The case against Decoherence

Gordie asked me to address decoherence which in my view was developed with the goal in mind of ridding the quantum world of spirits and souls. My normal snide self thinks this is mission impossible to demonstrate the false promise of decoherence when almost no one will read this, and some will not want to give up cherished beliefs. . But, the importance of this issue lies in the question, can the wavefunction collapse without an observer? We strongly say NO for the reasons below. Much of this will come from a hillarious and devastating critique of decoherence by Phillip Pearle, True Collapse and False Collapse in Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, Philadelphia, PA, USA, September 8-11, 1994, pp. 51-68. Edited by Da Hsuan Feng and Bei Lok Hu. Cambridge, MA: International Press, 1997. https://arxiv.org/pdf/quant-ph/9805049.pdf

First we need to understand what collapse is.

Quantum descriptions of things always involves at least two possible outcomes. The quantum description is a mixture of those two

Uncollapsed state= Reality 1 option AND Reality option 2 added together.

Collapsed state = Reality 1 OR Reality 2, no longer mixed. The state. it is one or the other. The famous physicist John Bell said:

" *The idea that elimination of coherence, in one way or another, implies the replacement of “and” by “or” is a very common one among solvers of the “measurement problem.” It has always puzzled me* ." John Bell (Pearle page 9)

As Pearle says: *What is wrong with standard quantum theory (SQT)? Doesn’t it give wonderful agreement with every experiment so far performed? Then why should anyone wish to change it?*

*…*

*What is wrong with SQT is its description of quantum events. It doesn’t describe them.* (Pearle p. 1)

and he goes on to describe the situation:

*While the experimenter turns on the apparatus and* *monitors its smooth functioning, the theoretician follows the smooth evolution* *of the statevector according to Schrodinger’s equation. Suddenly, the experimenter* *sings out “An event has occurred, and this is the result.” Abruptly,* *the theoretician stops his calculation, replaces the statevector, which has by* *now become the sum of states corresponding to different possible outcomes of* *the experiment, by the one state which the experimenter told him had actually* *occurred, and then continues his calculation of the smooth evolution of the* *statevector.*

*In other words, the practitioner of SQT must go outside the theory, to* *obtain additional information, in order to use the theory correctly. What is* *missing is that the theory doesn’t give the probability that an event occurs* *between t and t + dt."*

The Schrodinger equation has no collapse mechanism in it. It is an equation in which no reality is ever chosen. Left to its own mathematical devices the Schrodinger equation would crunch endlessly onward with no reality option ever chosen. **This is important: Decoherence specifically doesn’t change the Schrodinger equation** . Some approaches have tried to add terms to it, but decoherence ISN’T one of those approaches. Thus decoherence must live with the Schrodinger equation that never collapses. The entire problem is that when we observe a system we see ONE reality not a SUM of two realities; Schrodinger equation only works with at least 2 realities, never 1.

So with an unchanged Schrodinger equation, Pearle emphasizes the point above by:

*"It follows from Eqs.(2.1) and (3.1) that the statevector at any time T which* *evolves under a particular w(t) is*

[GRM: Schrodinger equation with 2 mixed states] (3.3)

*Now, according to Eq.(3.3), NOTHING has HAPPENED, i.e., the statevector* *is still a superposition of the states |a > and |b > with unchanged squared* *amplitudes. Of course, the phases of the amplitudes are changing, but certainly* *one cannot find in this statevector evidence that an event occurred or, supposing* *that an event occurred, whether it resulted in a or b." Pearle p. 6*

Decoherence advocates go on "unphased" as Pearle says, and create a density matrix each of whose elements describe a different possible environmental interaction. They whole matrix is said to represent all the possible quantum level environmental interactions, and when superposition is lost, all elements of the matrix go to zero except the diagonal values. As long as the off-diagonal elements are not zero, the superposition is still in effect.

The Stanford Dictionary of Philosophy says:

" *We are left with the following choice* *whether or not* *we include decoherence: either the composite system is not described by such a sum, because the Schrödinger equation actually breaks down and needs to be modified, or it is described by such a sum, but then we need to understand what that means, and this requires giving an appropriate interpretation of quantum mechanics. ***Thus, decoherence as such does not provide a solution to the measurement problem**, at least not unless it is combined with an appropriate interpretation of the theory

*"***Unfortunately, naive claims of the kind that decoherence gives a complete answer to the measurement problem are still somewhat part of the ‘folklore’ of decoherence**, and deservedly attract the wrath of physicists (e.g. Pearle 1997) and philosophers (e.g. Bub 1997, Chap. 8) alike ." https://plato.stanford.edu/entries/qm-decoherence/#ConApp

To head off an objection here about philosophers and feelings, raised in the forum above, philosophers of science are required as part of their degree program to take the course work of that area they want to be involved with–in other words, they do both math, physics and philosophy.

Problem 1. The off-diagonal values NEVER go exactly to zero–they contain small values

In other words decoherence is an approximation. They feel if the values of the off-diagonal elements are small enough we can ignore them—for all practical purposes.

" *The off diagonal elements contain multiple inner products whose magnitude is less than 1. As time goes on, the cat will have interacted with more and more mice, and new innerproducts will appear in 23.13. Hence, the off-diagonal elements will decrease exponentially with time until the cat’s state becomes indistinguishable from the mixture:*

*…*

**though, formally speaking, the system remains entangled at all moments, the eventual outcome is classical for all practical purposes** *. In other words, diagonal form of the density matrix is classically interpretable* ."" Moses Fayngold and Vadim Fayngold, Quantum Mechanics and Quantum Information: A guide through the Quantum World, Wiley 2013, equation 23.13

I used to tell my math and physics teachers, well I got the answer correct, for all practical purposes. It never got me the grade I erroneously thought I deserved.

In reality this means the superposition/entanglement continues forever. Others say the same thing:

*" For the second one, decoherence is only a way to show why ***no macroscopic superposed state can be observed,** so explaining the classical appearance of the macroscopic world, **while the quantum entanglement between the system, the apparatus and the environment never disappears** ." Herve Zwirn The Measurement Problem: Decoherence and Convivial Solipsism, https://arxiv.org/ftp/arxiv/papers/1505/1505.05029.pdf p.1

In the above, the small probabilities in the off-diagonal locations are a sign that entanglement remains and no collapse to a single reality has happened. Collapse isn’t a partial thing that we get to ‘for all practical purposes’. It is an event that happened or didn’t happen, period. We see one reality or multiple realities, and since we NEVER see multiple realities, collapse must be complete and decoherence doesn’t provide that.

" **Decoherence is, formally, never complete. There always remain exponentially small non-diagonal terms in the reduced density matrix** *, reminding us that an initial pure state remains pure according to basic quantum mechanics. Does it mean that decoherence is only a phenomenological theory[26],or is there some deeper way of interpreting very small probabilities [25]?"* Roland Omnes, Results and Problems in Decoherence Theory, Brazilian Journal of Physics, vol. 35, no. 2A, June, 2005, p. 210

If quantum entanglement/superposition doesn’t disappear completely, there is NO collapse!!!

Problem 2. The environmental interactions in the off diagonal positions are just as quantum as any other event, and thus are in mixed states themselves without having a definite reality. That is, they represent interactions that may never be real.

Pearle says this of the idea:

" *The False Collapse claim is that, at some large time, when the off-diagonal density matrix elements are suitably small (in current parlance, at the decoherence *time), an event (a or b) occurs for any system.*

*This claim makes no sense, for two reasons. **The behavior of an ensemble of evolutions which have not taken place cannot be crucial in determining the occurrence of an event in one evolution which did take place (what is not real cannot have an effect upon what is real).**If no individual statevector describes events, an ensemble of these statevectors cannot do so (a property missing in each element of a collection must be missing in the collection). Therefore this scheme cannot solve the events problem. Although the density matrices (2.7)and (3.4) have the same form, this does not mean that the arguments leading to these expressions are equally sound."* Pearle, p.7

Problem 3. A contradiction

Adler points out a stunning contradiction in his strongly argued critique of decoherence. Due to the nature of Hilbert space (go look it up), the off diagonal terms do which are performing an inner tensor sum, have nothing to do with the evolution of the main object of the experiment. Adler says:

*"Thus, when quantum mechanics is applied uniformly at all levels, to the apparatus and its environment as well as to the system, we are faced with a contradiction. This contradiction is in no way ameliorated by decoherence, since the inner product of Eq. (5b) plays no role in the ﬁnal state vector of Eq. (6a) or Eq. (6b) that describes the outcome of the measurement."* *Stephen L. Adler http://cds.cern.ch/record/531385/files/0112095.pdf p. 7,8*

I refer people to that for details.

Problem 4 basis dependence of decoherence.

According to Fayngold and Fayngold, one must pick which parameter to write the density matrix on, and then that parameter or basis, will collapse. Other basis’s won’t collapse! What kind of deal is it when an object only collapses on one basis but not another?

" *There is a common misconception that td is the time for a system to become ‘classical.’ The truth is that decoherence is basis dependent. You must make your choice of basis before writing the density matrix. Then decoherence will cause the off-diagonal elements to vanish, ***but only in that basis!**

*…*

*Suppose now that the qubit has fully relaxed and decohered and the thermal equilibrium state rho has been attained. This system (we should now call that, instead of qubit) is classical in the energy basis. No energy measurement will show any correlation between |R> and |L>. ***However, the superposition still exists in the computational basis** {|^>,|v>} **as shown by the nonzero off-diagonal elements, so the resulting equilibrium state can be highly quantum-mechanical!** " Moses Fayngold and Vadim Fayngold, Quantum Mechanics and Quantum Information: A guide through the Quantum World, Wiley 2013, Box 23:1

Braun agrees:

" *Decoherence is a basis-dependent phenomenon. Obviously, if a reduced density matrix has become diagonal in a given basis ***, it will contain off-diagonal elements (i.e. ‘coherences’) in another basis.** " Daniel Braun, Dissipative Quantum Chaos and Decoherence," Springer, 2003, p. 53

Decoherence, even when it diagonalizes the matrix, doesn’t provide an explanation of why we see those other ‘coherences’ in the other basis’s. If part of the object is uncollapsed, we should observe multiple realities and we don’t. Decoherence doesn’t collapse objects to one chosen reality.

The Feyngolds talk about how only special circumstances will allow all basis’s to be simultaneously classical.

There are several basis’s one can chose from, obviously from above, one can chose energy as the basis, or computation as the basis, there is positional basis, momentum basis etc. If decoherence only affects one aspect of the particle, that is a huge problem in my view.

Problem 5. Time.

To me this is one of the best arguments against decoherence. Td is the time of decoherence. Te is the time of the collapse, or in decoherence terms, time to classical behavior. Pearle says that that time Td is chosen in an ad hoc manner.

" *In order to properly assess the meaning of Eq.(3.4), I believe it is salutary to avoid using the phrase “decoherence” time because that seems to imply that there is a physical process called decoherence which takes this amount of time to be completed. I suggest that the phrase “No One Will Ever kNow” time, or NOWEN time for short, is more apt for the following reason.*

*"If the statevector evolution is as described in the previous section, then an event does not occur at any time* ." Pearle p. 7

Pearle then lays out the mathematical argument and concludes:

" *The NOWEN time is, by definition, a time at which the last term in Eq.(3.5) is small enough to be beyond experimental resolution. Only if you chose Te to be equal to or greater than the NOWEN time will the experimental result be the same as if your claim were right. With this choice, No One Will Ever kNow that you were wrong.* " Pearle p. 7

We conclude that decoherence does not replace the observer with a naturalistic collapse mechanism. Indeed, even strong advocates of decoherence admit that metaphysical bias’s enter in to the choice to teach decoherence and admit that it is a bit of a sleight of hand:

" *Decoherence offers a theoretical framework in which the measurement problem can be swept under the carpet (pushed into a system larger than that which we can observe). The effect is that quantum mechanics can be studied and presented to a student without the need for the ad hoc ``wave collapse’’ being presented as a primary tool of the theory. One can achieve, in many cases, the same apparent effect of a wave collapse without recourse to von Neumann’s mysterious first intervention.*

*Thus we clarify that decoherence is not a new theory unto itself, but is instead an efficient and fruitful repackaging of theory ***. It does not solve the measurement problem** , and most certainly wouldn’t have satisfied the reservations of Einstein in his later years " Tim Jones, https://www.physics.drexel.edu/~tim/open/main/node2.html

Similarly,Breuer and Petruccione, say the same thing:

" *Since decoherence, as it is understood here, is ultimately linked to the tracing over degrees of freedom of the environment, it cannot, of course, solve the measurement problem. This means that ***decoherence cannot be used to deduce the reduction of the state vector and the statistical interpretation of quantum mechanics from the unitary evolution given by the Schrodinger equation** . " Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002 p. 270

The observer remains as a ghost outside of the machinations of physics.