Decoherence and standard formalism

[fanieh]

In the standard mathematical formalism, the environment were treated classically, this is because observers (being macroscopic recording mechanisms) are treated classically, so the system is isolated. Decoherence is about open system, so how is decoherence compatible with Copenhagen or the standard formalism at all?
How can you make the standard formalism have an open system-environment too? Or perhaps is it correct to think the standard mathematical formalism is already updated and Copenhagen is already outdated? How do you link the two if you were to give a lecture about this in class (which I'll do)?

 

[PeterDonis]

In the standard mathematical formalism, the environment were treated classically

Please provide a reference for this statement. It doesn't look correct to me.

 

[fanieh]

Please provide a reference for this statement. It doesn't look correct to me.

I mean, in the Copenhagen interpretation, the environment were treated classically because observers were treated classically. But I'm not interested in interpretations but in the standard mathematical formalism. How does the standard mathematical formalism deal with the environment? In the density matrix approach, the environment is traced out.. but this after there is entanglement between the system and environment. In the Copenhagen, there is no entanglement between the system and environment because the environment is treated classically and only the quantum system is treated in isolated. So I'm confused how many physicists could say the Copenhagen is the standard mathematical formalism when it treated the environment as classical and there is no entanglement between environment and system in the Copenhagen. How do you resolve the two?

 

[Simon Phoenix]

In the standard mathematical formalism, the environment were treated classically, this is because observers (being macroscopic recording mechanisms) are treated classically, so the system is isolated. Decoherence is about open system, so how is decoherence compatible with Copenhagen or the standard formalism at all?

That's definitely not correct. One of the motivations for the decoherence treatment was to explain how we can get something that looks like a non-unitary irreversible process from a theory (QM) in which the basic interactions are all unitary (and reversible). It's a similar problem to explaining the second law of thermodynamics within classical physics where all the fundamental processes at an individual level are obeying time-reversible evolutions.

So the basic question here is that if the laws of QM govern everything then surely they also govern the bits and pieces that make up a measurement device. So where does this discontinuous and irreversible change that we call a 'measurement' come from? The system and the measuring device are all made of things that obey the laws of QM which mean that all of the interactions between the things that make up our system and the things that make up our measuring device are unitary interactions.

So the very essence of the decoherence treatment is to treat everything quantum mechanically, not classically.

If, for example, we had a single EM field mode inside a cavity then there's going to be some leakage of that radiation inside the cavity to the outside world. How do we model this? Well, if some radiation is getting out of the cavity then there has to be a coupling of the field mode inside the cavity to the field modes outside the cavity. Field modes in QM are essentially just quantum harmonic oscillators and so we can model this system as a single (cavity) field mode coupled to a number of field modes outside the cavity. Essentially just a system then of coupled harmonic oscillators.

We can then make some reasonable assumptions about the form of that coupling and assume there are an infinite number of discrete field modes outside the cavity (all distinguished by some frequency). With this we can write down a Hamiltonian for the cavity mode plus outside modes. What we're after is an equation that describes the evolution of the field mode inside the cavity, because that's our system of interest. The field modes outside the cavity are our environment and we're looking for a kind of 'averaged' equation of motion for our cavity mode as a result of all of the myriad interactions with the environmental (outside) modes.

To proceed further we can take the continuum limit for the environmental mode (a continuous distribution of frequencies), assume some basic initial state for the environmental modes (thermal states, for example) and then do a coarse-grained averaging procedure to end up with a master equation for our cavity field mode. What we then have is something that can model a dissipative process in QM fully quantum mechanically. It's no different in spirit to the treatment of spontaneous emission in a fully quantum way.

What's interesting is that if we construct such a model then for certain initial environmental conditions the master equation for the cavity mode can be solved exactly and we find that the effect of the environment is to rapidly drive the cavity field mode into a diagonal density operator - which is the density operator we get from performing a measurement in which we remain ignorant of the result.

For this cavity field mode example if we begin with a cavity field mode prepared in a superposition of two coherent states then there is an exponential decay of the off-diagonal elements of the cavity field density matrix with a decay rate that is proportional to the square of the 'distance' between them - so any macroscopic (big difference between the coherent states) superpositions get driven to mixtures very, very quickly.

Zeh and Zurek did some magnificent work to look at this kind of thing in more general terms and showed that this decoherence is actually a more general feature of interactions of a 'small' quantum system with a large (but still quantum) environment. The idea being that pretty much any 'sensible' quantum environment is going to have this diagonalizing effect on the system of interest. So this leads to the idea that a measuring device can be modeled as a quantum object that is coupled to a large environment and if we treat it this way then we can get something out of it that kind of looks like a quantum measurement (it has the right density matrix for an ensemble).

As a way to model dissipative effects in QM - wonderful; as a solution to the 'measurement problem' I'm less convinced but clearly some 'decoherence' process must be happening in any measuring device - which is a suitably 'large' quantum object, of course. Whether that's quite enough to explain all the features of what constitutes a quantum measurement I don't think so - but it's a fantastic step closer to fully solving that puzzle I reckon.

 

[fanieh]

That's definitely not correct. One of the motivations for the decoherence treatment was to explain how we can get something that looks like a non-unitary irreversible process from a theory (QM) in which the basic interactions are all unitary (and reversible). It's a similar problem to explaining the second law of thermodynamics within classical physics where all the fundamental processes at an individual level are obeying time-reversible evolutions.

So the basic question here is that if the laws of QM govern everything then surely they also govern the bits and pieces that make up a measurement device. So where does this discontinuous and irreversible change that we call a 'measurement' come from? The system and the measuring device are all made of things that obey the laws of QM which mean that all of the interactions between the things that make up our system and the things that make up our measuring device are unitary interactions.

So the very essence of the decoherence treatment is to treat everything quantum mechanically, not classically.

If, for example, we had a single EM field mode inside a cavity then there's going to be some leakage of that radiation inside the cavity to the outside world. How do we model this? Well, if some radiation is getting out of the cavity then there has to be a coupling of the field mode inside the cavity to the field modes outside the cavity. Field modes in QM are essentially just quantum harmonic oscillators and so we can model this system as a single (cavity) field mode coupled to a number of field modes outside the cavity. Essentially just a system then of coupled harmonic oscillators.

We can then make some reasonable assumptions about the form of that coupling and assume there are an infinite number of discrete field modes outside the cavity (all distinguished by some frequency). With this we can write down a Hamiltonian for the cavity mode plus outside modes. What we're after is an equation that describes the evolution of the field mode inside the cavity, because that's our system of interest. The field modes outside the cavity are our environment and we're looking for a kind of 'averaged' equation of motion for our cavity mode as a result of all of the myriad interactions with the environmental (outside) modes.

To proceed further we can take the continuum limit for the environmental mode (a continuous distribution of frequencies), assume some basic initial state for the environmental modes (thermal states, for example) and then do a coarse-grained averaging procedure to end up with a master equation for our cavity field mode. What we then have is something that can model a dissipative process in QM fully quantum mechanically. It's no different in spirit to the treatment of spontaneous emission in a fully quantum way.

What's interesting is that if we construct such a model then for certain initial environmental conditions the master equation for the cavity mode can be solved exactly and we find that the effect of the environment is to rapidly drive the cavity field mode into a diagonal density operator - which is the density operator we get from performing a measurement in which we remain ignorant of the result.

For this cavity field mode example if we begin with a cavity field mode prepared in a superposition of two coherent states then there is an exponential decay of the off-diagonal elements of the cavity field density matrix with a decay rate that is proportional to the square of the 'distance' between them - so any macroscopic (big difference between the coherent states) superpositions get driven to mixtures very, very quickly.

Zeh and Zurek did some magnificent work to look at this kind of thing in more general terms and showed that this decoherence is actually a more general feature of interactions of a 'small' quantum system with a large (but still quantum) environment. The idea being that pretty much any 'sensible' quantum environment is going to have this diagonalizing effect on the system of interest. So this leads to the idea that a measuring device can be modeled as a quantum object that is coupled to a large environment and if we treat it this way then we can get something out of it that kind of looks like a quantum measurement (it has the right density matrix for an ensemble).

As a way to model dissipative effects in QM - wonderful; as a solution to the 'measurement problem' I'm less convinced but clearly some 'decoherence' process must be happening in any measuring device - which is a suitably 'large' quantum object, of course. Whether that's quite enough to explain all the features of what constitutes a quantum measurement I don't think so - but it's a fantastic step closer to fully solving that puzzle I reckon.

Appreciated your explanations. What I was saying was that Copenhagen was not compatible with the idea the environment is also quantum. Hence just saying the two are not compatible and whether we should make Copenhagen outdated once and for all.

By the way. Your first paragraph was not exactly right in that decoherence doesn't automatically means the born rule applied. It just delocalized the phases.. to get one outcome.. you still need the born rule.

 

[PeterDonis]

What I was saying was that Copenhagen was not compatible with the idea the environment is also quantum.

And I asked you for a reference for this statement. Either provide one or stop making this claim.

The standard theory of decoherence, which Simon Phoenix described, is interpretation neutral. It doesn't take any position on whether collapse happens or not.

 

[Simon Phoenix]

What I was saying was that Copenhagen was not compatible with the idea the environment is also quantum. Hence just saying the two are not compatible and whether we should make Copenhagen outdated once and for all.

By the way. Your first paragraph was not exactly right in that decoherence doesn't automatically means the born rule applied. It just delocalized the phases.. to get one outcome.. you still need the born rule.

Yes I misunderstood your initial post slightly - sorry. The issue is to understand how we can get something that looks like 'collapse' from a theory in which all the interactions are governed by time-reversible laws of evolution - so strictly speaking that's mathematically impossible. What we get is something that, for all practical purposes, looks a lot like collapse. It neatly explains why we don't see 'macroscopic' superpositions in the real world.

The issue of why we get one particular outcome (an eigenstate) is not wholly explained within the decoherence treatment - which is one of the reasons I'm not convinced it's a full solution to the so-called 'measurement problem' in QM. I would say that it has to be a big part of that solution though.

 

[bhobba]

Appreciated your explanations. What I was saying was that Copenhagen was not compatible with the idea the environment is also quantum.

That is false.

It is silent on the issue. It simply assumes a classical world exists and observations appear in it. It says nothing about what that world is ie if its quantum or not. We now know everything is quantum even the classical world of Copenhagen.

Thanks
Bill

 

[fanieh]

And I asked you for a reference for this statement. Either provide one or stop making this claim.

The standard theory of decoherence, which Simon Phoenix described, is interpretation neutral. It doesn't take any position on whether collapse happens or not.

Here:

https://arxiv.org/pdf/quant-ph/0312059v4.pdf

"The Copenhagen interpretation additionally postulates
that classicality is not to be derived from quantum
mechanics, for example, as the macroscopic limit
of an underlying quantum structure (as is in some sense
assumed, but not explicitely derived, in the standard interpretation),
but instead that it be viewed as an indispensable
and irreducible element of a complete quantum
theory—and, in fact, be considered as a concept prior to
quantum theory. In particular, the Copenhagen interpretation
assumes the existence of macroscopic measurement
apparatuses that obey classical physics and that
are not supposed to be described in quantum mechanical
terms (in sharp contrast to the von Neumann measurement
scheme, which rather belongs to the standard
interpretation); such a classical apparatus is considered
necessary in order to make quantum-mechanical phenomena
accessible to us in terms of the “classical” world of
our experience. This strict dualism between the system
S, to be described by quantum mechanics, and the apparatus
A, obeying classical physics, also entails the existence
of an essentially fixed boundary between S and A,
which separates the microworld from the macroworld (the
“Heisenberg cut”). This boundary cannot be moved significantly
without destroying the observed phenomenon
(i.e., the full interacting compound SA).Especially in the light of the insights gained from decoherence
it seems impossible to uphold the notion of a
fixed quantum–classical boundary on a fundamental level
of the theory."

How is Copenhagen based on a classical environment compatible with environment based on quantum? I'm running out of word to say in a lecture so hope you can share how they are still related. Maybe can one say Copenhagen has less explanatory power even if the mathematics can be fitted to either?

 

[bhobba]

so strictly speaking that's mathematically impossible. What we get is something that, for all practical purposes, looks a lot like collapse.

Technically its how an improper mixture becomes a proper one. The details can't be discussed in a post - you really need to study a book. THE standard one is:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

Thanks
Bill

 

[bhobba]

"The Copenhagen interpretation additionally postulates
that classicality is not to be derived from quantum
mechanics,

That is false - it makes no such assumption.

I can give a link that carefully explains the details but unfortunately the source strictly speaking doesn't meet our guidelines.

The textbook I mentioned does meet our guidelines and explains it all in excruciating detail as well as some of the issues that still remain.

Thanks
Bill

 

[Simon Phoenix]

Technically its how an improper mixture becomes a proper one. The details can't be discussed in a post - you really need to study a book. THE standard one is:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

Ooh - that looks like a fantastic book - I'm sure you've linked to it before but in my typically slow and plodding way this is the first time I've clicked on the link

 

[bhobba]

Ooh - that looks like a fantastic book - I'm sure you've linked to it before but in my typically slow and plodding way this is the first time I've clicked on the link

It is THE book on the issue - worth every cent IMHO. But it is no easy read - you need at least the background of Griffiths or similar to understand it, but likely with effort get by with Susskind.

Thanks
Bill

 

[Simon Phoenix]

But it is no easy read - you need at least the background of Griffiths or similar to understand it, but likely with effort get by with Susskind

I think I might just about manage

 

[fanieh]

That is false - it makes no such assumption.

I can give a link that carefully explains the details but unfortunately the source strictly speaking doesn't meet our guidelines.

The textbook I mentioned does meet our guidelines and explains it all in excruciating detail as well as some of the issues that still remain.

Thanks
Bill

Ey, that paper was written by the same author as the book! The paper was a condensed version of his book. I have the book. I'll find the relevant passages.

 

[fanieh]

That is false - it makes no such assumption.

I can give a link that carefully explains the details but unfortunately the source strictly speaking doesn't meet our guidelines.

The textbook I mentioned does meet our guidelines and explains it all in excruciating detail as well as some of the issues that still remain.

Thanks
Bill

Ok. Here's the very passage from the textbook you suggested to Simon https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20
or Maximilian Schlosshauer "Decoherence and the Quantum-to-Classical Transition". In page 335. Maximilian quoted

"The Copenhagen interpretation additionally postulates that classicality is not to be derived from quantum mechanics..."

So Maximilian is wrong??

 

[bhobba]

"The Copenhagen interpretation additionally postulates that classicality is not to be derived from quantum mechanics..." So Maximilian is wrong??

On the surface yes - but context is everything eg their are a number of variants of Copenhagen and different contexts on what classically means eg some take classical to mean exactly how macro objects emerged which Copenhagen does not explain.

That book is my bible on decoherence and I know for a fact that is not his view which is that there are 3 parts to the measurement problem.

1. The problem of non observance of interference
2. How the preferred basis emerges.This is why, for example, classical objects nearly always have a definite position
3. How an improper mixed state becomes a proper one.

The first 2 is explained by decoherence, the third some interpretations simply assume, while others explain.

The three taken together imply classicality because they explain how objects for all practical purposes have definite values which is what classical physics is.

Now I could give you the page he says that but in this type of situation where context can be a problem I would like you to explain to me, in your own words, not a quote, but in your own words, how objects with definite values is not classical. In particular exactly what do you mean by classicality.

Just as a hint to what might be going on is the so called factorisation problem which IMHO is way over hyped, but if its your worry requires another thread:
https://arxiv.org/abs/1210.8447

Thanks
Bill

 

[bhobba]

Especially in the light of the insights gained from decoherenceit seems impossible to uphold the notion of afixed quantum–classical boundary on a fundamental levelof the theory."

The above is very very true.

Now I ask you to think a bit. If there is no fixed boundary, and there isn't, (everything is quantum) what's a classical object to begin with?

Once you nut it (not a quote but in your own words) out please let us know what a classical object is for your claims to make sense.

Hint - the divide is a human construct and lies outside the theory - its also related to the factorisation issue. Either way its a very difficult problem but usually has some common characteristics such as exact position etc.

In modern times we don't worry about it - its simply when decoherence occurs.

Thanks
Bill

 

[bhobba]

Please provide a reference for this statement. It doesn't look correct to me.

Its not. Its treated via QM and randomises phases so its irreversible.

Thanks
Bill

 

[Demystifier]

When I see a discussion of the form
- Copenhagen says this.
- No, Copenhagen says that.
I like to point out that there is no Copenhagen interpretation:
https://www.physicsforums.com/threads/there-is-no-copenhagen-interpretation-of-qm.332269/

 

[fanieh]

The above is very very true.

Now I ask you to think a bit. If there is no fixed boundary, and there isn't, (everything is quantum) what's a classical object to begin with?

Once you nut it (not a quote but in your own words) out please let us know what a classical object is for your claims to make sense.

Hint - the divide is a human construct and lies outside the theory - its also related to the factorisation issue. Either way its a very difficult problem but usually has some common characteristics such as exact position etc.

In modern times we don't worry about it - its simply when decoherence occurs.

Thanks
Bill

Thank you for your message. I will reread the book and consider what you were saying. Then I'll get back to you after reading it with broader frame of reference.

 

[bhobba]

I like to point out that there is no Copenhagen interpretation:
https://www.physicsforums.com/threads/there-is-no-copenhagen-interpretation-of-qm.332269/

True - oh so very true.

Consistent histories, which some think Copenhagen done right, is much better to discuss in modern times:
http://quantum.phys.cmu.edu/CHS/histories.html

Thanks
Bill

 

[bhobba]

Then I'll get back to you after reading it with broader frame of reference.

I would suggest forgetting Copenhagen to begin with. It really is old hat:
http://scitation.aip.org/content/aip/magazine/physicstoday/article/58/11/10.1063/1.2155755

We now know both Bohr and Einstein were wrong so its a rather hard issue to debate.

Thanks
Bil

 

[fanieh]

I would suggest forgetting Copenhagen to begin with. It really is old hat:
http://scitation.aip.org/content/aip/magazine/physicstoday/article/58/11/10.1063/1.2155755

We now know both Bohr and Einstein were wrong so its a rather hard issue to debate.

Thanks
Bil

Is there any solid experimental proof that the environment is really entangled with the system? Something that would convince Bohr (if he were alive today) that the environment should also be treated quantumly?

 

[bhobba]

Is there any solid experimental proof that the environment is really entangled with the system? Something that would convince Bohr (if he were alive today) that the environment should also be treated quantumly?

We have tons of models, but I do not know their experimental status. For example a few stray photons from the CBMR has been calculated to give a dust particle a definite position. Has it been experimentally confirmed

Thanks
Bill

 

[Demystifier]

Is there any solid experimental proof that the environment is really entangled with the system? Something that would convince Bohr (if he were alive today) that the environment should also be treated quantumly?

There are experiments in mesoscopic systems (i.e. systems somewhere between microscopic and macroscopic) which show that decoherence is an actual continuous process. Experiments demonstrate that after a finite time of interaction with environment the system can be in a partially decohered state.

 

[fanieh]

There are experiments in mesoscopic systems (i.e. systems somewhere between microscopic and macroscopic) which show that decoherence is an actual continuous process. Experiments demonstrate that after a finite time of interaction with environment the system can be in a partially decohered state.

When one is walking in the street and one wears a hat.. is that hat supposed to be entangled with all the cars and all the buildings and everything in the environment even the clouds.. or is the hat supposed to be entangled only with the CBMR and air or some photons? I know photons from all the objects in the environment hit our retina so we can see them. But what would it mean our hat is entangled with all the things out there.. is the hat really entangled with every particle of the car surface or only the photons reflected by the car and would this be enough to cause superposition of hat and car such that it form an entanglement?

 

[Demystifier]

is that hat supposed to be entangled with all the cars and all the buildings and everything in the environment even the clouds..

Yes, your hat is entangled with pretty much everything, but with none of it very much. In practice, entanglement can only be observed if it is not too far from maximal entanglement. Therefore, your hat appears as if it was not entangled at all.

 

[fanieh]

Yes, your hat is entangled with pretty much everything, but with none of it very much. In practice, entanglement can only be observed if it is not too far from maximal entanglement. Therefore, your hat appears as if it was not entangled at all.

The hat can only be entangled maximally if there were only one particle in the environment.. but with billions and billions in the environment.. the pure state is divided into billions too. So can this be enough to put the hat in classical state instead of slight superposition of the position observable? Then why worry how improper mixed state of the hat becomes proper mixed state as per Maximillian.. the billions of entanglement is enough to put it in proper mixed state (position that is classical)?

 

[Demystifier]

The hat can only be entangled maximally if there were only one particle in the environment.. but with billions and billions in the environment.. the pure state is divided into billions too. So can this be enough to put the hat in classical state instead of slight superposition of the position observable? Then why worry how improper mixed state of the hat becomes proper mixed state as per Maximillian.. the billions of entanglement is enough to put it in proper mixed state (position that is classical)?

You are mixing two different meanings of the word "classical state". This can be best understood with an example. Consider a classical coin with the classical probability distribution
$$p(head)=\frac{1}{2}, \;\;\; p(tail)=\frac{1}{2}.$$
Knowing only this, can you tell what is the classical state of the coin?

 

[fanieh]

You are mixing two different meanings of the word "classical state". This can be best understood with an example. Consider a classical coin with the classical probability distribution
$$p(head)=\frac{1}{2}, \;\;\; p(tail)=\frac{1}{2}.$$
Knowing only this, can you tell what is the classical state of the coin?

1/2?
I was asking whether you no longer need the born rule when the hat was entangled with billions in the environment.. it can artificially create the born rule or trapping the hat particle from all sides such that it's like being in a iegenposition?

 

[Demystifier]

1/2?

Nope.

I was asking whether you no longer need the born rule when the hat was entangled with billions in the environment

You need the Born rule even then.

 

[fanieh]

Nope.

What has this got to do with my statement "the billions of entanglement is enough to put it in proper mixed state (position that is classical)?".. in your example.. proper mixed state is the head or tail.. maybe.. but what's the 1/2.. in the hat.. the proper mixed state in the position basis is position.. how's there two meanings of "classical state".. kindly elaborate.. thank you.

You need the Born rule even then.

 

[Demystifier]

it can artificially create the born rule or trapping the hat particle from all sides such that it's like being in a iegenposition?

No, the entanglement does not work in that way. Entanglement always involves a superposition, which is precisely what you want to avoid to get a "classical" state.

 

[fanieh]

Nope.

Let me resend the following. I think when you received messages directly from physicsforums.. any edited won't be updated to you so you might have missed the following because of wrong "quote" parenthesis. About your nope..

What has this got to do with my statement "the billions of entanglement is enough to put it in proper mixed state (position that is classical)?".. in your example.. proper mixed state is the head or tail.. maybe.. but what's the 1/2.. in the hat.. the proper mixed state in the position basis is position.. how's there two meanings of "classical state".. kindly elaborate.. thank you.