Measurement problem in simple experiment

[msumm21]

TL;DR Summary

Trying to better understand the measurement problem in the context of a simple experiment

I’m trying to understand the measurement problem using the simplest experiment I can think of--passing a particle P through a 50/50 beam splitter S, sending it down “path A” or “path B” with equal probability. Each path has a detector that can tell us if P was in that path. The detectors “amplify the signal” to eventually reach our "brain" through a series of interactions. Consider path A. P first interacts with some particle A1, which then interacts with A2, then A3, … until something AN reaches our "brain" to tell us the particle was detected in path A. Likewise in path B: B1,B2,...,BN.

Simple Analysis

P passes S and enters the state A+B before encountering any part of a detector. Upon encountering A1/B1 it becomes entangled in the state $A\otimes A1 + B\otimes B1$ (abusing notation here so that A1 here represents P interacting with the 1st part of the detector in path A). At this point entanglement destroys the pure state of P itself (P itself is in a mixed state, disallowing interference), but the pure state of the larger system remains: interference is possible using the bigger system of P, A1 and B1.

After the next interaction with A2/B2 the state becomes $A \otimes A1 \otimes A2 + B \otimes B1 \otimes B2$. Now P with the first portion of the detectors decohered, but by including the 2nd part of the detectors we could still produce interference between the two alternatives A/B.

Above repeats to A3/B3, A4/B4, ... until we "see a flash" from detector A or B (via AN or BN).

Interpretations / Questions

Decoherence
Some people reportedly believe decoherence solves the measurement problem. In the context of this experiment, do some believe that decoherence of P after interacting with the first portion of the measuring device constitutes collapse? If so, how do they explain the interference that can occur in the larger system P,A1,B1? If not, how do they mark a particular step where collapse occurred?

Pilot-wave theory
If, at any time in this experiment, interference between the two paths can be created, then when & how do you reveal which path P actually went down? When does the pilot wave "disappear" from the path that doesn't contain P?

Any other interesting interpretations or points about this experiment?

 

[PeterDonis]

Some people reportedly believe decoherence solves the measurement problem.

"Some people reportedly believe" is not a valid reference. You need to give a reference to an actual valid source (textbook or peer-reviewed paper) where this claim is made.

 

[PeterDonis]

If, at any time in this experiment, interference between the two paths can be created

Once decoherence has happened, interference can't be created between the two paths any more.

When does the pilot wave "disappear" from the path that doesn't contain P?

Never. The pilot wave doesn't "collapse" in the pilot wave interpretation.

 

[StevieTNZ]

Some people reportedly believe decoherence solves the measurement problem.

They are wrong. Like Chris Gerry is wrong in his book, "The Quantum Divide" (published by OUP), that the cat is either dead or alive as a result of quantum decoherence (see pages 162 - 164).

 

[msumm21]

Once decoherence has happened, interference can't be created between the two paths any more

After P interacts with A1/B1, the state of P itself is mixed, but the more complete system of P, A1 and B1 is pure until it interacts with A2/B2, so interference can occur, right? For example see operation A in this post which apparently introduces interference after "Wigner's friend" measures a qubit.

 

[PeterDonis]

After P interacts with A1/B1, the state of P itself is mixed, but the more complete system of P, A1 and B1 is pure until it interacts with A2/B2, so interference can occur, right?

If the total number of degrees of freedom is small enough in A2/B2 that they can all be tracked, yes. But at some point in the chain that will no longer be true; for some N, the interaction with AN/BN will spread among a number of degrees of freedom that is too large for them all to be tracked. That is when decoherence occurs. And N is going to be well before the interaction spreads to the point of including an observer's brain. A macroscopic photon detector by itself has enough degrees of freedom.

For example see operation A in this post which apparently introduces interference after "Wigner's friend" measures a qubit.

Such claims are hand-waving, without any actual math to back them up. Basically, "Wigner's friend" in such thought experiments is treated as though he were a qubit himself, instead of a human being with a human brain and human sensory system that has some $10^{28}$ or more degrees of freedom. (Not to mention that even the measuring device he uses to measure the qubit will have enough degrees of freedom to decohere before the interaction even gets to his brain and sensory system.)

 

[msumm21]

for some N, the interaction with AN/BN will spread among a number of degrees of freedom that is too large for them all to be tracked. That is when decoherence occurs.

For simplicity let me take a qubit 0+1. What I'd thought was that decoherence occurred when it interacted with anything X so that the resulting states of X were orthogonal, i.e. system state $0\otimes X_0 + 1\otimes X_1$ where $X_0 \bot X_1$. So that decoherence may occur with the subsystem (qubit) but never in the larger system including all things that have been interacted with. If true, there's never decoherence in the full system, it's only a property of subsystems. I realize that, in practice, things may get overwhelmingly complicated quick, but I'm aiming for what happens in quantum theory.

 

[PeroK]

They are wrong. Like Chris Gerry is wrong in his book, "The Quantum Divide" (published by OUP), that the cat is either dead or alive as a result of quantum decoherence (see pages 162 - 164).

Was that in Her Brittanic Majesty's Christmas message?

 

[PeterDonis]

What I'd thought was that decoherence occurred when it interacted with anything X so that the resulting states of X were orthogonal

No, that's not nearly enough. $X$ could just be another qubit, set up to interact with the first so that, for example, they end up in an entangled state with total spin zero. Such an interaction could easily be reversed (in fact such interactions and reversals between qubits are commonplace in quantum computing).

decoherence may occur with the subsystem (qubit) but never in the larger system including all things that have been interacted with.

No. I have no idea where you're getting this from. The whole point of decoherence is that it happens because of interactions spreading among "the larger system including all things that have been interacted with", because as the number of things (more precisely, degrees of freedom) interacted with continues to increase, it becomes impossible to keep track of all of them.

If true, there's never decoherence in the full system, it's only a property of subsystems.

Again, I have no idea where you're getting this from. It's wrong. See above.

 

[msumm21]

No, that's not nearly enough. X could just be another qubit, set up to interact with the first so that, for example, they end up in an entangled state with total spin zero. Such an interaction could easily be reversed (in fact such interactions and reversals between qubits are commonplace in quantum computing)

In the case of two qubits that become entangled to the singlet the density matrix of either one is diagonal, a mixed state, right? When the density matrix of a system becomes diagonal by interaction with another, is this not referred to as decoherence? If not, when do we call it decoherence? As far as I know it's just a repeat of the same logic whether the interaction is with 2, 3, 4, or 10^10 other particles--in all cases the density matrix of the qubit becomes diagonal iff the state can be written $0\otimes E_0 + 1\otimes E_1$ where $E_0 \bot E_1$.

 

[PeterDonis]

In the case of two qubits that become entangled to the singlet the density matrix of either one is diagonal, a mixed state, right?

For each qubit taken individually, yes.

When the density matrix of a system becomes diagonal by interaction with another, is this not referred to as decoherence?

No.

If not, when do we call it decoherence?

I already answered that. Go back and read my previous post again, carefully.

As far as I know it's just a repeat of the same logic whether the interaction is with 2, 3, 4, or 10^10 other particles

No, it isn't. You are looking at the wrong thing. Go back and read my previous post again, carefully.

in all cases the density matrix of the qubit becomes diagonal

As far as the density matrix is concerned, when the literature on decoherence talks about off diagonal terms in the density matrix disappearing, they are not talking about the reduced density matrix of an individual subsystem. They are talking about the density matrix of the combined system; the disappearance of off diagonal terms means the elimination of the possibility of interference between the different alternative configurations of the combined system.

 

[manyworldsBob]

Was that in Her Brittanic Majesty's Christmas message?

Cheeky!

 

[CoolMint]

As far as the density matrix is concerned, when the literature on decoherence talks about off diagonal terms in the density matrix disappearing, they are not talking about the reduced density matrix of an individual subsystem. They are talking about the density matrix of the combined system; the disappearance of off diagonal terms means the elimination of the possibility of interference between the different alternative configurations of the combined system.

This decoherence takes place in Hilbert place, where the state vectors of QM resides, right? And thus has nothing to do with an environment.

 

[msumm21]

As far as the density matrix is concerned, when the literature on decoherence talks about off diagonal terms in the density matrix disappearing, they are not talking about the reduced density matrix of an individual subsystem. They are talking about the density matrix of the combined system; the disappearance of off diagonal terms means the elimination of the possibility of interference between the different alternative configurations of the combined system.

We are always talking about reduced matrices right, neither of us is referring the entire universe? If we were, there always would be the ability to create interference, right?

On the original question, I guess I'm getting the feeling there's no answer to "where" decoherence occurred in the chain of events A1, A2, ... It's just a subjective thing?

 

[PeterDonis]

This decoherence takes place in Hilbert place, where the state vectors of QM resides, right?

Yes.

And thus has nothing to do with an environment.

No. The Hilbert space in question includes the degrees of freedom contained in the environment. That's the whole point.

 

[PeterDonis]

We are always talking about reduced matrices right

Not if we're talking about decoherence.

If we were, there always would be the ability to create interference, right?

Not once decoherence has taken place.

On the original question, I guess I'm getting the feeling there's no answer to "where" decoherence occurred in the chain of events A1, A2, ... It's just a subjective thing?

No. It occurs when I said it would in previous posts. Go read them.

 

[msumm21]

No. It occurs when I said it would in previous posts. Go read them.

Are you talking about the quote below? Neither here nor anywhere else do I see a precise definition of when decoherence occurs (my question in the OP). Don't you agree the quote below is subjective? Or if you have a precise definition of "number of degrees of freedom that is too large to track" could you provide it?

for some N, the interaction with AN/BN will spread among a number of degrees of freedom that is too large for them all to be tracked. That is when decoherence occurs.

 

[PeroK]

Or if you have a precise definition of "number of degrees of freedom that is too large to track" could you provide it?

That is the measurement problem!

 

[PeterDonis]

Are you talking about the quote below?

Yes.

Don't you agree the quote below is subjective?

No. The fact that there is no cookie cutter procedure for determining exactly how many degrees of freedom is "too many to track" does not mean it's subjective. There is plenty of literature on experiments that measure decoherence happening. The deficiency is in our theory: we don't have a good theoretical way of predicting in advance exactly when decoherence will happen in a given case. In other words, we have a lot more to learn. But that doesn't make what we have a lot more to learn about subjective.

 

[PeterDonis]

That is the measurement problem!

No, it isn't. The measurement problem is the problem of why we only observe a single result, even though the math, including the math of decoherence, does not show a single result; it shows multiple "branches", one for each possible result, which don't interfere with each other once decoherence has happened. Having a theory that told us in detail, in every case, exactly when decoherence would happen, i.e., how many degrees of freedom were "too many to track" in that case, would still not solve the problem of why we only observe a single result.

 

[msumm21]

There is plenty of literature on experiments that measure decoherence happening. The deficiency is in our theory: we don't have a good theoretical way of predicting in advance exactly when decoherence will happen in a given case.

Just to be clear, these experiments are finding processes that eliminate interference? Deviating from QT? OR do you mean something else?

We can't predict because QT implies something different or because we just can't do the math accurately enough with all the DOF, ...?

Thanks

 

[PeterDonis]

these experiments are finding processes that eliminate interference?

They're measuring decoherence. They can't measure the individual interactions that cause decoherence, because if they could, it wouldn't be decoherence; the whole point of decoherence is that you can no longer track the individual interactions with the environment.

We can't predict because QT implies something different

No.

or because we just can't do the math accurately enough with all the DOF, ...?

Yes.

 

[msumm21]

No, that's not nearly enough. X could just be another qubit, set up to interact with the first so that, for example, they end up in an entangled state with total spin zero. Such an interaction could easily be reversed (in fact such interactions and reversals between qubits are commonplace in quantum computing).

This doesn't seem consistent with e.g. "Theory of Decoherence" (section 1) explanation here. They use the example of an electron double-slit experiment, and claim decoherence occurs if the electron interacts with something at the slits that suppresses interference. They even point out that interference is still possible in the "larger system," and decoherence is merely a property of looking only at the subsystem of the electron.

 

[PeterDonis]

"Theory of Decoherence" (section 1) explanation here.

This is a philosophy website, not a physics textbook or peer-reviewed paper. See further comments on that below.

decoherence occurs if the electron interacts with something at the slits that suppresses interference

By that they mean (though they don't say so--see my comment below about philosophy websites and physics) "something at the slits that entangles the electron with a large number of degrees of freedom that cannot be kept track of". That's what "suppresses interference", by which they mean "results in no interference pattern being observed".

They even point out that interference is still possible in the "larger system," and decoherence is merely a property of looking only at the subsystem of the electron.

And what you are failing to recognize (and the article does not make it clear, which is why articles on a philosophy website are not good sources for learning physics) is that there is no way to do the relevant experiments on the "larger system", because the large number of degrees of freedom involved cannot be kept track of. (This is also a common omission in discussions of "Wigner's friend" type experiments, where it is often blithely assumed that one can treat human beings as if they were single qubits to which arbitrary unitary operations can be applied.) If in a particular scenario you can do the relevant experiments on the larger system to show interference, then decoherence has not yet occurred.

 

[msumm21]

That's what "suppresses interference", by which they mean "results in no interference pattern being observed".

Not sure what they meant, but suppression of an interference pattern on the screen only requires e.g. qubits to mark the slot through which the electron passed (not a large, untrackable DOF).

As far as the density matrix is concerned, when the literature on decoherence talks about off diagonal terms in the density matrix disappearing, they are not talking about the reduced density matrix of an individual subsystem. They are talking about the density matrix of the combined system; the disappearance of off diagonal terms means the elimination of the possibility of interference between the different alternative configurations of the combined system.

Here's another reference:

Decoherence, the measurement problem, and interpretations of quantum mechanics​

On page 7 I read:
"Environment-induced decoherence. The fast local suppression of interference between different states of the system. However, since only unitary time evolution is employed, global phase coherence is not actually destroyed—it becomes absent from the local density matrix that describes the system alone, but remains fully present in the total system-environment composition."

 

[Nugatory]

but suppression of an interference pattern on the screen only requires e.g. qubits to mark the slot through

Can you describe a specific experimental setup in which that statement is true?

 

[PeterDonis]

suppression of an interference pattern on the screen only requires e.g. qubits to mark the slot through which the electron passed (not a large, untrackable DOF).

Wrong. It requires some kind of irreversible interaction at the slits. A simple interaction with a qubit is not irreversible; you could track the qubit and reverse its effect and make the interference pattern re-appear.

If, OTOH, you measured the qubit in order to see which slit the electron passed through, then the interaction is not just with the qubit; you have a large number of untrackable degrees of freedom in the measuring device you're using to measure the qubit. That would be an irreversible interaction and would make the interference pattern disappear and not be recoverable. (Note that this "measurement" would not have to be observed by a human; just having the qubit interact with a macroscopic measuring device, even if its result were thrown away, would be enough.)

since only unitary time evolution is employed, global phase coherence is not actually destroyed

Yes, but this "global phase coherence" cannot be used to reverse any interactions or undo any measurement results. That's the key criterion. The "only unitary time evolution is employed" is why decoherence does not solve the measurement problem. It doesn't mean decoherence is just the same as two qubits getting entangled.

 

[stevendaryl]

[About whether interference is possible]

Not once decoherence has taken place.

Just for clarification, this is a “for all practical purposes” type claim, right? The full density matrix never actually evolves to be diagonal, it’s just that the off-diagonal terms become too small to be of any practical relevance. Interference between macroscopically distinguishable states is impossible for all practical purposes in a similar sense that fixing a broken egg by shaking the pieces is impossible. It’s not mathematically impossible, but so unlikely that it’s practically impossible.

 

[PeterDonis]

this is a “for all practical purposes” type claim, right?

That depends on which QM interpretation you adopt. In a no collapse interpretation, yes. In a collapse interpretation, no.

The full density matrix never actually evolves to be diagonal, it’s just that the off-diagonal terms become too small to be of any practical relevance.

As I understand the basic math of decoherence, yes, that is all you can conclude. The rest is a matter of interpretation, as above.

 

[msumm21]

Wrong. It requires some kind of irreversible interaction at the slits.

This is not correct, once the slit has been marked by an interaction, the interference pattern at the screen will not occur. Of course you can always reverse this in QT (unitary transformation are reversible), but I didn't say "mark it and then erase it."

 

[PeterDonis]

Of course you can always reverse this in QT (unitary transformation are reversible)

But that does not mean that what happens when a measurement is made and a permanent record is formed (even if it's not readable by humans, for example if it's stored in a large number of untrackable degrees of freedom that do not include any "pointer" or other indication that humans can read) can be reversed. Our actual experimental observation is that it can't; nobody has ever reversed decoherence, nor is there any prospect of anyone doing so. Certain interpretations of QM might claim that decoherence can in principle be reversed, but that is not the same as having actual evidence that it can.

 

[PeterDonis]

once the slit has been marked by an interaction, the interference pattern at the screen will not occur.

"Marked by an interaction" is not the same as "interacted with qubits". If you want to claim that interaction with a single qubit at one of the slits can suppress the interference pattern, please give a reference to a paper describing an experiment that has shown this.

 

[gentzen]

This doesn't seem consistent with e.g. "Theory of Decoherence" (section 1) explanation here.

This is a philosophy website, not a physics textbook or peer-reviewed paper. See further comments on that below.

decoherence occurs if the electron interacts with something at the slits that suppresses interference

By that they mean (though they don't say so--see my comment below about philosophy websites and physics) [... large number of degrees of freedom ...] by which they mean "results in no interference pattern being observed".

You are mistaken in believing that they mean [... large number of degrees of freedom ...], but you are right that they mean "results in no interference pattern being observed":

Such a phenomenon of suppression of interference is what is called decoherence.

This was somewhat surprising for me, because I expected "decoherence" to be the same as "environmental decoherence". For example, decoherence expert Maximilian Schlosshauer in 2011 gave the following definition in a Glossary:

decoherence A quantum-mechanical process whereby interactions of a quantum system with its environment lead to uncontrollable and practically irreversible entanglement between the two partners. Decoherence explains why it is so difficult in practice to prepare certain quantum states and to observe interference effects -- especially in the case of mesoscopic and macroscopic systems, for which decoherence is extremely fast and virtually inescapable. Decoherence is an application of the standard quantum formalism to open quantum systems; as such, it is neither an interpretation nor a new theory. Yet it is often invoked in foundational discussions, for example, when addressing aspects of the measurement problem. It’s also a cornerstone of Everett-style interpretations. Decoherence is a lively subject of experimental investigation and a feared enemy of quantum computers.

However, he also wrote in the same Glossary:

The Stanford Encyclopedia of Philosophy, online at http://plato.stanford.edu, is also an authoritative source of information. It has comprehensive entries -- some written by our interviewees -- on staples such as EPR, the Bell and Kochen–Specker theorems, the measurement problem, entanglement, quantum information, decoherence, quantum logic, and the common interpretations (Copenhagen, Everett, collapse theories, Bohmian mechanics, and modal and relational interpretations).

Interestingly, the first version of that SEP entry from 2003 did include "environment" as part of decoherence:

It is this phenomenon of suppression of interference through suitable interaction with the environment that we refer to by ‘suppression of interference’, and that is studied in the theory of decoherence. For completeness, we mention the overlapping but distinct concept of decoherent (or consistent) histories.

In 2012 Bacciagaluppi included sections on decoherent histories in the entry and weakened his statement to "It is this phenomenon of suppression of interference through suitable interaction with the environment that we call ‘dynamical’ or ‘environmental’ decoherence." Finally in 2020, he fully embraced that "suppression of interference" is what is meant by "decoherence" (if further qualifications are omitted).

My impression is that while the inclusion of "decoherent histories" might have been Bacciagaluppi's personal decision, there was also a real shift in the meaning of "decoherence" over time. What you can measure is the "suppression of interference" (within a well defined subsystem), so as control of decoherence became important for quantum computers and other quantum technologies, it made sense to separate the well defined "measurable" concept from the less well defined "explanatory" concept.

 

[PeterDonis]

My impression is that while the inclusion of "decoherent histories" might have been Bacciagaluppi's personal decision, there was also a real shift in the meaning of "decoherence" over time.

If there has been such a shift as you describe, it means that the term "decoherence" as it is now being used is not relevant to the measurement problem, since "decoherence" as it is now being used is no longer considered to be irreversible (since it is easy to reverse the operations in quantum computing that "suppress interference"). All of my statements about "decoherence" in this thread apply to the "environmental decoherence" concept which is what the term "decoherence" used to mean. If it doesn't mean that now, I'll change the term I'm using, not the arguments I'm making, which are still valid for "environmental decoherence" (or whatever we're supposed to call it now).

 

[gentzen]

but suppression of an interference pattern on the screen only requires e.g. qubits to mark the slot through

Can you describe a specific experimental setup in which that statement is true?

You need at least one qubit per electron to suppress an interference pattern in that way.
(a) The most straightforward experimental setups would be those that use quantum degrees of freedom of the electrons themselves for those qubits. Both electron spin and electron energy suggest themselves, but neither is easy/obvious to manipulate locally at a slit. (And since the electron wavelength is very small even at low energies, the "slits" must be very small and close to each other too, making local manipulations even more impossible.)
(b) Less straightforward experimental setups could try to use quantum degrees of freedom of random molecules put it the way of the electrons for those qubits. But this is problematic, because the interaction with such molecules risks to destroy any interference pattern, even in cases where no which-way information was ever present in the entanglement with those molecules.

once the slit has been marked by an interaction, the interference pattern at the screen will not occur.

"Marked by an interaction" is not the same as "interacted with qubits". If you want to claim that interaction with a single qubit at one of the slits can suppress the interference pattern, please give a reference to a paper describing an experiment that has shown this.

If "marked by an interaction" means "entangled with the state of some qubit", then the statement is not really false, but just misleadingly abstract. See my attempt above to describe some experimental setup in which that statement would be true.