Local hidden-variable theory: Difference between revisions

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{{short description|Interpretation of quantum mechanics}}

A local [[hidden variable theory]] is one in which distant events are assumed to have no ''instantaneous'' effect on local ones. The term is most often used in relation to the [[EPR paradox]] and [[Bell's theorem|Bell's inequalities]]. It is effectively synonymous with the concept of [[local realism]], which has always been considered a desirable property by physicists.

{{quantum|cTopic=Interpretations}}

In the [[Interpretations of quantum mechanics|interpretation of quantum mechanics]], a '''local hidden-variable theory''' is a [[hidden-variable theory]] that satisfies the [[principle of locality]]. These models attempt to account for the probabilistic features of [[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]] were explored by physicist [[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 [[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=[[New York Times]] |url=https://www.nytimes.com/2015/10/22/science/quantum-theory-experiment-said-to-prove-spooky-interactions.html}}</ref>

===Local hidden variables and the Bell tests===

== Models ==

The principle of "locality" enables the assumption to be made in [[Bell test experiments]] that the probability of a coincidence can be written in "factorisable" form:

=== Single qubit ===

:: (1) &nbsp;&nbsp; ''P'' ('''a''', '''b''') = &int; d&lambda; &rho;(&lambda;) ''p''_A ('''a''', &lambda;) ''p''_B('''b''', &lambda;),

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]].<ref name=Bell1964>{{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 }}</ref> Bell's model was later simplified by [[N. David Mermin]], and a closely related model was presented by [[Simon B. Kochen]] and [[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=[[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 |bibcode=2010FoPh...40..125H |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 |arxiv=2102.13036|bibcode=2022RvMP...94d5007B }}</ref>

=== Bipartite quantum states ===

where ''p''_A ('''a''', &lambda;) is the probability of detection of particle A with hidden variable &lambda; by detector A, set in direction '''a''', and similarly ''p''_A ('''b''', &lambda;) is the probability at detector B for particle B, sharing the same value of &lambda;. The source is assumed to produce particles in the state &lambda; with probability &rho;(&lambda;).

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.<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 |bibcode=1992PhRvL..68.3663O |issn=0031-9007|url-access=subscription }}</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]]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 [[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 [[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

Using (1), various "Bell inequalities" can be derived, giving restrictions on the possible behaviour of local hidden variable models.

<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>.

[[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>

When [[John Stewart Bell|John Bell]] originally derived his inequality, it was in relation to pairs of indivisible "spin-1/2" particles, every one of those emitted being detected. In these circumstances it is found that local realist assumptions lead to a straight line prediction for the relationship between [[quantum correlation]] and the angle between the settings of the two detectors. It was soon realised, however, that real experiments were not feasible with spin-1/2 particles. They were conducted instead using "photons". The local hidden variable prediction for these is not a straight line but a sine curve, similar to the quantum mechanical prediction but of only half the "visibility".

==Time-dependent variables==

The difference between the two predictions is due to the different functions ''p''_A and ''p''_B involved. By assuming different functions, a great variety of other realist predictions can be derived, some very close to the quantum-mechanical one. The choice of function, however, is not arbitrary. In optical experiments using polarisation, for instance, the natural assumption is that it is a cosine-squared function, corresponding to adherence to [[Etienne-Louis Malus|Malus' Law]].

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.<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|volume=57|issue=6|pages=775–781|doi=10.1209/epl/i2002-00578-y|s2cid=250792546|issn=0295-5075|url-access=subscription}}</ref><ref>{{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 }}</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==

===Bell tests with no "non-detections"===

* [[EPR paradox]]

* [[Bohr–Einstein debates]]

===References===

Consider, for example, David Bohm's thought-experiment (Bohm, 1951), in which a molecule breaks into two atoms with opposite spins. Assume this spin can be represented by a real vector, pointing in any direction. It will be the "hidden variable" in our model. Taking it to be a unit vector, all possible hidden variables are represented by all points on the surface of a unit sphere.

<references/>

[[Category:Quantum measurement]]

Suppose the spin is to be measured in the direction '''a'''. Then the natural assumption, given that all atoms are detected, is that all atoms the projection of whose spin in the direction '''a''' is positive will be detected as spin up (coded as +1) while all whose projection is negative will be detected as spin down (coded as &minus;1). The surface of the sphere will be divided into two regions, one for +1, one for &minus;1, separated by a great circle in the plane perpendicular to '''a'''. Assuming for convenience that '''a''' is horizontal, corresponding to the angle ''a'' with respect to some suitable reference direction, the dividing circle will be in a vertical plane. So far we have modelled side A of our experiment.

[[Category:Hidden variable theory]]

Now to model side B. Assume that '''b''' too is horizontal, corresponding to the angle ''b''. There will be second great circle drawn on the same sphere, to one side of which we have +1, the other &minus;1 for particle B. The circle will be again be in a vertical plane.

The two circles divide the surface of the sphere into four regions. The type of "coincidence" (++, &minus;&minus;, +&minus; or &minus;+) observed for any given pair of particles is determined by the region within which their hidden variable falls. Assuming the source to be "rotationally invariant" (to produce all possible states &lambda; with equal probability), the probability of a given type of coincidence will clearly be proportional to the corresponding area, and these areas will vary linearly with the angle between '''a''' and '''b'''. (To see this, think of an orange and its segments. The area of peel corresponding to a number n of segments is roughly proportional to n. More accurately, it is proportional to the angle subtended at the centre.)

The formula (1) above has not been used explicitly &mdash; it is hardly relevant when, as here, the situation is fully deterministic. The problem ''could'' be reformulated in terms of the functions in the formula, with &rho; constant and the probability functions step functions. The principle behind (1) has in fact been used, but purely intuitively.

[[Image: StraightLines.png|300px|thumb|right|Fig. 1: The realist prediction (solid lines) for quantum correlation when there are no non-detections. The quantum-mechanical prediction is the dotted curve.]]

Thus the local hidden variable prediction for the probability of coincidence is proportional to the angle (''b'' &minus; a) between the detector settings. The quantum correlation is defined to be the expectation value of the product of the individual outcomes, and this is

::(2) &nbsp;&nbsp; ''E'' = ''P''₊₊ + ''P''_{&minus;&minus;} &minus; ''P''_+&minus; &minus; ''P''_&minus;+

where ''P''₊₊ is the probability of a '+' outcome on both sides, ''P''_+&minus; that of a + on side A, a '&minus;' on side B, etc..

Since each individual term varies linearly with the difference (''b'' &minus; ''a''), so does their sum.

The result is shown in fig. 1.

===Optical Bell tests===

In almost all real applications of Bell's inequalities, the particles used have been "photons". It is not necessarily assumed the the photons are particle-like. They may be just short pulses of classical light (Clauser, 1978). It is not assumed that every single one is detected. Instead the hidden variable set at the source is taken to determine only the ''probability'' of a given outcome, the actual individual outcomes being partly determined by other hidden variables local to the analyser and detector. It is assumed that these other hidden variables are independent on the two sides of the experiment (Clauser, 1974; Bell, 1971).

In this "stochastic" model, in contrast to the above deterministic case, we do need equation (1) to find the local realist prediction for coincidences. It is necessary first to make some assumption regarding the functions ''p''_A and ''p''_B, the usual one being that these are both cosine-squares, in line with Malus' Law. Assuming the hidden variable to be polarisation direction (parallel on the two sides in real applications, not orthogonal), equation (1) becomes:

:: (3) &nbsp;&nbsp; ''P'' (''a'', ''b'') = &int; d&lambda; &rho;(&lambda;) ''cos''² (''a'' &minus; &lambda;) ''cos''<sup>2</sub>(''b'' &minus; &lambda;)

::::: = 1/8 + 1/4 ''cos''² &phi;,

where &phi; = ''b'' &minus; ''a''.

The predicted quantum correlation can be derived from this and is shown in fig. 2.

[[Image: MalusQC.png|300px|thumb|right|Fig. 2: The realist prediction (solid curve) for quantum correlation in an optical Bell test. The quantum-mechanical prediction is the dotted curve.]]

In optical tests, incidentally, it is not certain that the quantum correlation is well-defined. Under a classical model of light, a single photon can go partly into the '+' channel, partly into the '&minus;' one, resulting in the possibility of simultaneous detections in both. Though experiments such as Grangier et al.'s (Grangier, 1986) have shown that this probability is very low, it is not logical to assume it is actually zero. The definition of quantum correlation is adapted to the idea that outcomes will always be +1, &minus;1 or 0. There is no obvious way of including any other possibility, which is one of the reasons why [[Clauser and Horne's 1974 Bell test]], using single-channel polarisers, should be used instead of the [[CHSH Bell test]]. The "CH74" inequality concerns just probabilities of detection, not quantum correlations.

===Generalisations of the models===

By varying the assumed probability and density functions in equation (1) we can arrive at a considerable variety of local realist predictions.

====Chaotic Ball model====

An interesting modification of the deterministic case is obtained if we assume that the dividing great circles are not just thin lines but broader bands, within which there are ''no'' detections. This model has been developed in some detail by C. H. Thompson (Thompson, 1996). The predicted quantum correlations violate the [[CHSH Bell test]] when applied in the currently-accepted fashion. With deterministic "missing bands", the prediction consists of segments of straight lines, but it would clearly be possible to assume the probability of detection to vary gradually, changing the lines into curves and approaching closely to the quantum-mechanical formula.

====Optical models deviating from Malus' Law====

If we make realistic (wave-based) assumptions regarding the behaviour of light on encountering polarisers and photodetectors, we find that we are not compelled to accept that the probability of detection will reflect Malus' Law exactly.

We might perhaps suppose the polarisers to be perfect, with output intensity of polariser A proportional to ''cos''²(''a'' &minus; &lambda;), but reject the quantum-mechanical assumption that the function relating this intensity to the probability of detection is a straight line through the origin. Real detectors, after all, have "dark counts" that are there even when the input intensity is zero, and become saturated when the intensity is very high. It is not possible for them to produce outputs in exact proportion to input intensity for ''all'' intensities.

By varying our assumptions, it seems possible that the realist prediction could approach the quantum-mechanical one within the limits of experimental error (Marshall, 1983), though clearly a compromise must be reached. We have to match both the behaviour of the individual light beam on passage through a polariser and the observed coincidence curves. The former would be expected to follow Malus' Law fairly closely, though experimental evidence here is not so easy to obtain. We are interested in the behaviour of very weak light and the law may be slightly different from that of stronger light.

===Effects of other experimental imperfections===

When searching for a realist alternative that will match observations, it must not be forgotten that the published observations are not obtained under perfect conditions. A realist model needs to allow for known deviations from perfection. See the page on [[Loopholes in optical Bell test experiments|loopholes in optical Bell tests]] and (Thompson, 2004).

===References===

* '''Bell, 1971''': J. S. Bell, in ''Foundations of Quantum Mechanics'', Proceedings of the International School of Physics “Enrico Fermi”, Course XLIX, B. d’Espagnat (Ed.) (Academic, New York, 1971), p. 171 and Appendix B. Pages 171-81 are reproduced as Ch. 4, pp 29-39, of J. S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'' (Cambridge University Press 1987)

* '''Bohm, 1951''': D. Bohm, ''Quantum Mechanics'', Prentice-Hall 1951

* '''Clauser, 1974''': J. F. Clauser and M. A. Horne, ''Experimental consequences of objective local theories'', Physical Review D, '''10''', 526-35 (1974)

* '''Clauser, 1978''': J. F. Clauser and A. Shimony, ''Bell’s theorem: experimental tests and implications'', Reports on Progress in Physics '''41''', 1881 (1978)

* '''Grangier, 1986''': P. Grangier, G. Roger and A. Aspect, ''Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences'', Europhysics Letters '''1''', 173-179 (1986)

* '''Marshall, 1983''': T. W. Marshall, E. Santos and F. Selleri, ''Local Realism has not been Refuted by Atomic-Cascade Experiments'', Physics Letters A, '''98''', 5-9 (1983)

* '''Thompson, 1996''': C. H. Thompson, [http://arxiv.org/abs/quant-ph/9611037 ''The Chaotic Ball: An Intuitive Analogy for EPR Experiments''], Found. Phys. Lett. '''9''', 357 (1996)

* '''Thompson, 2004''': C. H. Thompson, [http://freespace.virgin.net/ch.thompson1/Papers/TheRec/TheRecord.htm ''Setting the Record Straight on Quantum Entanglement''] (2004). This repeats the ealier description of the Chaotic Ball model, adding updated information on the validity of the various Bell tests and the experimental loopholes.

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