The Notion of Slip in Coherence

In 1976, Philip Pearle proposed that collapse could be the outcome of a Brownian process, rather than being sudden and total, (Pearle 1976). The random character of collapse, as well as its agreement with Born's fundamental probability law, would come out then almost automatically.

It turns out that the explicit mechanism of collapse, which one describes here, shares some features with the CSL model and proceeds also in a Brownian way, like in Pearle's approach. An essential difference of CSL theories with the present approach is however that although this approach stands out of the domain of the standard interpretation (because of its reliance on local entanglement), it relies anyway on standard quantum dynamics and takes as its unique basis the Schro­dinger equation.

Like CSL theories (Pearle 1976; Ghirardi et al. 1990), the present theory relies on elementary events, which have only little individual effects, but which can bring out collapse by their accumulation in very large numbers. An essential difference between the two approaches stands however in the origin of these phenomena: This origin is unexplained in CSL theories, and their nature remains mysterious with only one identifiable character, which is that they break down the Schrodinger equation. The present theory relies on the contrary on explicit elementary events, which draw their existence from this equation, and it escapes a verdict of impos­sibility by the origin of these events in local entanglement.

One will call one of these elementary events a “slip in coherence”, to mean that its effect is a little slip-up in the evolution of a wave function, like a little mishap in the standard interpretation. Its main effect is to transfer small amounts of quantum probabilities between different measurement channels.

Some formalism is necessary for defining such an event and showing how it works. As usual in measurement theory, one considers two quantum systems and one denotes them by A and B. The system A is the measured one and B the mea­suring one. One considers necessary that the system B be macroscopic. The physical quantity, which must be measured, is an observable X belonging to the system A. One denotes by \k) the state vectors of the system A, which are associated with different values of this observable, and one writes down the initial state of the system A as a superposition of these state vectors,

\^A) = Sk^ck\^k). (³)

A quantum probability p_k, equal to \c_k\², is associated with every channel in this expression.

When discussing a measurement of this quantum system, one can use the model of a Geiger counter B containing an atomic gas. It turns out then that the phenomena acting as slips in coherence, along the way to collapse, are simply collisions of two atoms in the gas, say a and b, which satisfy the three following conditions:

(i) The collision is incoherent.

(ii) The initial wave function of Atom a is locally entangled with some state | j) of A.

(iii) The wave function of Atom b is not locally entangled with the system A.

Condition (i) regarding incoherence is the most significant one and must be specified with special care.

As usual, incoherence is meant as a random phase. One encountered already this kind of phases when describing the action of environment, but it must be now analyzed more carefully.

Every collision by an atmospheric molecule on the box containing the gas is associated with the previous random quantum phase a = Ap.x/h.

The main effect of one collision is to produce an associated wave of local entanglement (with the state of the molecule, which bounced away after collision). The wave is carried by phonons when it crosses the solid box and then by atoms in the gas. One saw that the geometry of this wave (i.e. the position and the orientation of its wave front, at some time) is governed by the place x where the wave origi­nated and with the time when it started. The associated phase a is an independent random parameter.

It might seem that one is emphasizing thus rather minute details, but this necessity is characteristic of quantum phases: One never sees them, but they can turn out essential: Quantum mechanics is so subtle.

One already mentioned another aspect of the waves of local entanglement with fluctuations in environment, which is the long time A? during which they take to cross the system B, while continuing to affect the state of B. Without coming to needless details, one may mention an obvious consequence of this information, which is that one knows the number N_t of molecules, which have hit the box during that time. One knows also the number of them, which contribute either to positive or to negative fluctuations, namely Nf = (N_t)^1/2. Knowing that, and also considering the random character of the phase, linked with the uncontrollable parameter Ap, one knows much as a matter of fact, about this vast hubbub.

One knows even still more than meets the eye: This is a convenient place for calling attention to a property, which could be called pompously a “principle of indeterminacy in fluctuations”. As a matter of fact, it consists only in a trivial comment on the notion of fluctuation, but one that will turn out so essential that it is worth attention.

What calls attention is the natural occurrence in the present formalism of two matrices, p_{AB +} and p_AB_, which are associated with excesses and with shortages in the action of the external flux, during the time At. The obvious property, or prin­ciple, which seems worth attention, is that absolutely no criterion could allow deciding: “This event is a positive fluctuation” or “Some events, among those that were predicted by averages and did not happen, can be identified as the missing ones.” That would of course be awkward, but it teaches a lesson, which is that, since one can predict the numbers of excesses and of lacks here or there, the expected values of positive and of negative fluctuations make sense and one can compute their effects, as if their two sets were distributed randomly.

One uses this property of indeterminacy in the present study and one considers it not as a principle but as a character, which is inherent to the concept of fluctuations.

The main tool allowing conclusions from this notion of incoherence can be then derived. It consists in the splitting of every wave functions y of the macroscopic system B into a sum of mutually incoherent components y_n, as in Eq. (1) in Proposition 2.

One can use it by looking at the combination of two effects, which both rely on local entanglement. On one hand, there is a property of incoherence in the sate of the measuring system, which is due to its local entanglement with fluctuations in the environment. On the other hand, there is a direct local entanglement between the systems A and B. When taken together, they imply an expression of a wave function (or eigenvector) of the density matrix p_{AB +}, which is given by

\Wab ) = Y_nk^Ck\WBin')\^k>. (⁴)

This relation has essentially the same meaning as Eq. (1) regarding random phases, except that some subcomponents y_Bkn occur within the previous component wave functions y_Bn. The same random phase occurs in two wave functions, such as y_Bj_n and y_Bj_n (with j j ), where the same index n characterizes this common phase. Different random phases occur on the contrary in wave functions, such as y_Bkn and y_Bkn, with different phase indices n n. (It may be useful to recall, to avoid misunderstanding, that these phases are not the elementary phases of the previous type a, but sums of several of them, and usually of many of them).

One can then easily derive the consequences of a collision between two atoms, when it satisfies the conditions (ii) and (iii) for a slip. One may call attention on the fact that Condition (iii), which requires that Atom b is not locally entangled, implies that its state is equally present in all the component wave functions y_Bkn sharing the same index n, and share accordingly the same random phase although they are algebraically entangled with different states \k) of A.

Two cases can happen then. In the first one, the two atoms belong to component wave functions with the same random phase (i.e. with the same index n). The collision is then coherent. Condition (i) for a slip does not hold and the (a, b) collision yields only an increase of local entanglement with the state of A with index j, by addition of Atom b to the set of locally entangled ones, as discussed at beginning of this paper. There is of course no transfer of quantum probabilities between different channels in that case.

The other case, which is the interesting one, occurs when all the conditions of a slip are satisfied, namely when the collision is incoherent in addition to conditions

(ii) and (iii). The state of Atom a belongs then to a component wave function (notice the index j, which is the one occurring in Condition (ii)). As for Condition

(iii), it requires that the state of Atom b be not locally entangled. It is therefore equally present in all the wave functions y_Bkm, where the index of phase m is different of n whereas the index k for measurement channels can take any value.

One can write down finally two simple formulas, which are necessary for a few last comments. They express the variations Sp_k in the various quantum probabilities p_k, when the slips under consideration obey the conditions (ii-iii), with a definite index j in Condition (ii), they occur during a short time interval St in a small region with volume SV, centered at a position x and containing a number SN of atoms. One gets then

^SPj = + Wpj(Spj =+ Wpj (1 - pj)^(St/²r)f-(x)fo(x)(SN/Nc), (5a)

Spj^ = - Wpjp_} (St/2x)fj(x)f₀(x)(SN/N_c), forjj. (5b)

The number N_c was shown in Eq. (2). The + sign in (5a) means that there has been an increase in the probability pj for the channel j, with which Atom a was locally entangled. The minus sign in (5b) means that all other channels suffered then a decrease in their probabilities.

The existence of these variations in quantum probabilities goes along with a significant first step towards collapse, by showing that these probabilities can vary. One sees that the sum of the right-hand side in (5a) and of the right-hand sides of (5b), for all values of j, vanishes, which means that the sum of all probabilities remains equal to 1.

The factor W in Eqs. (5a and 5b) is the probability for incoherence, which one can take as the trace of the matrix p_{AB +}. Its exact value is unknown, but one can establish an upper bound for it, which is: W < 4/3n = 0.4.... This bound provides also presumably a sensible order of magnitude for W. Finally, the factors fj(x) and f₀(x) are respectively the probabilities for local entanglement with Channel j and for non-local entanglement.

When the matrix—p_AB- governs this type of slip, in place of p_{AB +}, the plus sign in (5a) is replaced by a minus sign and the minus sign in (5b) by a plus sign.

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