Introducing the hydrodynamic equations (Madelung [416]) associated with the evolution of the wave function, and using them to guide the evolution of the additional variables (positions of particles), is a natural idea. In the dynamics of fluids, hydrodynamic equations can be obtained by taking averages of microscopic quantities over positions and velocities of point-like particles; for instance, the Navier-Stokes macroscopic equations can be derived from the Boltzmann transport equation by appropriate microscopic averages (Chapman-Enskog method); conversely, the hydrodynamic variables will influence the motion of individual particles. Moreover, there is some analogy between the guiding term and the force term in a Landau type kinetic equation, where each particle is subject to an average force proportional to the gradient of the density of the others. Nevertheless, here we are dealing with a single particle, so that the guiding term cannot be associated with interactions between particles. Moreover, we also know from the beginning that rather unusual properties must be contained in the guiding equations, at least if we wish to exactly reproduce the predictions of usual quantum mechanics: the Bell theorem states that the additional variables have to evolve non-locally in ordinary three-dimensional space (they evolve locally only in the configuration space of the system, exactly as for the state vector). In other words, in real space the additional variables must be able to influence each other at an arbitrary distance. Indeed, in the Bohmian equation of motion of the additional variables, the velocity of a particle contains an explicit dependence on its own position, as expected, but also a dependence on the positions of all the other particles (assuming that the particles are entangled).

In this appendix, we introduce a few simple models involving modified Schrödinger dynamics with stochasticity, in order to illustrate how such models may lead to an evolution that reproduces the reduction of the state vector during a measurement (emergence of a single eigenvalue during a single realization, with a random value). For the sake of simplicity, weignore the usual Hamiltonian evolution during the time of measurement, assuming for instance that this time is too short for this evolution to be significant; otherwise, it would be necessary to use the interaction representation with respect to the Hamiltonian, which does not change the calculations much, except that this introduces a time dependence of the operators.

Single operator

We consider the measurement of some quantum observable associated with an Hermitian operator A; we look for an equation of evolution containing a state vector reduction process associated with this particular measurement. Since the final eigenvector must vary randomly from one realization to the next, the evolution equation necessarily contains a random component. In our case, it will take the form of a random function of time (as opposed to the GRWtheory where the stochasticity is introduced by the discontinuous “hitting processes”, see §10.8.1.b).

Equation of evolution

We assume that the state vector ∣ψ(t)〈 evolves according to:

where w(t) is a real random function of time. In order to simplify the model as much as possible, we may discretize time into small finite intervals Δt, during which we assume that w(t) remains constant; moreover, we may also assume that the possible values of w(t) belong to a finite discrete ensemble w1, w2, …, wN.

The process of measurement plays an important role in quantum theory. Measurements can be direct, if the physical system S interacts directly with the measurement apparatus M (as we have assumed until now), or indirect. In the latter case, the physical system first interacts with an ancillary system B, which may have a space of states that is very different from that of S, for instance much larger; after this interaction has finished, M is used to perform a measurement on B, without any direct interaction with A. Because S is then “protected” from any direct interaction with the measurement apparatus, the state of S is not necessarily strongly modified, and may even be only weakly affected. In both cases, the process implies entanglement between several physical systems. In this chapter, we study how this entanglement is created and used for measurements as well as the notion of weak and continuous measurements. These questions play an important role in several of the interpretations of quantum mechanics that we discuss in Chapter 10.

Direct measurements

The Von Neumann model of quantum measurement [4] provides a general frame for describing the process in terms of correlations appearing (or disappearing) in the state vector associated with the whole system S +M. In this model, the two systems S andM are initially described by a product state ∣ψ0〈 and interact during the time of measurement, so that they become entangled; they then reach a final state ∣ψ′〈 and do not interact anymore.

The Bell theorem can take the form of several inequalities, as we have seen in §4.2. Moreover, since it was discovered, the theorem has stimulated the discovery of several other mathematical contradictions between the predictions of quantum mechanics and those of local realism. We review a few of them in this chapter: GHZ contradictions (§5.1) and their generalization (§5.2), Cabello's inequality (§5.3), and Hardy's impossibilities (§5.4). Finally, in §5.5, we discuss the notion of contextuality and introduce the BKS theorem.

GHZ contradiction

For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism [188, 189]. They involve a sign contradiction (100% violation) for perfect correlations, while the BCHSH inequalities are violated by about 40% (Cirelson bound) and deal with situations where the results of measurements are not completely correlated. In this section, we discuss three-particle systems, but generalizations to N particles are possible (§5.2).

Derivation

GHZ contradictions may occur in various systems, not necessarily involving spins. Initially, they were introduced in the context of entanglement swapping (§6.3.2) for four particles [188] or entanglement of three spinless particles [189].

More than 70 years after its publication, the article by Einstein, Podolsky, and Rosen (EPR) [88] is still cited hundreds of times every year in the literature; this is a very exceptional case of longevity for a scientific article! There is some irony in this situation since, for a long time, the majority of physicists did not pay much attention to the EPR reasoning. They considered it as historically interesting, but with no precise relevance to modern quantum mechanics; the argument was even sometimes completely misinterpreted. Astriking example is given in the Einstein-Born correspondence [89] where Born, even in comments that he wrote after Einstein's death, clearly shows that he never completely understood the nature of the objections raised by EPR. Born went on thinking that the point of Einstein was a stubborn rejection of indeterminism (“look, Albert, indeterminism is not so bad!”), while actually the major concern of EPR was locality and/or separability (we come back later to these terms, which are related to the notion of space-time). If giants like Born could be misled in this way, no surprise that, later on, many others made similar mistakes!

This is why, in what follows, we will take an approach that may look elementary, but at least has the advantage of putting the emphasis on the logical structure of the arguments and their generality. Doing so, we will closely follow neither the historical development of the ideas nor the formulation of the original article, but rather will emphasize the generality of the EPR reasoning.

We give an example of such a non-deterministic local theory that looks similar to quantum mechanics, and actually even makes use of its formalism, while it is in fact significantly different. This theory includes the non-determinism of quantum mechanics, but gives to the state vector a role that is more local than in standard quantum mechanics. For this, we consider a physicist who has well understood the basic rules of quantum mechanics concerning non-determinism, but who remains sceptical about non-locality (or non-separability; for a detailed discussion of the meaning of these terms, see §3.3.3.b as well as, for instance, [24, 47]). This physicist thinks that, if measurements are performed in remote regions of space, it is more natural to apply the rules of quantum mechanics separately in these two regions. In order to calculate the probability of any measurement result, he/she will then apply the rules of quantum mechanics, in a way that is locally perfectly correct, but that also assumes that it is possible to reason separately in the two regions of space. If for instance the two measurements take place in two different galaxies, our sceptical physicist is prepared to apply quantum mechanics at the scale of a galaxy, but not at an intergalactic scale!

How can one then treat the measurement process that takes place in the first galaxy? It is very natural to assume that the spin it contains is described by a state vector (or by a density operator, it makes no difference here) that may be used to apply the orthodox formula for obtaining the probabilities of each possible result.

In this chapter, we study the properties of quantum entanglement, and more generally the way correlations can appear in quantum mechanics. Quantum entanglement is an important notion that we have already discussed, for instance in the context of the Von Neumann chain or of the Schrödinger cat, but here we give more details on its properties.

In classical physics, the notion of correlation is well known. It hinges on the calculation of probabilities and on linear averages over a number of possibilities. A distribution gives the probability of having the first system in a some given state and the second in another state. If this distribution is not a product, the two systems are correlated. If it is a product, they are uncorrelated; measuring the properties of one system does not bring any information on the other. This is in particular the case if the state of each of the two systems is perfectly defined (which also defines the state of the whole system perfectly well). The notion of correlation between sub-systems therefore stems from the multiplicity of possible states of the whole system; fluctuations of this state are necessary to give its full meaning to the classical notion of correlation.

In quantum mechanics, the situation is different: as we have seen (in particular in Chapter 4), even a physical system that is perfectly defined by a given state vector already contains fluctuations.

From a fundamental point of view we must acknowledge that, since about 1935, our conceptual understanding of quantum mechanics has not progressed so much. Really new ideas are few and far between – except of course the major line initiated by the contribution of Bell [6]. This is in big contrast with the rest of physics, where new theoretical and experimental discoveries in many fields have flourished, very often with the help of the tools of quantum mechanics. The fantastic evolution of the experimental techniques has completely changed the situation. At the beginning of quantum mechanics, the observation of the tracks of single particles in Wilson chambers [348] played the essential role in the introduction of the postulate of state vector reduction, but otherwise it was impossible to observe continuously a single electron, atom, or ion; the experiments that theorists were proposing in discussions on the principles of quantum measurement were therefore “thought experiments” (“Gedanken Experiment”), as for instance in the famous Solvay meetings [1, 21]. But nowadays, after almost of century of experimental progress, experiments that were then unthinkable have become a reality.

A huge number of contemporary experiments involves the laws of quantum mechanics in general; several books would not be sufficient to describe all of them. Nevertheless, in the majority of experiments, what is really observed is the sum over a very large number of particles of one individual microscopic observable (sum of atomic dipoles for instance), which is accurately described by the average value of this observable.

Time correlation functions

In standard quantum mechanics, the calculation of any two-time correlation function has to include the evolution of the system between the two times considered; this evolution is contained in the unitary evolution operator U(t′, t), as for instance in relation (10.9). In Bohmian theory, it is important to take into account the effect of the first measurement, which correlates the system under study to a measurement apparatus and creates “empty waves” (§10.6.1.c). Otherwise one obtains contradictions with the standard predictions.

For instance the author of [443] considers a one-dimensional harmonic oscillator that is initially in a stationary state, and studies the correlation function of the position at times t and time t′, in the particular case where t′ − t is equal to half the period 2π/ω of the oscillator. In standard quantum mechanics, it is easy to show that the corresponding position operators 〉X(t) and X(t′) are then opposite, so that the correlation function 〉X(t)X(t′〈 is equal to − 〉[X(t)]2〈, therefore negative. In Bohmian mechanics, the particle is initially static since the wave function is real. If one ignores the effect of the first measurement, the position of the particle will remain at the same place, which corresponds to a correlation function equal to 〉[X(t)]2〈, therefore positive; one reaches an apparent complete contradiction. But, if one takes into account the effect of the first measurement on the particle, one finds that, just after this measurement, each position of the oscillator becomes correlated with a different Bohmian position of the pointer.