View source for Local hidden-variable theory

{{short description|Interpretation of quantum mechanics}}
{{quantum|cTopic=Interpretations}}
In the [[Physics:Interpretations of quantum mechanics|interpretation of quantum mechanics]], a '''local hidden-variable theory''' is a [[Hidden-variable theory|hidden-variable theory]] that satisfies the [[Physics:Principle of locality|principle of locality]]. These are models, usually [[Determinism|deterministic]], that attempt to account for the probabilistic features of [[Physics:Quantum mechanics|quantum mechanics]] via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent. 

The mathematical implications of a local hidden-variable theory with regards to [[Quantum entanglement|quantum entanglement]] were explored by physicist [[Biography:John Stewart Bell|John Stewart Bell]], who in 1964 [[Bell's theorem|proved]] that broad classes of local hidden-variable theories cannot reproduce the correlations between measurement outcomes that quantum mechanics predicts, a result since confirmed by a range of detailed [[Physics:Bell test|Bell test]] experiments.<ref>{{cite news |last=Markoff |first=Jack |date=21 October 2015 |title=Sorry, Einstein. Quantum Study Suggests 'Spooky Action' Is Real. |work={{wipe|New York Times}} |url=https://www.nytimes.com/2015/10/22/science/quantum-theory-experiment-said-to-prove-spooky-interactions.html}}</ref>

== Models ==

=== Single qubit ===
A [[Bell's theorem|collection of related theorems]], beginning with Bell's proof in 1964, show that quantum mechanics is incompatible with local hidden variables. However, as Bell pointed out, restricted sets of quantum phenomena ''can'' be imitated using local hidden-variable models. Bell provided a local hidden-variable model for quantum measurements upon a spin-1/2 particle, or in the terminology of quantum information theory, a single [[Qubit|qubit]].<ref name=Bell1964>{{cite journal | last1 = Bell | first1 = J. S. |  year = 1964 | title = On the Einstein Podolsky Rosen Paradox | url = https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf | journal = [[Physics:Physics Physique Физика|Physics Physique Физика]] | volume = 1 | issue = 3| pages = 195–200 | doi = 10.1103/PhysicsPhysiqueFizika.1.195 }}</ref> Bell's model was later simplified by [[Biography:N. David Mermin|N. David Mermin]], and a closely related model was presented by [[Biography:Simon B. Kochen|Simon B. Kochen]] and [[Biography:Ernst Specker|Ernst Specker]].<ref>{{cite journal|first1=S. |last1=Kochen |author-link1=Simon B. Kochen |first2=E. |last2=Specker |author-link2=Ernst Specker |journal=Journal of Mathematics and Mechanics |volume=17 |year=1967 |title=The Problem of Hidden Variables in Quantum Mechanics |number=1 |pages=59–87 |jstor=24902153}}</ref><ref name="mermin1993">{{Cite journal|last=Mermin|first=N. David|author-link=David Mermin|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><ref>{{Cite journal |last1=Harrigan |first1=Nicholas |last2=Spekkens |first2=Robert W. |date=2010-02-01 |title=Einstein, Incompleteness, and the Epistemic View of Quantum States |url=https://doi.org/10.1007/s10701-009-9347-0 |journal=Foundations of Physics |language=en |volume=40 |issue=2 |pages=125–157 |arxiv=0706.2661 |doi=10.1007/s10701-009-9347-0 |s2cid=32755624 |issn=1572-9516}}</ref> The existence of these models is related to the fact that [[Gleason's theorem]] does not apply to the case of a single qubit.<ref>{{Cite journal |last1=Budroni |first1=Costantino |last2=Cabello |first2=Adán |last3=Gühne |first3=Otfried |last4=Kleinmann |first4=Matthias |last5=Larsson |first5=Jan-Åke |date=2022-12-19 |title=Kochen-Specker contextuality |url=https://link.aps.org/doi/10.1103/RevModPhys.94.045007 |journal=Reviews of Modern Physics |language=en |volume=94 |issue=4 |page=045007 |doi=10.1103/RevModPhys.94.045007 |hdl=11441/144776 |s2cid=251951089 |issn=0034-6861|hdl-access=free }}</ref>

=== Bipartite quantum states ===
Bell also pointed out that up until then, discussions of [[Quantum entanglement|quantum entanglement]] focused on cases where the results of measurements upon two particles were either perfectly correlated or perfectly anti-correlated. These special cases can also be explained using local hidden variables.<ref name=Bell1964/><ref>{{Cite journal |last1=Ou |first1=Z. Y. |last2=Pereira |first2=S. F. |last3=Kimble |first3=H. J. |last4=Peng |first4=K. C. |date=1992-06-22 |title=Realization of the Einstein-Podolsky-Rosen paradox for continuous variables |url=https://link.aps.org/doi/10.1103/PhysRevLett.68.3663 |journal=Physical Review Letters |language=en |volume=68 |issue=25 |pages=3663–3666 |doi=10.1103/PhysRevLett.68.3663 |pmid=10045765 |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Bartlett |first1=Stephen D. |last2=Rudolph |first2=Terry |last3=Spekkens |first3=Robert W. |date=2012-07-10 |title=Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction |url=https://link.aps.org/doi/10.1103/PhysRevA.86.012103 |journal=Physical Review A |language=en |volume=86 |issue=1 |page=012103 |arxiv=1111.5057 |bibcode=2012PhRvA..86a2103B |doi=10.1103/PhysRevA.86.012103 |s2cid=119235025 |issn=1050-2947}}</ref>

For [[Separable state|separable state]]s of two particles, there is a simple hidden-variable model for any measurements on the two parties. Surprisingly, there are also [[Quantum entanglement|entangled states]] for which all [[Physics:Measurement in quantum mechanics|von Neumann measurements]] can be described by a hidden-variable model.<ref>{{cite journal |author1=R. F. Werner | title=Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model |journal=[[Physical Review A]]| year=1989 |volume=40 | issue=8 | doi=10.1103/PhysRevA.40.4277 | pages=4277–4281 |bibcode=1989PhRvA..40.4277W | pmid=9902666 }}</ref>  Such states are entangled, but do not violate any Bell inequality. The so-called [[Physics:Werner state|Werner state]]s are a single-parameter family of states that are invariant under any transformation of the type <math>U \otimes U,</math> where <math>U</math> is a unitary matrix. For two qubits, they are noisy singlets given as 
<math display="block">\varrho = p \vert \psi^- \rangle \langle \psi^-\vert + (1 - p) \frac{\mathbb{I}}{4},</math>
where the singlet is defined as <math>\vert \psi^-\rangle = \tfrac{1}{\sqrt{2}}\left(\vert 01\rangle - \vert 10\rangle\right)</math>.

[[Biography:Reinhard F. Werner|Reinhard F. Werner]] showed that such states allow for a hidden-variable model for <math>p \leq 1/2</math>, while they are entangled if <math>p > 1/3</math>. The bound for hidden-variable models could be improved until <math>p = 2/3</math>.<ref>{{cite journal |author1=A. Acín |author2=N. Gisin |author3=B. Toner | title=Grothendieck's constant and local models for noisy entangled quantum states |journal=[[Physical Review A]]| year=2006 |volume=73 |issue=6 | doi=10.1103/PhysRevA.73.062105 | pages=062105 |arxiv=quant-ph/0606138 |bibcode=2006PhRvA..73f2105A |s2cid=2588399 }}</ref> Hidden-variable models have been constructed for Werner states even if positive operator-valued measurements ([[POVM]])  are allowed, not only von Neumann measurements.<ref>{{cite journal |author1=J. Barrett | title=Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality |journal=[[Physical Review A]]| year=2002 |volume=65 | issue=4 | doi=10.1103/PhysRevA.65.042302 | pages=042302 |arxiv=quant-ph/0107045 |bibcode=2002PhRvA..65d2302B | s2cid=119390251 }}</ref> Hidden variable models were also constructed to noisy maximally entangled states, and even extended to arbitrary pure states mixed with white noise.<ref>{{cite journal |last1=Almeida |first1=Mafalda L. |last2=Pironio |first2=Stefano |last3=Barrett |first3=Jonathan |last4=Tóth |first4=Géza |last5=Acín |first5=Antonio |title=Noise Robustness of the Nonlocality of Entangled Quantum States |journal=Physical Review Letters |date=23 July 2007 |volume=99 |issue=4 |pages=040403 |doi=10.1103/PhysRevLett.99.040403|pmid=17678341 |arxiv=quant-ph/0703018|s2cid=7102567 }}</ref> Beside bipartite systems, there are also results for the multipartite case.  A hidden-variable model for any von Neumann measurements at the parties has been presented for a three-qubit quantum state.<ref>{{cite journal |author1=G. Tóth | author2=A. Acín | title=Genuine tripartite entangled states with a local hidden-variable model |journal=[[Physical Review A]]| year=2006 |volume=74 | issue=3 | doi=10.1103/PhysRevA.74.030306 | pages=030306 |arxiv=quant-ph/0512088 |bibcode=2006PhRvA..74c0306T | s2cid=4792051 }}</ref>

==Time-dependent variables==
Previously some new hypotheses were conjectured concerning the role of time in constructing hidden-variables theory. One approach was suggested by K. Hess and W. Philipp and relies upon possible consequences of time dependencies of hidden variables; this hypothesis has been criticized by [[Biography:Richard D. Gill|Richard D. Gill]], {{Ill|Gregor Weihs|lt=|de}}, [[Biography:Anton Zeilinger|Anton Zeilinger]] and Marek Żukowski, as well as D. M. Appleby.<ref>{{Cite journal|last1=Hess|first1=K|last2=Philipp|first2=W|date=March 2002|title=Exclusion of time in the theorem of Bell|url=https://iopscience.iop.org/article/10.1209/epl/i2002-00578-y|journal=Europhysics Letters (EPL)|volume=57|issue=6|pages=775–781|doi=10.1209/epl/i2002-00578-y|s2cid=250792546|issn=0295-5075}}</ref><ref>{{Cite journal|last1=Gill|first1=R. D.|last2=Weihs|first2=G.|last3=Zeilinger|first3=A.|author-link3=Anton Zeilinger|last4=Zukowski|first4=M.|date=2002-11-12|title=No time loophole in Bell's theorem: The Hess-Philipp model is nonlocal|journal=Proceedings of the National Academy of Sciences|language=en|volume=99|issue=23|pages=14632–14635|arxiv=quant-ph/0208187|doi=10.1073/pnas.182536499|issn=0027-8424|pmc=137470|pmid=12411576|doi-access=free }}</ref><ref>{{cite journal|last=Appleby |first=D. M. |title=The Hess-Philipp Model is Non-Local |journal=International Journal of Quantum Information |volume=1 |number=1 |pages=29–36 |year=2003 |doi=10.1142/S021974990300005X |arxiv=quant-ph/0210145 |bibcode=2002quant.ph.10145A}}</ref>

==See also==
* [[Physics:EPR paradox|EPR paradox]]
* [[Physics:Bohr–Einstein debates|Bohr–Einstein debates]]

==References==
<references/>

[[Category:Quantum measurement]]
[[Category:Hidden variable theory]]

{{Sourceattribution|Local hidden-variable theory}}