ''z''-axis: according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the ''x'' direction.<ref name=Laloe>{{cite journal|last=Laloe|first=Franck|title=Do We Really Understand Quantum Mechanics|journal=American Journal of Physics|volume=69|issue=6|pages=655–701|year=2012|arxiv=quant-ph/0209123|bibcode=2001AmJPh..69..655L|doi=10.1119/1.1356698|s2cid=123349369 }} {{erratum|doi=10.1119/1.1466818|checked=yes}}</ref>{{rp|318}}

Whatever axis their spins are measured along, they are always found to be opposite. In quantum mechanics, the ''x''-spin and ''z''-spin are "incompatible observables", meaning the [[uncertainty principle|Heisenberg uncertainty principle]] applies to alternating measurements of them: a quantum state cannot possess a definite value for both of these variables. Suppose Alice measures the ''z''-spin and obtains ''+z'', so that the quantum state collapses into state I. Now, instead of measuring the ''z''-spin as well, Bob measures the ''x''-spin. According to quantum mechanics, when the system is in state I, Bob's ''x''-spin measurement will have a 50% probability of producing +''x'' and a 50% probability of -''x''. It is impossible to predict which outcome will appear until Bob actually ''performs'' the measurement. Therefore, Bob's positron will have a definite spin when measured along the same axis as Alice's electron, but when measured in the perpendicular axis its spin will be uniformly random. It seems as if information has propagated (faster than light) from Alice's apparatus to make Bob's positron assume a definite spin in the appropriate axis.