Pre-print a cura dell’autore. Originale pubblicato su:
Problems in Quantum Physics, GDANSK 1987, Editors World Scientific
ON THE PHYSICAL MEANING OF BELL’S INEQUALITY
AND THE RELATED EXPERIMENTAL TESTS
Dipartimento di Fisica –Università di Catania – 95129 Catania, Italy
ABSTRACT: It is shown that the experiments performed so far in order to test Bell’s inequality are not suited to test for determinism and locality, because of an intrinsic impossibility of factorizing the inputs of the required coincidence circuit. On the other hand the factorizability hypothesis is a necessary ingredient for proving the inequality and implies the need for ideal apparatuses which, however, cannot be easily realized in practice without a detailed deterministic model which is, in principle, rejected by the usual interpretation of quantum mechanics.
After Bell’s result 1] and the experimental verification of the quantum mechanical previsions, 23] it seems that the following propositions can be safely accepted:
Proposition A: Any classical separable physical system must obey Bell’s inequality
and because quantum physical systems do exist which violates Bell’s inequality and, moreover, such violations are experimentally well confirmed, we are entitled to assert the following:
Proposition B: No local hidden variable theory can reproduce all the results of quantum theory which are experimentally well verified.
We shall clarify, in the sequele, the precise meaning of the terms used in propositions A and B, but by now we accept their usual meanings and observe that, certainly, A implies B but, of course, if A should not be true then B may not be true or, at any rate, Bell’s inequality cannot be used to assert B. In order to falsify proposition A, given its logical structure, it will be enough to exhibit a single concrete example of a separable classical physical system which violates Bell’s inequality. Such an example, in fact, has already been given in a previous paper.4] Here, we shall briefly summarize it in order to examine some logical consequences which may have some bearing on the physical interpretation of quantum mechanics.
In order to present the essential point without introducing unnecessary metaphysical presumptions, we shall try to describe the paradigmatic physical situation avoiding any term which cannot be reconducted to experimental observations or manipulations. In such terms we consider a black box, S, representing our physical system, defined by certain physical manipulations. On opposite sides of S, in a given spatial direction which we call the z‑axis, we have a pair of objects P1 and P2 which we call polarizers, each characterized by a unit vector in the plane normal to the z‑axis. The directions of these vectors can be varied by opportune knobs. We have, in series to the polarizers, two detectors D1 and D2 which register counts in the presence of S. We also have a coincidence system which gives a count if and only if, separately, two counts will be registered in D1 and D2 within a fixed interval of time wich we call the resolving time. The polarizers affect the counting intensity of the two detectors and and, hence, of the coincidence in dependence of the setting of the knobs.
Given such an experimental situation, it is easy, by using Bell’s hypotheses, to derive his inequality 4] and to verify that all experiments performed so far to test it 2,3] fall under this scheme. However, we may also fill in the black box with a classical separable (actually: separated) physical system S, which reproduce the quantum correlation curve and so violates Bell’s inequality. Such a system is constituted by two identical subsystems S1 and S2 each made of a one-dimensional lattice of n + 2 classical particles with linear interactions between nearest neighbours only. The equilibrium positions of all the particles in both subsystems are aligned along the z‑axis, but the particles are free to oscillate in the (x, y)‑plane. It is supposed, for simplicity, that in both subsystems: a) the elastic constants in the longitudinal directions are stiff enough so that we have, practically, only transverse vibrations; b) the transverse couplings and masses are identical for all particles; c) the particles with index ρ = 0 and ρ = n + 1 are constrained to their equilibrium positions. The effect of the polarizers can be realized by putting constraints on the two particles (one for each subsystem) with ρ = σ , being σ a specified value of the index, comprised in the range 1 £ σ £ n. These constraints will impede vibrations along a specifìed direction (hard direction) in the (x, y)‑plane, letting the particle vibrate only along the orthogonal direction (easy direction), say ; this implies the following equation for the constraints: xσ sin = yσ cos, on each side. Such constraints will act on the propagating motion of the particles like a calcite cristal acts for light waves. In fact, if we give non-zero initial conditions only to particles with ρ < σ, the vibration state of all particles will be modified with respect to the non-constrained state with same initial conditions but the vibration component along the easy direction will not be affected by the constraints and, by observing the particles with ρ > σ, it wiIl be the only one actually present, so that the constraints generate, for such particles, a state of linear polarization along the easy direction . As detectors, we may assume devices such that, without perturbing in any sensible way the system, will give a count every time some appropriate dynamical variable crosses a given threshold with positive slope (in our particular case we have chosen the square modulus of the vibration amplitude along the easy direction of the two particles, one for each side, with ρ = n). Of course, such combined system of polarizer-detectors obeys the separability (locality) requirement, used to derive Bell’s inequality, namely: the registering of a count on one side is not influenced by the setting of the knob (constraint) on the other side. Here, of course, there is no problem with any possible superluminal communication because the two subsystems are actually separated at whatever relative distance of systems and detectors.
The motion of each particle results in a sovrapposition of circular motions whose frequencies have irrational ratios. It is obvious that, given the stationarity of the motion, our system can be replaced by a symmetrical statistical ensemble of couples of identical, physically independent but correlated, spinning particles with identical precession motion (but for the initial phases) realized by suitable external forces; in such a way, we can better simulate an experiment of the Bohm-Aharonov type.5] By a computer simulation of the described experiment, 4] by choosing particular values for the various parameters of the system, we were able to obtain the quantum correlation curve for an ideal experiment realized with an atomic cascade, 2,3] thus violating Bell’s inequality.
In order to see more perspicuously the reason for such a violation we may consider a much simpler classical separable system constituted by two classical particles describing identical simple circular motions (also here, we can think of a statistical ensemble of couples of precessing classical spins) and observe the coincidences between the threshold crossings of the square modulus of the amplitude at an angle , which in this case is given, for each of the two particles, by A2cos2(ωt - θ). If T is the value of the threshold, the crossing times will be tr(θ) = c(T) + (θ + r π)/ω with c(T) independent on θ, being r an integrai number. For three fixed directions θ1 = θ, θ2 = 3θ and θ3 = 2θ and for a given r, we have: tr(θ2) - tr(θ3) = tr(θ3) - tr(θ1) = θ/ω. By choosing a resolving time for the coincidence such that θ/ω < 2τ < 2θ/ω, Bell’s inequality will be systematically violated because we bave a coincidence in channel pairs (θ1, θ3) and (θ2, θ3) but not in channel pair (θ1, θ2), in spite of the fact that the three events, in the different channels, are totally correlated. We shall see that this is a general fact for any usual coincidence circuit, so that such a device is not suited to test Bell’s inequality which is derived, as we shall show, under an implicit hypothesis of factorizability of the coincidence inputs.
In fact, to describe the working of a coincidence circuit,6] as those used so far,2,3]we may consider two trains of electrical pulses incoming on the two inputs of the coicidence system: let A(t) and B(t) be the electrical voltages, as functions of the time, at the two inputs of the coincidence system; let α º [ t1, t1 + Δ1] and β º [ t2, t2 + Δ2] be two time intervals; let fA(α,χ1) and fB(β,χ2), with χ1 Î α a and χ2 Î β, be two dichotomic random variables taking the value 1 if, at least, a pulse (threshold level crossing) generated by A(t) or B(t) lies respectively in α or β and taking the value 0, otherwise; let fAB(α,β,χ) be a binary random variable taking the value 1 if a pulse generated by A(t) has arrived at χ1 Î α and, simultaneously, a pulse generated by B(t) has arrived at χ Î β with the condition t2 = χ1 or, alternatively, the same sequence of pulses has come but with an inverted role of the two inputs, while taking the value 0 in all other cases; the variable fAB, by definition, represents 6] the output of the coincidence circuit after the inputs fA and fB, and cannot be expressed as the product of the two inputs, i.e.: fAB(α,β,χ) ¹ fA(α,χ) fB(β,χ). But the named factorizability hypothesis is necessary to demonstrate Bell’s inequality, as we shall see, so that the output of a coincidence system may violate this inequality, independently of the local or nonlocal definition of the inputs and therefore is not suited to test factorizability and, a fortiori, cannot say anything about the less restrictive classical concept of locality as defined by Einstein 7] . A coincidence system without such unwanted features can be devised by fixing, a priori, a time window and counting a coincidence only if two pulses, coming from the two inputs, happen to stay in such a fixed time interval, but, unluckily enough, such a device is bound to obey Bell’s inequality by the very defìnition of a coincidence that it has been given in this case and, also here, independently of the local or non‑local definition of the inputs.
But let us see if we can find out a more convenient experimental situation which may allow us to verify Bell’s inequality. In order to explore such a possibility it is convenient to analyse more carefully Bell’s mathematical hypotheses and compare them with the usual definitions of the concepts of determinism or causality on one side and separability or locality on the other side (the two terms in each couple are, usually, treated as synonymous in the physical literature).
In the words of Heisenberg, which were intended to express the old conception of determinism as mantained by Einstein but to be substituted by a new one compatible with Quantum Mechanics, the law of causality is formulated as follows: 8] “If at a certain time all data are known for a given system then it is possible to predict unambiguously the physical behavior of the system also for the future.”
EPR added to this a request for realism:7] “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”
The preceding definitions imply also the principle of statistical determinism as expressed by d’Espagnat: 9] “If two statistical ensembles of physical systems are submitted to identical treatments, and if subsequent observation reveals significant statistical difference between them, the implication is that the two ensembles were not identical at the start.”
The notion of locality, referring to two physical systems which are correlated by some physical interaction which they may have undergone in the past but anyway no longer interacting, is introduced by Einstein with the following words: 10] “The factual situation of the system S2 is independent of what is done with the system S1, which is spatially separated from the former.”
In the following we shall use the term classical for any system or observable compatible with the previous standard definitions.
Bell’s mathematical formulation of determinism is stated at p. 195-6 of his paper 1] by pretending that the Q.M. operators, one of which yields the value +1 upon a measurement of the component of the spin of the first particle, where is some unit vector, when the other yields the value -1 upon a measurement of the spin component of the second particle and viceversa, are represented, in a more complete speciflcation (i.e. deterministically), by two observable , , λ being the hidden-variable. So he is postulating a correspondence between the deterministic observables and and the quantum operators and (we have indicated with the symbol f the mapping which gives the right correspondence, from the set of the bounded quantum operators to its counterpart represented by the set of the deterministic observables 11]). On p.195 he says: “The result A of measuring is then determined by and λ, and the result B of measuring in the same instance is determined by and λ”, so that the measured values of the deterministic observables for each possible value of the unit vector must belong to the spectrum of the corresponding quantum operators ; we summarize this assertion by the formula:11]
Bell’s mathematical formulation of locality is explicitly worded at p.196 by pretending “that the result of B for particle 2 does not depend on the setting , of the magnet for particle 1, nor A on ”; but implicitly assuming in his formula (2) that the elementary event, determined by a specified element λ in the statistical ensemble Λ of the hidden variables, is given by the product , or, in his own words: “If ρ(λ) is the probability distribution of λ then the expectation value of the product of the two components and is
(formula (2) of Bell’s paper). This should equal the quantum mechanical expectation value, which for the singlet state is
in other words the deterministic observable corresponding to the quantum operator given by the product corresponds also to the product of the two deterrninistic observables A and B corresponding respectively to the quantum operators and or, in a formula: 11]
Besides: formula (2) of Bell’s paper implies that, for the quantum operator , its quantum expectation value is equal to the expectation value of the corresponding deterministic observable, for any quantum state , thus also for any distribution density of the hidden variables i.e.
with ρ independent on and f independent on . All this implies that is (modulo an unobservable * functional F, such that , for any and any ) a linear functional of . We remind here that, according to a generalization of the Gleason‑Kahane‑Zelazko theorem, for any invertible operator either in the Banach or in the Jordan algebra of the bounded quantum operators, formula (2) implies (1) and also 11]
which corresponds to von Neumann’s definition of hidden variables as the determinants of the dispersion free states. 13]
But now we can see that many classical observables exist which do not obey Bell’s axioms and may violate his inequality. We remind that it has always been asserted that the formalism of Quantum Mechanics takes into account the role of the observing apparatus on the observed physical system. There are many ways in which such a role can be introduced in a classical model. The simplest way is to suppose that any classical observing apparatus performs a time averaging on any classical dynamical variable ** and that the quantum operator corresponds to such phenomenological observable and not directly to the classical dynamical variable. We may represent such a phenomenological observable by:
with x(χ) a point in phase space at time χ . We remind, here, that such an integration is always performed in any experimental observation by the high fequency cut introduced by any concrete physical measuring device. Obviously such classical observables obey all the standard definitions of determinism and locality given before but cannot obey conditions (1), (2), or (3). In fact, given any two classical dynamical variables f and g, for the corresponding classical phenomenological observables we have, in general (excluding exceptional cases as, for instance, the case in which one or both variables take constant values in the course of the time), (we may suppose, here, that the observable is being measured by a simple correlator). Also (3) and, for classical observables corresponding to bounded quantum operators, (1) are, obviously, violated. We note that such violations apply also to the dichotomic variables envisaged by Bell 1] and the violation may be very strong in dependence of the average time duration of the two alternating values of the variables, as compared with the range of integration, as determined by the parameter τ ; but such detailed knowledge can only be obtained by means of a fully deterministic model which, however, is, in principle, excluded by the usual interpretation of Quantum Mechanics. So that, in general, Bell’s hypotheses can be applied only for ideal observations not realized in practice.
Of course, many other ways can be introduced, classically, to take into account the role of the observing apparatus as the number of zero-crossings of a fluctuating but deterministic time function, with the same effect on Bell’s inequality as considered in the previous example, whose detailed study has been described in Ref.4, showing the intrinsic non-factorizability of any coincidence circuit used so far in order to test Bell’s inequality.
In conclusion, from what we bave shown, it emerges: Bell’s inequality is, of course, a correct mathematical theorem under his hypotheses. Two of such hypotheses (namely, determinism and locality) have been exphicitly considered by Bell; a third hypothesis (factorizability) was implicitly advanced, certainly because he was considering ideal measuring apparatuses. The theorem is physically relevant on the theoretical side because it forbids any local classical interpretation of Q.M. which does not take into account the role of the finite resolution of any physical device. However, on the empirical side, Bell’s theorem cannot forbid a local classical interpretation of Q.M., when the role of the measuring device is taken into account for a more complete specification, as envisaged by EPR, of the experiments. Of course, our considerations cannot tell how such a complete specification could be achieved but simply tell that the experiments devised for this purpose, at date, cannot forbid it.
Notes and References
* See axiom two of Mackey axiomatics 12] TORNA
** This hypothesis is a generalization of the phenomenological derivative introduced to cope with Loschimdt and Zermelo paradoxes in classical statistical mechanics; see 14] TORNA
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