The Notion of Local Entanglement

One first recalls the concept of entanglement between two physical systems, which is essential in quantum physics. Briefly expressed, it shows that, when two physical systems interact, their two states can never return to their initial independence, and their wave functions remain mingled together by algebraic knots, which nothing can unravel.

Schrodinger said that this property is the one by which quantum mechanics differs most from classical mechanics. The essence of Schrodinger's famous “cat problem”, was that the state of a cat and the state of a radioactive source, together enclosed in a box, came out in an entangled state, out of which nothing could take them.

Mathematically, entanglement means that a wave function involving the cat and the radioactive source is a sum of two distinct wave functions, the first one being a product of a wave function of the cat, alive, with a wave function of the radioactive source, still intact. The second function is also a product of two wave functions involving a dead cat and a decayed source.

The algebraic character of entanglement is obvious, since its expression holds into a sum of two products (of wave functions). This character is hard to “see” intuitively, but it stands as the fortress in which one will have to find a weak point.

A noticeable event occurred in 1972, when Eliot Lieb and Derek Robinson discovered new properties of entanglement (Lieb and Robinson 1972). These were local properties, contrary to all known other ones. They did not involve all the atoms in a system, whereas everything was involved together in the box where Schrodinger’s cat was locked up. Lieb-Robinson's discovery occurred however in the field of spins lattices, which is of importance in statistical physics but rather far from the domain of research regarding collapse, where it went practically unno­ticed.

Perhaps this indifference of the community of research on collapse, was due to a belief, which John Bell expressed clearly: He wrote, essentially what most people believed more or less, which was that the deep problem of uniqueness of Reality was certainly not accessible by means of the standard methods of physics, which are only valid “for all practical purposes” (Bell 1987).

Since then, a remarkable book by Serge Haroche and Jean-Michel Raimond reported many experiments, mostly in quantum optics, which made sure that col­lapse never occurs at a microscopic scale, but only at a macroscopic one (Haroche and Raimond 2006). But since macroscopic physics is the domain of “fapp” methods (valid only for practical purposes), what does that mean?

There is certainly a lesson there: Collapse is entirely restricted to macroscopic conditions, and this is also true of the uniqueness of physical Reality, which can be seen as an emergence of the quantum world into the classical one.

One main point of the present work is that local entanglement holds the keys for collapse (I use the plural because one will see that there are two keys and the corresponding doors will have to be opened one after the other). The first one is concerned with local entanglement between a measuring system and a measured system.

One can introduce this notion of local entanglement in a case far away from quantum mechanics. This example involves a macroscopic gas, which is made of atoms: for instance an argon gas acting as the detecting part of a Geiger counter. One denotes it by B. It can express by a signal that an energetic charged particle has crossed it. One denotes this “measured microscopic system” by A.

One knows since Boltzmann how to describe the atoms in a gas by the equations of mechanics. One adopts for a moment this viewpoint, by assuming that the position and velocity of all atoms is given at some initial time 0, before the arrival of the external particle A.

As a preliminary, and an alternative to the notion of local entanglement, one will think of an “influence” of the particle A on the atoms, in the following sense: When the particle collides with one atom, the motion of this atom is strongly modified and its later motion conserves ever after an influence of this event. When comparing the difference, or lack of difference, between the state of motion of every atom in the case when A interacted with the gas and when it did not, one can distinguish whether a definite atom was “influenced”, of not influenced by A. If the two data differ, one says that the atom was influenced by the particle A.

One can illustrate this phenomenon by imagining that all atoms in B carry ini­tially a white color, whereas A is colored red. Influence is expressed then by the idea that, when the red particle A collides with a white atom a, it makes this atom become red (“influenced”). This red color is carried then to other atoms, either under direct collisions of atoms with A, or under collisions of an already red atom with a white one. When two white atoms (“non influenced”) collide, they remain white.

One can conceive easily that the red color expands then, starting from the trajectory of the particle A, but a mathematical formulation shows much more. When seen from the standpoint of classical statistical mechanics, the collisions between atoms bring an expansion of the red color. The mathematical formulation of this expansion is governed by a diffusion equation (like the one, which describes the diffusion of heat, for instance).

If one denotes by f₁ (x, t) the probability for the atoms to be red, near a point x in the gas, and by f₀(x, t) their probability for being white, the transfer of redness (which looks much like a contagion of measles) is due to collisions involving one red atom and one white one. Its probability is therefore proportional to the product fi(x)fo(x) and yields a nonlinear process (since f₀ = 1 — f₁).

This type of equation is known in the theory of nonlinear waves, where it is often characterized by a moving front, which separates in the present case a red region from a white one. More precisely, the gas is white in an outside region, which the front has not yet reached; behind the front, it shows shades of pink until it reaches full red at a distance of order of one mean free path of atoms. The front moves at a fixed velocity, which coincides with the velocity of sound in the case of a dilute gas.

Quite remarkably, these qualitative and semi-quantitative properties remain valid in the quantum case. It is more proper then to speak of “local entanglement” between A and B, rather than of an influence by A on B. One will use sometimes the abbreviation “LE” for “local entanglement”, or for “locally entangled” when used in place of an adjective.

As a matter of fact, the existence and propagation of LE are predictions of the Schrddinger equation!

This prediction can look surprising (at least), since it is completely at variance with the standard interpretation of quantum mechanics. The reason is that LE cannot be associated with quantum observables, whereas the standard interpretation claims that everything possessing a physical meaning should necessarily be expressible by observables. There is no inconsistency in the difference, but there is certainly novelty.

There is something more: The previous probabilities of influence and of no influence, fi (x) and f₀(x), can be still defined (by using “locally entangled quantum fields”) and they make sense in quantum mechanics for a wave function yr. They are then positive, smaller than 1 and sometimes equal to 1, but their sum is generally not equal to 1: It can be larger or smaller. In spite of their formal existence, these quantities cannot be therefore considered as having a meaning of probabilities.

But there is still more: When one considers a many-body system (like the gas, which one takes as an example), one must deal no more with an individual wave function, but with a “mixed state” (or “density matrix”), which involves an extre­mely high number of wave functions (“eigenfunctions” or “eigenvectors”). There are then compensations, between the contributions of different eigenfunctions and also of pairs of them, of the deviations from 1 of f1 (x) + f₀(x). This sum becomes 1, with very high precision, when referred to the “real” mixed state of a many-body system. One can therefore consider them, legitimately, as being respectively local probabilities of LE and of no LE.

There is something startling in this extremely simple result, not because of its restriction to conditions of high complexity, but because it is valid in these realistic conditions. It forces one to get out of the domain of the standard interpretation and, at the same time, to discover that there is new physics in the domain of LE. One began to explore it some years ago, and found it much richer than could be expected at first sight.

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