Local hidden variable theory

In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no "instantaneous" (or at least faster-than-light) effect on local ones. According to the quantum entanglement theory of quantum mechanics, on the other hand, distant events may under some circumstances have instantaneous correlations with local ones. As a result of this it is now generally accepted that there can be no interpretations of quantum mechanics which use local hidden variables. (There are those who dispute this. Their arguments are called "loophole theories", referring to a loophole in the theory of quantum mechanics.) The term is most often used in discussions of the EPR paradox and Bell's inequalities. It is effectively synonymous with the concept of local realism, which can only correctly be applied to classical physics and not to quantum mechanics.

Local hidden variables and the Bell tests

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:

:: (1) $P(a,b)\; =\; int\; d\; lambda\; cdot\; ho(lambda)\; cdot\; p\_A(a,\; lambda)\; cdot\; p\_B(b,\; lambda)$

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

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

When 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".

The difference between the two predictions is due to the different functions $p\_A(a,\; lambda)$ and $p\_B(a,\; lambda)$ 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 Malus's Law.

Bell tests with no "non-detections"

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 values of the hidden variable are represented by all points on the surface of a unit sphere.

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 −1). The surface of the sphere will be divided into two regions, one for +1, one for −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.

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 −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" (++, −−, +− or −+) 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 λ 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 — 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 ρ constant and the probability functions step functions. The principle behind (1) has in fact been used, but purely intuitively.

Thus the local hidden variable prediction for the probability of coincidence is proportional to the angle ("b" − 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) "E" = "P"₊₊ + "P"₋₋ − "P"₊₋ − "P"₋₊

where "P"₊₊ is the probability of a '+' outcome on both sides, "P"₊₋ that of a + on side A, a '−' on side B, etc..

Since each individual term varies linearly with the difference ("b" − "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 that 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 $p\_A(a,\; lambda)$ and $p\_B(a,\; lambda)$, 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) $P(a,b)\; =\; int\; d\; lambda\; cdot\; ho(lambda)\; cdot\; cos^2(a\; -\; lambda)\; cdot\; cos^2(b\; -\; lambda)\; =\; frac\{1\}\{8\}\; +\; frac\{cos^2\; phi\}\{4\}$, where $phi\; =\; b\; -\; a$.

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

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 '−' 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 that it is actually zero. The definition of quantum correlation is adapted to the idea that outcomes will always be +1, −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.

Generalizations 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.

Time effects

Previously some new hypotheses were conjectured concerning the role of time in constructing hidden variables theory. One approach is suggested by K. Hess and W. Philipp (Hess, 2002) and discusses possible consequences of time dependences of hidden variables, previously not taken into account by Bell's theorem. This hypothesis has been criticized by R.D. Gill, G. Weihs, A. Zeilinger and M. Żukowski (Gill, 2002).

Another hypothesis suggests to review the notion of physical time (Kurakin, 2004). Hidden variables in this concept evolve in so called 'hidden time', not equivalent to physical time. Physical time relates to 'hidden time' by some 'sewing procedure'. This model stays physically non-local, though the locality is achieved in mathematical sense.

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" − λ), 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.

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 Theory", 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)
* Gill, 2002: R.D. Gill, G. Weihs, A. Zeilinger and M. Żukowski, [http://arxiv.org/abs/quant-ph/0208187 "No time loophole in Bell's theorem; the Hess-Philipp model is non-local"] , quant-ph/0208187 (2002)
* 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)
* Hess, 2002: K. Hess and W. Philipp, Europhys. Lett., 57:775 (2002)
* Kurakin, 2004: Pavel V. Kurakin, [http://www.geocities.com/bellstheorem/ "Hidden variables and hidden time in quantum theory"] , a preprint #33 by Keldysh Inst. of Appl. Math., Russian Academy of Sciences (2004)
* 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)

ee also

* Hidden variable theory