local hidden-variable theory

A 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.{{cite journal | last1 = Bell | first1 = J. S. | author-link = John Stewart Bell | year = 1964 | title = On the Einstein Podolsky Rosen Paradox | url = https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf | journal = Physics Physique Физика | volume = 1 | issue = 3| pages = 195–200 | doi = 10.1103/PhysicsPhysiqueFizika.1.195 }} Bell's model was later simplified by N. David Mermin, and a closely related model was presented by Simon B. Kochen and Ernst Specker.{{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}}{{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=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}}{{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}} The existence of these models is related to the fact that Gleason's theorem does not apply to the case of a single qubit.{{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 |arxiv=2102.13036}}

Bell also pointed out that up until then, discussions of 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.{{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|url-access=subscription }}{{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}}

For separable states of two particles, there is a simple hidden-variable model for any measurements on the two parties. Surprisingly, there are also entangled states for which all von Neumann measurements can be described by a hidden-variable model.{{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 }} Such states are entangled, but do not violate any Bell inequality. The so-called Werner states are a single-parameter family of states that are invariant under any transformation of the type $U\; \backslash otimes\; U,$ where $U$ is a unitary matrix. For two qubits, they are noisy singlets given as

$$\backslash varrho\; =\; p\; \backslash vert\; \backslash psi^-\; \backslash rangle\; \backslash langle\; \backslash psi^-\backslash vert\; +\; (1\; -\; p)\; \backslash frac\{\backslash mathbb\{I\}\}\{4\},$$

where the singlet is defined as $\backslash vert\; \backslash psi^-\backslash rangle\; =\; \backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\backslash left(\backslash vert\; 01\backslash rangle\; -\; \backslash vert\; 10\backslash rangle\backslash right)$.

Reinhard F. Werner showed that such states allow for a hidden-variable model for $p\; \backslash leq\; 1/2$, while they are entangled if $p\; >\; 1/3$. The bound for hidden-variable models could be improved until $p\; =\; 2/3$.{{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 }} Hidden-variable models have been constructed for Werner states even if positive operator-valued measurements (POVM) are allowed, not only von Neumann measurements.{{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 }} Hidden variable models were also constructed to noisy maximally entangled states, and even extended to arbitrary pure states mixed with white noise.{{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 }} 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.{{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 }}

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 Richard D. Gill, {{Ill|Gregor Weihs|lt=|de}}, Anton Zeilinger and Marek Żukowski, as well as D. M. Appleby.{{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|volume=57|issue=6|pages=775–781|doi=10.1209/epl/i2002-00578-y|s2cid=250792546|issn=0295-5075|url-access=subscription}}{{Cite journal|last1=Gill|first1=R. D.|author-link=Richard D. Gill|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 }}{{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}}

• EPR paradox
• Bohr–Einstein debates

Category:Quantum measurement

Category:Hidden variable theory