Decoherence, Coherence, Pure State, Mixed State

[rodsika]

Hi, I'm confused by subtle differences between the concept. Let's take the example of a Schrodinger Cat. Supposed you could make a box that can isolate anything inside from say gravity, microwave radiation, is in 0 kelvin, etc. or let's just accept (for sake of discussion) that a box can totally and completely isolate what is put inside. Now supposed you put a cat inside it. Questions:

1. Is the cat completely 100% isolated from any enviromental in pure state or mixed state?

2. If the answer is mixed state, how come they say the universe is in pure state? I think complex thing can be in pure state too, isn't it. Does pure state means things are in quantum coherence? But the universe in pure state are not be in coherence at all (because all our phases can't be made coherence) hence pure state doesn't mean coherence?

3. Would the cat suffer decoherence in the atoms inside his body even if his entire body was completely isolated from the environment hence avoiding in principle any decoherence between it and environment?*

4. If yes to question number 3, how come they keep giving example where if a cat is totally enclosed or isolated in a box... it can form superposition giving rise to Many Worlds where each world has each separate history?

5. If the cat completely isolated is in pure state, yet its internal body is in decoherence. Then it is not in superposition. Here being in pure state can occur without superposition?

 

[JesseM]

In principle it should be possible to prepare the contents of the box in a pure state, which I think just means you would perform a measurement that would give you the maximum possible information about the particles in the box allowed by quantum physics (like a precise measurement of a complete set of commuting observables), allowing you to construct a state vector for the system.

As for decoherence, as I understand it this only applies to some subsystem of a larger system. So if you have the contents of the box in a pure state at the moment the box is sealed, then the complete state vector of everything in the box should remain in a pure state forever as long as the box remains isolated...but if you consider the state of the cat subsystem as separate from the state of the remaining contents of the box (air molecules, cat toys, etc.), then the interaction of the cat with the environment will cause the cat subsystem to go to a mixed state, with the interference terms approaching zero. From p. 8 of the Maximilian Schlosshauer paper you linked to on the other thread:

Note that decoherence derives from the presupposition
of the existence and the possibility of a division of the
world into “system(s)” and “environment.” In the decoherence
program, the term “environment” is usually understood
as the “remainder” of the system, in the sense
that its degrees of freedom are typically not (cannot be,
do not need to be) controlled and are not directly relevant
to the observation under consideration (for example,
the many microsopic degrees of freedom of the system),
but that nonetheless the environment includes “all those
degrees of freedom which contribute significantly to the
evolution of the state of the apparatus” (Zurek, 1981,
p. 1520).

This system–environment dualism is generally associated
with quantum entanglement, which always describes
a correlation between parts of the universe. As long as
the universe is not resolved into individual subsystems,
there is no measurement problem: the state vector | i of the entire universe5 evolves deterministically according
to the Schr¨odinger equation i~ @
@t | i = bH | i, which
poses no interpretive difficulty. Only when we decompose
the total Hilbert-state space H of the universe into
a product of two spaces H1 ⊗ H2, and accordingly form
the joint-state vector | i = | 1i| 2i, and want to ascribe
an individual state (besides the joint state that describes
a correlation) to one of the two systems (say, the apparatus),
does the measurement problem arise.

Sorry about the equations being illegible, couldn't be bothered to translate everything into LaTeX but you can look over that section of the linked paper to see what they should look like. Also see p. 9 where he talks about the "reduced density matrix" for a subsystem (like the cat alone), and says:

The typical situation in which the reduced density matrix
arises is this: Before a premeasurement-type interaction,
the observer knows that each individual system is
in some (unknown) pure state. After the interaction, i.e.,
after the correlation between the systems is established,
the observer has access to only one of the systems, say,
system 1; everything that can be known about the state
of the composite system must therefore be derived from
measurements on system 1, which will yield the possible
outcomes of system 1 and their probability distribution.
All information that can be extracted by the observer
is then, exhaustively and correctly, contained in the reduced
density matrix of system 1, assuming that the Born
rule for quantum probabilities holds.

He also makes some distinction on that page between the "reduced density matrix" for a subsystem and a regular mixed state, so perhaps I am not quite right in saying above that the state of the cat would become a mixed state, although apparently he says it would look like a mixed state so I'm not sure the distinction is very relevant (if the MWI is true it seems to me there would be no 'true' mixed states anywhere in the universe according to his way of speaking, but that doesn't stop us from using the formalism):

However, note that the formal identification of the reduced
density matrix with a mixed-state density matrix
is easily misinterpreted as implying that the state of the
system can be viewed as mixed too (see also the discussion
by d’Espagnat, 1988). Density matrices are only a
calculational tool for computing the probability distribution
of a set of possible outcomes of measurements;
they do not specify the state of the system.7 Since the
two systems are entangled and the total composite system
is still described by a superposition, it follows from
the standard Rules of Quantum Mechanics that no individual
definite state can be attributed to one of the systems.
The reduced density matrix looks like a mixedstate
density matrix because, if one actually measured
an observable of the system, one would expect to get a
definite outcome with a certain probability; in terms of
measurement statistics, this is equivalent to the situation
in which the system is in one of the states from the set of
possible outcomes from the beginning, that is, before the
measurement. As Pessoa (1998, p. 432) puts it, “taking
a partial trace amounts to the statistical version of the
projection postulate.”

 

[rodsika]

In principle it should be possible to prepare the contents of the box in a pure state, which I think just means you would perform a measurement that would give you the maximum possible information about the particles in the box allowed by quantum physics (like a precise measurement of a complete set of commuting observables), allowing you to construct a state vector for the system.

As for decoherence, as I understand it this only applies to some subsystem of a larger system. So if you have the contents of the box in a pure state at the moment the box is sealed, then the complete state vector of everything in the box should remain in a pure state forever as long as the box remains isolated...but if you consider the state of the cat subsystem as separate from the state of the remaining contents of the box (air molecules, cat toys, etc.), then the interaction of the cat with the environment will cause the cat subsystem to go to a mixed state, with the interference terms approaching zero. From p. 8 of the Maximilian Schlosshauer paper you linked to on the other thread:

Sorry about the equations being illegible, couldn't be bothered to translate everything into LaTeX but you can look over that section of the linked paper to see what they should look like. Also see p. 9 where he talks about the "reduced density matrix" for a subsystem (like the cat alone), and says:

He also makes some distinction on that page between the "reduced density matrix" for a subsystem and a regular mixed state, so perhaps I am not quite right in saying above that the state of the cat would become a mixed state, although apparently he says it would look like a mixed state so I'm not sure the distinction is very relevant (if the MWI is true it seems to me there would be no 'true' mixed states anywhere in the universe according to his way of speaking, but that doesn't stop us from using the formalism):

Since the whole universe is isolated. Can you say the whole universe is in pure state? I thought pure state meant coherence. Because you need to prepare the buckyball in pure state to exhibit coherence. But you can't obviously prepare the whole universe to exhibit coherence. I think coherence means the phases interfere, right. So how can all the phases of all objects in the universe interfere. Coherence only occur in laser, and simple things because they are easier to produce interference. But since pure state means coherence, and the Universe is in pure state, the universe should be in coherence, but it isn't. Thanks for assistance in understanding all this

 

[JesseM]

Since the whole universe is isolated. Can you say the whole universe is in pure state?

In the MWI yes, but probably not in Copenhagen.

I thought pure state meant coherence. Because you need to prepare the buckyball in pure state to exhibit coherence. But you can't obviously prepare the whole universe to exhibit coherence. I think coherence means the phases interfere, right. So how can all the phases of all objects in the universe interfere.

Here's how I think it works (as always take my explanation with a grain of salt): interference is in the probability distributions for different outcomes, and for an entangled multiparticle system, to see the interference you need to look at "outcomes" involving all the particles in the system in order to see the sort of statistical correlations which indicate interference. See for example my post discussing the "delayed choice quantum eraser" here, where an interference pattern is only seen if you measure both the 'signal' and 'idler' photons together, the signal photons alone don't show interference...the link to the paper in my post on that thread is outdated, but here is a working one. If you have a two-particle entangled system then if you measure both particles you will find correlations in their outcomes which indicate interference (as in the delayed choice quantum eraser with the 'signal' and 'idler'), but if you have a three-particle entangled system and you measure just two of them the outcomes may not show any sign of interference, you have to measure all three at once to find the correlations that demonstrate interference. So for the entire universe, I think even in the MWI the interference would be impossible to measure since you'd need to measure every particle, you couldn't find evidence that the whole universe was in a pure state just by measuring some subsystem of it.

 

[rodsika]

In the MWI yes, but probably not in Copenhagen.

Here's how I think it works (as always take my explanation with a grain of salt): interference is in the probability distributions for different outcomes, and for an entangled multiparticle system, to see the interference you need to look at "outcomes" involving all the particles in the system in order to see the sort of statistical correlations which indicate interference. See for example my post discussing the "delayed choice quantum eraser" here, where an interference pattern is only seen if you measure both the 'signal' and 'idler' photons together, the signal photons alone don't show interference...the link to the paper in my post on that thread is outdated, but here is a working one. If you have a two-particle entangled system then if you measure both particles you will find correlations in their outcomes which indicate interference (as in the delayed choice quantum eraser with the 'signal' and 'idler'), but if you have a three-particle entangled system and you measure just two of them the outcomes may not show any sign of interference, you have to measure all three at once to find the correlations that demonstrate interference. So for the entire universe, I think even in the MWI the interference would be impossible to measure since you'd need to measure every particle, you couldn't find evidence that the whole universe was in a pure state just by measuring some subsystem of it.

I just realized something. The universe may be in coherence just like a buckyball in double slit experiment is in coherence. I keep asking what is it like to be a buckyball. Maybe we are like a buckyball.. Oh no.. this goes right back to the Many Worlds... the buckyball going to the left is experiencing a branch separate from the right.. our classical world now may just be a branch.

Hmm... I'm looking for non-Many worlds interpretation of this all. Oh yes, Copenhagen gives us comfort. The entire world now is collapse dbecause it is observed. Who is observing it.. Oh no, God? I'm atheist. There may be better explanations. I'll contemplate on this but you agree that our universe is like the buckyball.. they are both in pure state and in coherence, correct?! So it's either Many World or Copenhagen.

The problem with Copenhagen is the wave is a wave of possibility and it collapsed (either by decoherence or self collapse although the former makes better sense). So when the buckyball is sent off, it doesn't have any definite position by principle. It is smear wave. Only upon detection does it possesses definite location. How to imagine this.. we can treat the buckyball as quantum object. Similary we can treat the entire universe as quantum object. Before our universe is measured (by maybe God, etc.). Maybe we can imagine we are all literally just a wave of possibility. This is not a stretch to imagine and won't overtake Many Worlds in bizarreness. What is your problem with this idea? I think i'd pick this over Many worlds.

 

[rodsika]

In principle it should be possible to prepare the contents of the box in a pure state, which I think just means you would perform a measurement that would give you the maximum possible information about the particles in the box allowed by quantum physics (like a precise measurement of a complete set of commuting observables), allowing you to construct a state vector for the system.

As for decoherence, as I understand it this only applies to some subsystem of a larger system. So if you have the contents of the box in a pure state at the moment the box is sealed, then the complete state vector of everything in the box should remain in a pure state forever as long as the box remains isolated...but if you consider the state of the cat subsystem as separate from the state of the remaining contents of the box (air molecules, cat toys, etc.), then the interaction of the cat with the environment will cause the cat subsystem to go to a mixed state, with the interference terms approaching zero. From p. 8 of the Maximilian Schlosshauer paper you linked to on the other thread:

Sorry about the equations being illegible, couldn't be bothered to translate everything into LaTeX but you can look over that section of the linked paper to see what they should look like. Also see p. 9 where he talks about the "reduced density matrix" for a subsystem (like the cat alone), and says:

He also makes some distinction on that page between the "reduced density matrix" for a subsystem and a regular mixed state, so perhaps I am not quite right in saying above that the state of the cat would become a mixed state, although apparently he says it would look like a mixed state so I'm not sure the distinction is very relevant (if the MWI is true it seems to me there would be no 'true' mixed states anywhere in the universe according to his way of speaking, but that doesn't stop us from using the formalism):

There is still a few concept that confuses me. In wikipedia it is said:

"Decoherence does not generate actual wave function collapse. It only provides an explanation for the appearance of wavefunction collapse. The quantum nature of the system is simply "leaked" into the environment. That is; components of the wavefunction are decoupled from a coherent system, and become identified with the immediate surroundings or material universe. A total superposition of the global or universal wavefunction still occurs (i.e. - it remains coherent)..."

Here's the confusing part.

Supposed you have a buckyball composing of 430 atoms prepared in pure state. It means a total superposition exists. Now if one just considers say interaction of 50 atoms inside the buckyball and ignores the rest, decoherence occurs?? How could that be. Or why doesn't the decoherence inside the system spread to the whole? Or is it like Special Relativity, where things seen from our frames of reference differs from other frames. We can measure our length as 1 meter while those who see us moving fast only see 0.8 meter length. Is this the context of decoherence where things seen in isolation is decohered while the whole view is in coherence? Hope you can give an actual example. Thanks.

 

[JesseM]

Supposed you have a buckyball composing of 430 atoms prepared in pure state. It means a total superposition exists. Now if one just considers say interaction of 50 atoms inside the buckyball and ignores the rest, decoherence occurs?? How could that be.

I don't really know what kind of answer you're looking for when you say "how could that be". Apparently it just follows from the math of QM when you calculate a reduced density matrix for the 50 atoms, I don't see why you think this should be so problematic?

Or why doesn't the decoherence inside the system spread to the whole?

Don't know what "spread to the whole" would mean, the rules of wavefunction evolution don't allow for a pure state to evolve into anything but another pure state, but the reduced density matrix for a subsystem would I think be calculated from this very pure state of the whole system.

Or is it like Special Relativity, where things seen from our frames of reference differs from other frames. We can measure our length as 1 meter while those who see us moving fast only see 0.8 meter length. Is this the context of decoherence where things seen in isolation is decohered while the whole view is in coherence?

I'd say they're pretty different, relativity is about viewing the same phenomena from the perspective of different frames of reference (different coordinate systems), not about viewing a subset of something larger. Maybe a slightly closer analogy would be to conditional probability, where P(A|B) would represent the probability that event A occurred on only the subset of trials where event B also occurred, which can be different from P(A) which represents the probability that A occurred on any trial regardless of what happened with B.

Hope you can give an actual example. Thanks.

Did you read over the delayed choice quantum eraser example in post #4? I'm not sure but I think it is actually close to what's going on with decoherence. In that example the signal and idler photons are entangled, if you just look at the total probability distribution for the signal photon alone (like looking at the probability of a subsystem being in different states) it does not show an interference pattern (like how decoherence destroys the possibility of seeing interference effects in the probability distribution of the subsystem), but if you look at probability distributions dealing with both signal and idler (like looking at the probability for both the 'system' and its 'environment'), like the conditional probability that the signal photon was detected at various locations given that the idler was detected at some specific detector, then you can see an interference pattern (like how the combined system + environment is still in a pure state where the interference terms haven't been driven down close to zero).

 

[rodsika]

I don't really know what kind of answer you're looking for when you say "how could that be". Apparently it just follows from the math of QM when you calculate a reduced density matrix for the 50 atoms, I don't see why you think this should be so problematic?

Don't know what "spread to the whole" would mean, the rules of wavefunction evolution don't allow for a pure state to evolve into anything but another pure state, but the reduced density matrix for a subsystem would I think be calculated from this very pure state of the whole system.

I'd say they're pretty different, relativity is about viewing the same phenomena from the perspective of different frames of reference (different coordinate systems), not about viewing a subset of something larger. Maybe a slightly closer analogy would be to conditional probability, where P(A|B) would represent the probability that event A occurred on only the subset of trials where event B also occurred, which can be different from P(A) which represents the probability that A occurred on any trial regardless of what happened with B.

Did you read over the delayed choice quantum eraser example in post #4? I'm not sure but I think it is actually close to what's going on with decoherence. In that example the signal and idler photons are entangled, if you just look at the total probability distribution for the signal photon alone (like looking at the probability of a subsystem being in different states) it does not show an interference pattern (like how decoherence destroys the possibility of seeing interference effects in the probability distribution of the subsystem), but if you look at probability distributions dealing with both signal and idler (like looking at the probability for both the 'system' and its 'environment'), like the conditional probability that the signal photon was detected at various locations given that the idler was detected at some specific detector, then you can see an interference pattern (like how the combined system + environment is still in a pure state where the interference terms haven't been driven down close to zero).

Thanks. I think I'm a bit confused about the meaning of Universal Wavefunction and Pure State. "universal wavefunction" is mentioned in the wikipedia Decoherence article:

"Decoherence does not generate actual wave function collapse. It only provides an explanation for the appearance of wavefunction collapse. The quantum nature of the system is simply "leaked" into the environment. That is; components of the wavefunction are decoupled from a coherent system, and become identified with the immediate surroundings or material universe. A total superposition of the global or universal wavefunction still occurs (i.e. - it remains coherent)..."

The words "universal wavefunction" is highlighted and if you click it.. the following comes out:

"The Universal Wavefunction or Universal Wave Function is a term introduced by Hugh Everett in his Princeton PhD thesis[1] The Theory of the Universal Wave Function, and forms a core concept in the relative state interpretation[2][3] or many-worlds interpretation[4][5] of quantum mechanics. However, it has also received more recent investigation from James Hartle and Stephen Hawking[6] in which they derive a specific solution to the Wheeler-deWitt equation to explain the initial conditions of the Big Bang cosmology.
Everett's thesis introduction reads:

Since the universal validity of the state function description is asserted, one can regard the state functions themselves as the fundamental entities, and one can even consider the state function of the entire universe. In this sense this theory can be called the theory of the "universal wave function," since all of physics is presumed to follow from this function alone.[7]
The universal wave function is the wavefunction or quantum state of the totality of existence, regarded as the "basic physical entity"[8] or "the fundamental entity, obeying at all times a deterministic wave equation"[9]"

Here's the confusing part.
In our classical universe which is in pure state, does universal wavefunction means the totality of our classical universe or is it in the context of "the wavefunction or quantum state of the totality of existence" that includes all the branches or worlds in Many worlds?
Or when you say our universe is in pure state. Are you referring to the classical universe now (our branch) or the totality of all the Everett branches?
We are jumping from the buckyball in pure state in superposition of left and right to our collapsed classical universe (equavalent to say left only or right), hence I don't know if pure state refers to our classical universe (now) or Everett totality of all branches. Well? I guess this is all I need to know for now. Many thanks.

 

[JesseM]

Here's the confusing part.
In our classical universe which is in pure state

Why would you say the classical universe is in a pure state? A pure state always involves superpositions of some form or another...even if you picked a position eigenstate where all the particles were in definite positions, they'd all be in a superposition of wildly different momentum states, maybe the closest you could come to "classical" would be a pure state where both position and momentum for each particle were confined to a microscopic range (as small as would be allowed by the uncertainty principle), but it seems dubious to call this classical since it still involves superpositions and also involves a lot more information then would be available to observers who are only aware of macroscopic objects. And in any case, even if you started from a pure state with these properties it would very quickly evolve into a state involving superpositions of very different macroscopic descriptions as in the Schroedinger's cat example.

does universal wavefunction means the totality of our classical universe or is it in the context of "the wavefunction or quantum state of the totality of existence" that includes all the branches or worlds in Many worlds?

The second one, when people talk about the wavefunction of the universe it always seems to be in the context of the MWI, and they're postulating a quantum state involving a superposition of very different macroscopic outcomes.

 

[rodsika]

Why would you say the classical universe is in a pure state? A pure state always involves superpositions of some form or another...even if you picked a position eigenstate where all the particles were in definite positions, they'd all be in a superposition of wildly different momentum states, maybe the closest you could come to "classical" would be a pure state where both position and momentum for each particle were confined to a microscopic range (as small as would be allowed by the uncertainty principle), but it seems dubious to call this classical since it still involves superpositions and also involves a lot more information then would be available to observers who are only aware of macroscopic objects. And in any case, even if you started from a pure state with these properties it would very quickly evolve into a state involving superpositions of very different macroscopic descriptions as in the Schroedinger's cat example.

The second one, when people talk about the wavefunction of the universe it always seems to be in the context of the MWI, and they're postulating a quantum state involving a superposition of very different macroscopic outcomes.

Oh. I was led to it by the following logic.
If we isolate an electron, it is in pure state.
If we isolate a buckyball, it is in pure state.
If we isolate a cat, it is in pure state.
If we isolate a classical universe, it is in pure state.

Since our classical universe is isolated with no external observer, I thought it is in pure state too. Now why is the classical universe an exception. What is the difference of classical universe to a classical cat. Hmm...

 

[JesseM]

Oh. I was led to it by the following logic.
If we isolate an electron, it is in pure state.
If we isolate a buckyball, it is in pure state.
If we isolate a cat, it is in pure state.
If we isolate a classical universe, it is in pure state.

Since our classical universe is isolated with no external observer, I thought it is in pure state too. Now why is the classical universe an exception. What is the difference of classical universe to a classical cat. Hmm...

I wouldn't say that mere isolation means it's in a pure state, the quantum state in some sense represents your knowledge of the system (except specifically in the MWI where the quantum state of the universe is assigned more of an objective reality), that's why I said in post #2 that you'd have to prepare the system in a pure state by performing some sort of measurement that gives you the maximum possible information allowed by QM, which you can use to construct a state vector:

In principle it should be possible to prepare the contents of the box in a pure state, which I think just means you would perform a measurement that would give you the maximum possible information about the particles in the box allowed by quantum physics (like a precise measurement of a complete set of commuting observables), allowing you to construct a state vector for the system.

If you didn't have that sort of maximal information I think you'd probably treat it as starting out in a mixed state, a statistical ensemble of different possible pure states reflecting your own uncertainty. But again it seems to me that outside of the MWI, the difference between a mixed state and a pure state is more epistemological (dealing with your own knowledge and best descriptions) than ontological (dealing with any kind of objective truth about the system's 'actual' state).

 

[rodsika]

I wouldn't say that mere isolation means it's in a pure state, the quantum state in some sense represents your knowledge of the system (except specifically in the MWI where the quantum state of the universe is assigned more of an objective reality), that's why I said in post #2 that you'd have to prepare the system in a pure state by performing some sort of measurement that gives you the maximum possible information allowed by QM, which you can use to construct a state vector:

If you didn't have that sort of maximal information I think you'd probably treat it as starting out in a mixed state, a statistical ensemble of different possible pure states reflecting your own uncertainty. But again it seems to me that outside of the MWI, the difference between a mixed state and a pure state is more epistemological (dealing with your own knowledge and best descriptions) than ontological (dealing with any kind of objective truth about the system's 'actual' state).

Schroedinger Cat is in my mind all day and even appear in my dreams at night. So let me write something about it to organize my thoughts and concerns so we can see clearly the issue.
You mentioned one must prepare the system in a pure state by performing some sort of measurement. This means at least using a radioactive source that can either decay or not inside the hypothetically isolated box. But I'm interested in the cat body internal environment and decoherence. Can't the cat's bones, bloodstream, organs be their own environment for the decoherence? Even if the cat can be totally isolated. Won't its internal part decohere the whole cat such that even if your have a radioactive source and poison, it won't go into superposition of dead or alive?

The confusion is in Many Worlds. Any random atomic occurence can split branches. So even if the Cat is completely isolated in the box with no other thing with him. His own body can split branches due to the random atomic processes in his body. So to avoid confusion. Let's ignore Many Worlds in the meantime.

Let's go to Modified Copenhagen. In Original Copenhagen. Wavefunction is collapsed by macroscopic object. But in modified enhanced Copenhagen, decoherence is what causing apparent collapse. In Copenagen, when the isotope is measured by the poison detector. Everything collapsed including the cat so end of story. In decoherence, things are interesting because collapse didn't occur by macroscopic contact. If the cat is totally isolated and in pure state with the isotopes/poison inside. Question is (this is the most important question I'd like to know for now) would the cat body internal parts like bones, bloodstream, organs be their own environment for the decoherence before anything else?* Thanks.

 

[JesseM]

You mentioned one must prepare the system in a pure state by performing some sort of measurement. This means at least using a radioactive source that can either decay or not inside the hypothetically isolated box.

Huh? The presence or absence of a radioactive source has nothing at all to do with whether you can treat the system in the box as being in a pure state, as I said you would prepare a pure state by making some sort of maximally detailed measurement of all the particles that compose whatever is inside the box the moment before it becomes isolated.

But I'm interested in the cat body internal environment and decoherence. Can't the cat's bones, bloodstream, organs be their own environment for the decoherence?

A bone cannot be the same bone's environment (unless you're talking about the bone's macrostate vs. the bone's microstate, see below), but the division into "subsystem" and "environment" can be done any way you like, for example the rest of the cat's body can be the "environment" for a bone.

Won't its internal part decohere the whole cat such that even if your have a radioactive source and poison, it won't go into superposition of dead or alive?

"Decohere the whole cat" doesn't mean anything as far as I can see, decoherence is a relative notion, a subsystem decoheres relative to some external environment. I think I already covered this in post #2:

As for decoherence, as I understand it this only applies to some subsystem of a larger system. So if you have the contents of the box in a pure state at the moment the box is sealed, then the complete state vector of everything in the box should remain in a pure state forever as long as the box remains isolated...but if you consider the state of the cat subsystem as separate from the state of the remaining contents of the box (air molecules, cat toys, etc.), then the interaction of the cat with the environment will cause the cat subsystem to go to a mixed state, with the interference terms approaching zero.

The confusion is in Many Worlds. Any random atomic occurence can split branches. So even if the Cat is completely isolated in the box with no other thing with him. His own body can split branches due to the random atomic processes in his body.

You need to be careful when you say "his own body can split branches"--if he's isolated then his body is in a giant superposition, and any subsystem of his body may be said to split into different "branches" in some approximate sense due to decoherence, but what does it mean to say the whole cat splits into branches? For decoherence you always need a subsystem + environment, with decoherence of the subsystem defined relative to the environment. The one trick I think you can use to talk about the whole cat is that you can just look at coarse-grained "macrostates" of the cat (i.e. descriptions differentiated by macroscopic details, like "alive" vs. "dead") which leave aside a lot of microscopic detail as in statistical mechanics (all that micro-detail would constitute the microstate), and then you can treat the microscopic degrees of freedom as a kind of "environment" for the macrostate, so in that sense the different possible macrostates of the whole cat could decohere relative to the microstates. That seems to be what's being talked about on p. 8 of http://users.ox.ac.uk/~mert0130/papers/proc_dec.pdf , where he writes:

The basic setup is probably familiar to most readers. We assume that the
Hilbert space H of the system we are interested in is factorised into “system”
and “environment” subsystems ... Here, the “environment” might be a genuinely external environment (such as the atmosphere or the cosmic microwave background); equally, it might be an “internal environment”, such as the microscopic degrees of freedom of a fluid.

Let's go to Modified Copenhagen. In Original Copenhagen. Wavefunction is collapsed by macroscopic object. But in modified enhanced Copenhagen, decoherence is what causing apparent collapse.

I don't think that notion of "modified enhanced Copenhagen" makes any sense, by definition decoherence says there is no actual collapse so the total system remains in a pure state, in which case you have the MWI. Anything that deserves the name "Copenhagen" must at least model measurement as involving a true "collapse" (though it doesn't necessarily have to say this collapse is an ontological reality, it can just say that it's a necessary part of the model we humans use to make predictions in QM while remaining agnostic about what's "really" going on).

 

[rodsika]

Huh? The presence or absence of a radioactive source has nothing at all to do with whether you can treat the system in the box as being in a pure state, as I said you would prepare a pure state by making some sort of maximally detailed measurement of all the particles that compose whatever is inside the box the moment before it becomes isolated.

A bone cannot be the same bone's environment (unless you're talking about the bone's macrostate vs. the bone's microstate, see below), but the division into "subsystem" and "environment" can be done any way you like, for example the rest of the cat's body can be the "environment" for a bone.

"Decohere the whole cat" doesn't mean anything as far as I can see, decoherence is a relative notion, a subsystem decoheres relative to some external environment. I think I already covered this in post #2:You need to be careful when you say "his own body can split branches"--if he's isolated then his body is in a giant superposition, and any subsystem of his body may be said to split into different "branches" in some approximate sense due to decoherence, but what does it mean to say the whole cat splits into branches? For decoherence you always need a subsystem + environment, with decoherence of the subsystem defined relative to the environment. The one trick I think you can use to talk about the whole cat is that you can just look at coarse-grained "macrostates" of the cat (i.e. descriptions differentiated by macroscopic details, like "alive" vs. "dead") which leave aside a lot of microscopic detail as in statistical mechanics (all that micro-detail would constitute the microstate), and then you can treat the microscopic degrees of freedom as a kind of "environment" for the macrostate, so in that sense the different possible macrostates of the whole cat could decohere relative to the microstates. That seems to be what's being talked about on p. 8 of http://users.ox.ac.uk/~mert0130/papers/proc_dec.pdf , where he writes:I don't think that notion of "modified enhanced Copenhagen" makes any sense, by definition decoherence says there is no actual collapse so the total system remains in a pure state, in which case you have the MWI. Anything that deserves the name "Copenhagen" must at least model measurement as involving a true "collapse" (though it doesn't necessarily have to say this collapse is an ontological reality, it can just say that it's a necessary part of the model we humans use to make predictions in QM while remaining agnostic about what's "really" going on).

I thought you were referring to creating quantum choices that can be measured hence I mentioned about isolates to produce dead and alive cat states. So you are just saying that*
one must make "some sort of maximally detailed measurement of all the particles that compose whatever is inside the box the moment before it becomes isolated." But if you don't do this. It doesn't affect the particles. The object is still in pure state!* You may not get any data of the state vector but it doesn't affect the object. It is still in pure state. For example. If I'm isolated in a 100% isolation box. My body is in pure state, independent if you do a detailed measurement of every particle in my body or what. Why would your measuring me affect me?

Anyway. If I'm in pure state inside the box. And there is a double slit inside. I guess I can appear in two places at once and enter it both, without any Many Worlds at all. I prefer this interpretation.
*
Speaking of Many worlds and your last paragraph.
But majority of physicists now accept the reality of decoherence, yet it doesn't necessarily they accept Many worlds. So Decoherence doesn't entail Many Worlds. So what is Decoherence like without Many worlds. Perhaps they treat decoherence are math-like akin to how they treat the wavefunction as mathematical entity. In Many worlds, the wave function means there are actual copies of the system. In Copenhagen, the wavefunction represent the properties of the system. So if you will take the Copenhagen mindset, Decoherence is just pure math like how their wave function is pure math. In the double slit experiment, when decoherence affects the setup. Interference disappear.. and nothing happens to the branch that doesn't register in the register. It just vanish. This is what majority of physicists who don't believe in Many Worlds believe what happens in Decoherence. So you can't say when a system is in pure state, it necessarly mean MWI.

 

[Einstein Mcfly]

Oh. I was led to it by the following logic.
If we isolate an electron, it is in pure state.
If we isolate a buckyball, it is in pure state.
If we isolate a cat, it is in pure state.
If we isolate a classical universe, it is in pure state.

No, technically speaking all states are most completely described by a mixed sate density matrix determined by the Boltzmann distribution at whatever finite temperature the system is at. Only by approximation (although in many cases totally justified) do we ever achieve a pure state.

 

[JesseM]

I thought you were referring to creating quantum choices that can be measured hence I mentioned about isolates to produce dead and alive cat states. So you are just saying that*
one must make "some sort of maximally detailed measurement of all the particles that compose whatever is inside the box the moment before it becomes isolated." But if you don't do this. It doesn't affect the particles. The object is still in pure state!*

Please read post #11 again, you are treating "pure state" vs. "mixed state" as ontological facts about the system rather than epistemological facts about our knowledge of the system and how we model it, outside of the specific context of the MWI I don't think it makes any sense to ask whether a system is "really" in a pure state or a mixed state.

Anyway. If I'm in pure state inside the box. And there is a double slit inside. I guess I can appear in two places at once and enter it both, without any Many Worlds at all. I prefer this interpretation.

But that pure state would include different versions of you whose brain each recorded either going through one slit or going through the other, I don't think it makes any sense to suggest you could somehow experience both at once, see my [post=3240295]post #70 on the other thread[/post].

Speaking of Many worlds and your last paragraph.
But majority of physicists now accept the reality of decoherence, yet it doesn't necessarily they accept Many worlds. So Decoherence doesn't entail Many Worlds. So what is Decoherence like without Many worlds.

Well, even if you don't accept MWI you can still see what the QM math says about the wavefunction evolution for an isolated system which can be broken up into a "subsystem" and its "environment", see [post=3178736]this comment of mine from a thread in March[/post]. In practice though I think most physicists who work on the issue of decoherence would reject the idea that any special "collapse" happens on measurement, even for a non-isolated system, regardless of whether they accept all the ideas associated with the MWI (they might accept the mathematical formalism of the MWI but be agnostic about whether other versions of the same human experimenters besides the ones they experience can really be considered 'real' for example...some might also prefer other no-collapse interpretations like Bohmian mechanics).

 

[JesseM]

The above says that laser, superconductivity, and superfluidity are examples of things in coherence (meaning pure state), hence this are ontological facts!*

That's a good question, I'm not sure about the answer. I would guess it's that we model the general phenomenon of superconductivity/lasers etc. by imagining we know the precise pure state of some hypothetical system and seeing what general behavior will result, but that doesn't mean we have sufficiently detailed measurements of any real-world laser/superconducting system to define a specific pure state for them. And defining the state of a particular real-world laser/superconducting system as a mixed state might not result in any different general predictions, a mixed state is just a statistical ensemble of pure states so for example if all the pure states that were part of the ensemble for a laser had the same laser frequency, then you might get the same broad predictions about "laser-like" behavior using the empirically-based mixed state. Alternately, it's possible that there's something about lasers and superconductors that makes it easier to actually measure sufficient information to determine a precise pure state for all the particles involved, though I doubt this is the answer.

 

[rodsika]

That's a good question, I'm not sure about the answer. I would guess it's that we model the general phenomenon of superconductivity/lasers etc. by imagining we know the precise pure state of some hypothetical system and seeing what general behavior will result, but that doesn't mean we have sufficiently detailed measurements of any real-world laser/superconducting system to define a specific pure state for them. And defining the state of a particular real-world laser/superconducting system as a mixed state might not result in any different general predictions, a mixed state is just a statistical ensemble of pure states so for example if all the pure states that were part of the ensemble for a laser had the same laser frequency, then you might get the same broad predictions about "laser-like" behavior using the empirically-based mixed state. Alternately, it's possible that there's something about lasers and superconductors that makes it easier to actually measure sufficient information to determine a precise pure state for all the particles involved, though I doubt this is the answer.

Isn't it more logical to say that even if we don't have maximal information or any information at all about the particles in the box. We can assume the cat or any object inside a 100% hypothetical isolation box is in pure state? Note the laser in our CD player works even if we don't measure all the particles in the electronics of the player. So similar to the laser in CD player, the cat is also in pure state. And both are ontological facts.

Anyway. Where (references, web site) did you hear that to prepare the contents of something in pure state, one has to (as you said) "perform a measurement that would give you the maximum possible information about the particles in the box allowed by quantum physics (like a precise measurement of a complete set of commuting observables), allowing you to construct a state vector for the system."

 

[JesseM]

Isn't it more logical to say that even if we don't have maximal information or any information at all about the particles in the box. We can assume the cat or any object inside a 100% hypothetical isolation box is in pure state? Note the laser in our CD player works even if we don't measure all the particles in the electronics of the player. So similar to the laser in CD player, the cat is also in pure state. And both are ontological facts.

I disagree, if you haven't actually measured the initial state to prepare it in a particular state, then the ontological claim "it is in a pure state" is exactly as unverifiable as "it is in a mixed state", no later measurement of the state will provide any evidence that one or the other was correct.

Incidentally, to elaborate a little on this suggestion:

I would guess it's that we model the general phenomenon of superconductivity/lasers etc. by imagining we know the precise pure state of some hypothetical system and seeing what general behavior will result, but that doesn't mean we have sufficiently detailed measurements of any real-world laser/superconducting system to define a specific pure state for them.

Thinking about this some more, maybe if we use QM to model a hypothetical isolated laser apparatus, and then split it up into a "subsystem" consisting of either just the photons or the photons plus the atoms in the gain medium (or some sort of macrostate of either one), and an "environment" consisting of the rest of the apparatus, perhaps we could show that in this case there is little to no decoherence of the "subsystem" due to its interaction with the "environment", and this would have important implications for how the laser behaves, so this could be another sense in which lasers are said to be in a coherent state even if we cannot perform the type of measurement that would define a pure state for any real-world laser.

Anyway. Where (references, web site) did you hear that to prepare the contents of something in pure state, one has to (as you said) "perform a measurement that would give you the maximum possible information about the particles in the box allowed by quantum physics (like a precise measurement of a complete set of commuting observables), allowing you to construct a state vector for the system."

I don't know where I learned this originally, but for a reference you can see p. 238 of the textbook Advanced Visual Quantum Mechanics, at the bottom it says:

We see that the mixed state $$\rho$$ expresses our ignorance about the details of the preparation process. A pure state can only be obtained by a simultaneously preparatory measurement of a complete set of commuting observables. Whenever the experiment leaves us ignorant of the eigenvalue of at least one of the observables, we have to describe the state by a density operator.

 

[rodsika]

I disagree, if you haven't actually measured the initial state to prepare it in a particular state, then the ontological claim "it is in a pure state" is exactly as unverifiable as "it is in a mixed state", no later measurement of the state will provide any evidence that one or the other was correct.

Incidentally, to elaborate a little on this suggestion:

Thinking about this some more, maybe if we use QM to model a hypothetical isolated laser apparatus, and then split it up into a "subsystem" consisting of either just the photons or the photons plus the atoms in the gain medium (or some sort of macrostate of either one), and an "environment" consisting of the rest of the apparatus, perhaps we could show that in this case there is little to no decoherence of the "subsystem" due to its interaction with the "environment", and this would have important implications for how the laser behaves, so this could be another sense in which lasers are said to be in a coherent state even if we cannot perform the type of measurement that would define a pure state for any real-world laser.

I don't know where I learned this originally, but for a reference you can see p. 238 of the textbook Advanced Visual Quantum Mechanics, at the bottom it says:

But the 100% hypothetical box is like Hilbert Space which entails pure state. According to Bruce Collin in Schroedinger Rabbit:

"The cat box, or philospher box as it is now, floats within this central shield, and within it is a further thin spherical shell like a Christmas bauble that is cooled to the lower temperature practicable, in the milli-or perhaps even micro-Kelvin range, to suppress the radiation of infrared photons. The inner shell contains a perfect vacuum except for the very ocassional very low energy infrared photon emitted by the walls. Because such photons have a wavelength of several meters, they reveal nothing about the position or state of the central chamber beyond the fact that is is there.

In theory, the space capsule now no longer contains a philosopher-astronaut, but a Hilbert space, a probability distribution of philosopher-astronauts doing increasingly divergent things, as their personal histories diverge depending on exactly how many photons hit each cell of their retinas and other quantum events that multiply into macorsopic consequences in various ways. If we could look inside the capsule (which is, by definition, impossible), we might imagine seeing something like a multiple-exposure photograph. Is the astonaut wring, or brushing her teeth, or just staring into space?

We have seemingly created a macroscopic bubble of Hilbert space, in which different probability histories of the astronaut, eventually diverging quite significantly, can trace themselves out."

----------------

So this box is automatically Hilbert Space and in pure state. Is it not? Of course in principle this can't be done in real world because of gravity. But let's say the box could be made (for sake of discussion although we know it can't be made), then pure state is automatically assume. Don't you agree?

 

[JesseM]

But the 100% hypothetical box is like Hilbert Space which entails pure state.

Why do you say that? A mixed state is just a statistical ensemble of different vectors in Hilbert Space, and if the system remains completely isolated, then the various vectors making up the mixed state should each evolve into massive superpositions of different classical states like Schroedinger's cat. So, Colin Bruce's statement seems to be equally applicable regardless of whether we model the initial state of the system at the moment it was sealed in "the box" as a pure state or a mixed state.

 

[rodsika]

Why do you say that? A mixed state is just a statistical ensemble of different vectors in Hilbert Space, and if the system remains completely isolated, then the various vectors making up the mixed state should each evolve into massive superpositions of different classical states like Schroedinger's cat. So, Colin Bruce's statement seems to be equally applicable regardless of whether we model the initial state of the system at the moment it was sealed in "the box" as a pure state or a mixed state.

Hmm.. I thought mixed state automatically entail no superposition. That is why I assume there is pure state inside the 100% hypothetical isolated box because it can experience superposition. If you say that mixed state can also experience superposition. Why can't you make my physical body which is in mixed state to experience superposition of sitting down and standing up?

But mixed state won't experience interference. You can't produce interference in the double slit experiment by using buckyball that is in mixed state. This means superposition can't occur in mixed state Buckyball. But you said it can. How. Unless. You are saying that superposition of other observables can happen? In the double slit, it's superposition of path that is not possible with a mixed state buckyball. So I automatically assume all kinds of superposition is not possible. So you are saying that in a Buckyball in thermal agitation and in mixed state. Superposition of other observable is possible? Please give an observable that can still experience superposition. Thanks.

 

[JesseM]

Hmm.. I thought mixed state automatically entail no superposition. That is why I assume there is pure state inside the 100% hypothetical isolated box because it can experience superposition. If you say that mixed state can also experience superposition. Why can't you make my physical body which is in mixed state to experience superposition of sitting down and standing up?

But mixed state won't experience interference. You can't produce interference in the double slit experiment by using buckyball that is in mixed state. This means superposition can't occur in mixed state Buckyball. But you said it can. How. Unless. You are saying that superposition of other observables can happen? In the double slit, it's superposition of path that is not possible with a mixed state buckyball. So I automatically assume all kinds of superposition is not possible. So you are saying that in a Buckyball in thermal agitation and in mixed state. Superposition of other observable is possible? Please give an observable that can still experience superposition. Thanks.

A mixed state just means you assign different classical probabilities to different pure states, then each pure state evolves normally according to the usual QM rules. The details of what you know about the system via measurements, and how the system interacts with its environment if it's not isolated (I think when decoherence is happening the environment basically acts as though it's 'measuring' the system even if you don't, which would explain why for example I can't treat you as being in a superposition of sitting and standing even I don't know whether you're sitting or standing), determines the details of the mixed state and what kind of superposition/interference you might see. For example, suppose you know the particle was measured as it went through the slits in a double-slit experiment, but you don't actually know which slit it was found to have gone through. Then you could have a mixed state that assigned a 50% probability to the pure state S1 that would result from it being measured to go through the left slit, and a 50% probability to the pure state S2 that would result from it being measured to go through the right slit. Then if you evolve S1 forward and find it predicts probability p1 that the particle will be detected at a particular position on the screen, and evolve S2 forward and find it predicts probability p2 that the particle will be detected at that same position, the total probability assigned by the mixed state to that position would be 0.5*p1 + 0.5*p2. So, the total probability distribution for the mixed state is just the sum of the probability distributions for S1 and S2 individually which doesn't show the same sort of interference pattern as seen when no measurement is made of which slit the particle goes through, see the diagrams with the patterns created by a gun firing through a slit near the top of this page.

As for what would still be in superposition--well, just think about what would be in superposition for the pure state S1 on its own, and for S2 on its own, it should still be in superposition in the mixed state. Both S1 and S2 involved measuring the position at the moment the particle passed through the slit, but afterwards there are no further position measurements until it hits the screen, so in either case it would be in a superposition of different positions with maximum amplitude closest to the slit it went through.

 

[rodsika]

A mixed state just means you assign different classical probabilities to different pure states, then each pure state evolves normally according to the usual QM rules. The details of what you know about the system via measurements, and how the system interacts with its environment if it's not isolated (I think when decoherence is happening the environment basically acts as though it's 'measuring' the system even if you don't, which would explain why for example I can't treat you as being in a superposition of sitting and standing even I don't know whether you're sitting or standing), determines the details of the mixed state and what kind of superposition/interference you might see. For example, suppose you know the particle was measured as it went through the slits in a double-slit experiment, but you don't actually know which slit it was found to have gone through. Then you could have a mixed state that assigned a 50% probability to the pure state S1 that would result from it being measured to go through the left slit, and a 50% probability to the pure state S2 that would result from it being measured to go through the right slit. Then if you evolve S1 forward and find it predicts probability p1 that the particle will be detected at a particular position on the screen, and evolve S2 forward and find it predicts probability p2 that the particle will be detected at that same position, the total probability assigned by the mixed state to that position would be 0.5*p1 + 0.5*p2. So, the total probability distribution for the mixed state is just the sum of the probability distributions for S1 and S2 individually which doesn't show the same sort of interference pattern as seen when no measurement is made of which slit the particle goes through, see the diagrams with the patterns created by a gun firing through a slit near the top of this page.

As for what would still be in superposition--well, just think about what would be in superposition for the pure state S1 on its own, and for S2 on its own, it should still be in superposition in the mixed state. Both S1 and S2 involved measuring the position at the moment the particle passed through the slit, but afterwards there are no further position measurements until it hits the screen, so in either case it would be in a superposition of different positions with maximum amplitude closest to the slit it went through.

Hmm... Isn't it that we prepare a Buckyball to be in pure state by simply shielding or isolating it from the environment? This is why I assume that merely isolating anything completely from the environment would automatically put it in pure state like the totally isolated Buckyball. So aside from complete isolation, what must you do to "prepare" the Buckyball to turn it into pure state? Must you arrange the atoms in certain orders or configurations.. is this what you meant.

Also I was assuming that by totally isolating a virus from the environment, it is automatically in pure state. But it seems you must do stuff to the virus to make it pure state, in this case. What must you do to the virus to make it pure state? By making sure the internal parts can interfere.. but this means altering the virus itself so it's no longer the original virus (?)

Thanks a lot for you assistance. Why didn't you become a professor at Universities :)

 

[rodsika]

After rereading the thread again and again. Here's what seems to be taking place.

Pure state and mixed state is only about our knowledge of the system. So if we don't measure the state of each particle, we won't know it is in pure state.

An unknown object in complete isolation. If we don't know what its particles are made up of. We won't know how to do measurements to detect interference (that is.. we won't know the separation of the slits to use for example).. is this what you mean? But we could do diagnostics by using variable separation of the slits until we can detect interference.

Also another thing. De Broglie wavelength is proportional to mass. So by merely knowing the mass of the Buckyball, we can know the de Broglie wavelength without knowing how exactly the particles inside are arranged. Hence we don't have to know every detail of what's inside the Buckyball to perform interference experiment. We only have to isolate it completely and knowing it's mass. Here one can treat it as pure state because we can know its behavior of mixed states by the right separation of the slits to be used to detect the interferences.

 

[JesseM]

Hmm... Isn't it that we prepare a Buckyball to be in pure state by simply shielding or isolating it from the environment? This is why I assume that merely isolating anything completely from the environment would automatically put it in pure state like the totally isolated Buckyball. So aside from complete isolation, what must you do to "prepare" the Buckyball to turn it into pure state? Must you arrange the atoms in certain orders or configurations.. is this what you meant.

Again, if you want our representation of the Buckyball to be a pure state, ideally I think you have to actually measure a complete set of commuting observables for all the particles that make it up. But see my above comments about the laser, it may be that even if you don't have a detailed pure state for a particular buckyball, you can still show that in any generic hypothetical buckyball some parts will remain coherent relative to other parts or to the external environment in some experimental setup, and this will have consequences for predictions about how buckyballs in general behave in that experimental setup, which you could test with real buckyballs even if you don't have enough information about them to represent them as being in a pure state.

What must you do to the virus to make it pure state? By making sure the internal parts can interfere.. but this means altering the virus itself so it's no longer the original virus (?)

No, nothing about the notion of a mixed state automatically implies there won't be interference. For example, before I gave an example of a mixed state based on knowing the particle was measured to go through one slit or the other, and that destroyed the interference on the screen, but suppose some other property of the particle was measured before it reached the slits (including a possible 'measurement' by the environment via decoherence), and again you know the measurement was performed but you don't know the result...in this case you could again have a mixed state, but the individual pure states that would make up the mixed state would all be ones where no information was collected about which slit the particle went through, so in this case you would predict an interference pattern on the screen even though it was a mixed state. So whether a given type of interference can be seen depends on the type of mixed state you have, what exactly the pure states are that you're taking a weighted average of to define the mixed state.

Thanks a lot for you assistance. Why didn't you become a professor at Universities :)

Never got around to doing the grad school thing, reviewing everything I ever learned for the GREs seems like a lot of work, maybe someday though...

 

[rodsika]

Again, if you want our representation of the Buckyball to be a pure state, ideally I think you have to actually measure a complete set of commuting observables for all the particles that make it up. But see my above comments about the laser, it may be that even if you don't have a detailed pure state for a particular buckyball, you can still show that in any generic hypothetical buckyball some parts will remain coherent relative to other parts or to the external environment in some experimental setup, and this will have consequences for predictions about how buckyballs in general behave in that experimental setup, which you could test with real buckyballs even if you don't have enough information about them to represent them as being in a pure state.

No, nothing about the notion of a mixed state automatically implies there won't be interference. For example, before I gave an example of a mixed state based on knowing the particle was measured to go through one slit or the other, and that destroyed the interference on the screen, but suppose some other property of the particle was measured before it reached the slits (including a possible 'measurement' by the environment via decoherence), and again you know the measurement was performed but you don't know the result...in this case you could again have a mixed state, but the individual pure states that would make up the mixed state would all be ones where no information was collected about which slit the particle went through, so in this case you would predict an interference pattern on the screen even though it was a mixed state. So whether a given type of interference can be seen depends on the type of mixed state you have, what exactly the pure states are that you're taking a weighted average of to define the mixed state.

Never got around to doing the grad school thing, reviewing everything I ever learned for the GREs seems like a lot of work, maybe someday though...

So pure state and mixed states are epistemological or "dealing with your own knowledge and best descriptions". How about the word "quantum coherence"? Is it epistemological or ontological (meaning "dealing with any kind of objective truth about the system's 'actual' state")?

In a cat inside a 100% hypothetical isolation box. Can we say the cat is in quantum coherence? Although I know we can't say if it is in pure state or mixed state because we weren't able to count all its particle before it got isolated as you emphased. But can we generally said the cat is in quantum coherence?

Now if you say that the cat is in quantum coherence inside the box. Can it experience a macrostate superposition of being in in all sleeping positions? You may say we have to invoke Many Worlds for this. And it is possible in Many worlds. But in pure Copenhagen. Anything in Superposition is smeared out. So in Copenhagen. Can we say that the cat is without positions or ghost like? Or does the indeterminacy of positions only has to do with microscopic scale or only his atoms would in indefinite locations while his macroscopic shape is still a cat?

 

[JesseM]

So pure state and mixed states are epistemological or "dealing with your own knowledge and best descriptions". How about the word "quantum coherence"? Is it epistemological or ontological (meaning "dealing with any kind of objective truth about the system's 'actual' state")?

As I said I don't really know the detailed math of decoherence vs. coherence so I'm not sure about this, but googling a little, p. 121 of this book seems to say "quantum coherence" is a matter of degree:

When $$0 < |\langle R \mid R' \rangle |^2 < 1$$, $$\mid \phi \rangle$$ will still be an entangled state, but $$\hat{\rho}_S$$ will be a mixture with some degree of quantum coherence left in it. This quantum coherence is revealed by the presence of off-diagonal components in (4.33) in this case. The amount of quantum coherence present will depend on how close to 1 $$|\langle R \mid R' \rangle |^2$$ is, covering a continuous range from a pure state to a mixture devoid of coherence.

Also, again think of my comment about the laser in post #19, it may be that talk of coherence in a type of subsystem of some larger type of system, like particles in the cavity of a laser apparatus, is based on imagining how the reduced density matrix of this subsystem would behave if we knew the full (pure) state of the larger apparatus (or the apparatus plus its own external environment), even if we don't actually know that for any particular real instance of the system.

(incidentally, if anyone reading this thread does understand this stuff at a more technical level, any thoughts on the matter of how a system like a laser can be said to be in a coherent state if we don't actually have enough information to define a pure state for any particular laser? Is this paper discussing whether or not quantum coherence in lasers is a 'myth' or a 'fact' at all relevant here?)

In a cat inside a 100% hypothetical isolation box. Can we say the cat is in quantum coherence? Although I know we can't say if it is in pure state or mixed state because we weren't able to count all its particle before it got isolated as you emphased.

I wouldn't say we "can't say", since we don't know enough to construct a pure state I would say we would have to model it as being in a mixed state since there'd be multiple possible pure states compatible with the information we do have. Of course for something like a cat I imagine it wouldn't really be practical to construct even a mixed state because the number of possible pure states would be so vast, but for a much smaller system put in isolation without our having enough info to construct a pure state, probably a mixed state could actually be used.

Now if you say that the cat is in quantum coherence inside the box. Can it experience a macrostate superposition of being in in all sleeping positions?

I don't think it can "experience" this, see my [post=3240295]post 70 on the other thread[/post], but I think our model of it would involve such superpositions, because even if our model of it was a mixed state this is just a sum of pure states and as long as it remains in isolation I'd expect each individual pure state to evolve into such a macrostate superposition, like I said in post #21 here.

 

[rodsika]

As I said I don't really know the detailed math of decoherence vs. coherence so I'm not sure about this, but googling a little, p. 121 of this book seems to say "quantum coherence" is a matter of degree:

Also, again think of my comment about the laser in post #19, it may be that talk of coherence in a type of subsystem of some larger type of system, like particles in the cavity of a laser apparatus, is based on imagining how the reduced density matrix of this subsystem would behave if we knew the full (pure) state of the larger apparatus (or the apparatus plus its own external environment), even if we don't actually know that for any particular real instance of the system.

(incidentally, if anyone reading this thread does understand this stuff at a more technical level, any thoughts on the matter of how a system like a laser can be said to be in a coherent state if we don't actually have enough information to define a pure state for any particular laser? Is this paper discussing whether or not quantum coherence in lasers is a 'myth' or a 'fact' at all relevant here?)

I wouldn't say we "can't say", since we don't know enough to construct a pure state I would say we would have to model it as being in a mixed state since there'd be multiple possible pure states compatible with the information we do have. Of course for something like a cat I imagine it wouldn't really be practical to construct even a mixed state because the number of possible pure states would be so vast, but for a much smaller system put in isolation without our having enough info to construct a pure state, probably a mixed state could actually be used.

I don't think it can "experience" this, see my [post=3240295]post 70 on the other thread[/post], but I think our model of it would involve such superpositions, because even if our model of it was a mixed state this is just a sum of pure states and as long as it remains in isolation I'd expect each individual pure state to evolve into such a macrostate superposition, like I said in post #21 here.

I thought macroscopic superposition only occurs in Many Worlds. When you say this statement that "as long as it remains in isolation I'd expect each individual pure state to evolve into such a macrostate superposition", are you referring to Many Worlds only or also pure Copenhagen? In pure Copenhagen, things are wave of possibility before being measured. So you are saying that in Copenhagen it is not just microstate superpositions in the scale of the atoms but also macrostate superpositions of all sleeping positions that literally produce a smeared cat??

 

[JesseM]

I thought macroscopic superposition only occurs in Many Worlds. When you say this statement that "as long as it remains in isolation I'd expect each individual pure state to evolve into such a macrostate superposition", are you referring to Many Worlds only or also pure Copenhagen? In pure Copenhagen, things are wave of possibility before being measured. So you are saying that in Copenhagen it is not just microstate superpositions in the scale of the atoms but also macrostate superpositions of all sleeping positions that literally produce a smeared cat??

If the cat could be kept totally isolated then according to the rules of the Copenhagen interpretation there should be no "collapse" due to observation, so it should evolve into macroscopic superpositions--that was exactly why Schroedinger thought up this example, to try to suggest there was a problem with the Copenhagen idea that isolated quantum systems have wavefunctions that evolve according to the deterministic Schroedinger equation until they are "observed".

 

[rodsika]

If the cat could be kept totally isolated then according to the rules of the Copenhagen interpretation there should be no "collapse" due to observation, so it should evolve into macroscopic superpositions--that was exactly why Schroedinger thought up this example, to try to suggest there was a problem with the Copenhagen idea that isolated quantum systems have wavefunctions that evolve according to the deterministic Schroedinger equation until they are "observed".

Hi, you wrote in the other thread the following:

"Sure, even in the Copenhagen interpretation you could have decoherence if you could keep a sufficiently complex system consisting of both a subsystem and its "environment" in isolation for a little while, so there'd be no external system to "collapse" it (like Schroedinger's cat, or a simulation on a large quantum computer)."

and you wrote above that "If the cat could be kept totally isolated then according to the rules of the Copenhagen interpretation there should be no "collapse" due to observation, so it should evolve into macroscopic superpositions"

Now a question that has been haunting me the whole day.

Decoherence can occur inside the cat body (subsystem of the whole which is what decoherence is all about). This means the cat bloodstream or bones can become classical with definite positions (a result of decoherence). So how could the cat suffer macroscopic superposition when part of its internal body had become classical?

This is what I meant in message #7 here where I asked:

"Supposed you have a buckyball composing of 430 atoms prepared in pure state. It means a total superposition exists. Now if one just considers say interaction of 50 atoms inside the buckyball and ignores the rest, decoherence occurs?? How could that be."

Your reply was:

I don't really know what kind of answer you're looking for when you say "how could that be". Apparently it just follows from the math of QM when you calculate a reduced density matrix for the 50 atoms, I don't see why you think this should be so problematic?

And I asked (in the same message)

"Or why doesn't the decoherence inside the system spread to the whole?"

Your reply was:

Don't know what "spread to the whole" would mean, the rules of wavefunction evolution don't allow for a pure state to evolve into anything but another pure state, but the reduced density matrix for a subsystem would I think be calculated from this very pure state of the whole system.

What I meant by spread was this. Since decoherence occurs in the 50 atoms in the 430 atom buckyball, the 50 atoms become classical. Why didn't the classicality spread to the entire buckyball such that it the position become definite and classical?? This is similar to the question why the cat in macroscopic superposition has the internal organ like liver suffer decoherence which becomes classical and yet it didn't spread to the entire cat making the whole cat classical. It's like seeing a smeared image of the cat with a solid liver inside.

Note very important that we are talking about Copenhagen in this message where before things are measured, they are in superposition. So pls don't mention Many Worlds because in Many worlds one can argue there are many branches and all are classical. I'm focusing on Copenhagen where before measurements things exist in smeared out superposition but decoherence can exist internally inside the smeared superposition just like you said in message #2:

As for decoherence, as I understand it this only applies to some subsystem of a larger system. So if you have the contents of the box in a pure state at the moment the box is sealed, then the complete state vector of everything in the box should remain in a pure state forever as long as the box remains isolated...but if you consider the state of the cat subsystem as separate from the state of the remaining contents of the box (air molecules, cat toys, etc.), then the interaction of the cat with the environment will cause the cat subsystem to go to a mixed state, with the interference terms approaching zero.

Here the same question can be asked why decoherence in a region (say cat liver) inside the pure state didn't spread the region of classicality (classical liver) to the entire object (in smeared out superposition) preventing or killing the superposition.

And lastly. How complex must be the thing inside the 100% isolated box before this spreading of region of decoherence spread to the entire making turning it classical (in position).

Thanks.

 

[JesseM]

Decoherence can occur inside the cat body (subsystem of the whole which is what decoherence is all about). This means the cat bloodstream or bones can become classical with definite positions (a result of decoherence).

Decoherence means that the reduced density matrix for a bone, say, would come to be a mixed state where the amplitude is concentrated on position eigenstates and the interference terms were close to zero. But that's not quite the same as saying the particles have become "classical with definite positions", the mixed state would still look like a statistical ensemble of different position eigenstates, each of which would feature the particles in different sets of positions. So for example one position eigenstate might involve a given leg bone being upright because the cat is in a standing position, another might involve it being horizontal because the cat is in a sleeping position, the reduced density matrix would include both. Decoherence doesn't provide a "collapse" that selects one of the many position eigenstates, it just means that it's approximately correct to treat it as a statistical mixture of these different states rather than a superposition where you have to worry about interference effects.

What I meant by spread was this. Since decoherence occurs in the 50 atoms in the 430 atom buckyball, the 50 atoms become classical.

I think it's misleading to say they "become classical", as I said they don't go to any single definite set of positions, and it's only approximately correct to treat them as a regular statistical ensemble if you're just looking at those 50 particles and ignoring the other 380, you would still find interference effects if you looked at all 430 at once. Again think of the delayed choice quantum eraser, where if you just look at the probability distribution for signal photons ignoring correlations with idlers, you find a non-interference pattern on the screen behind the double slit, but if you consider the conditional probability that a signal photon landed at various positions given that the idler was detected at some detector, then here there may be an interference pattern.

Note very important that we are talking about Copenhagen in this message where before things are measured, they are in superposition. So pls don't mention Many Worlds because in Many worlds one can argue there are many branches and all are classical. I'm focusing on Copenhagen where before measurements things exist in smeared out superposition but decoherence can exist internally inside the smeared superposition

Copenhagen is just sort of agnostic about what's "really" going on in the box before measurement, it just says that we model the inside of the box as a big superposition in order to figure out the probabilities of various outcomes when we open it. In terms of Copenhagen, say you could prepare some vast number of boxes with the identical cats measured to be in identical initial pure states at the moment the box was sealed, and then for each one you opened the box after some fixed time and made another exhaustive measurement of all the particles, and then you looked at the statistical patterns in all these different experiments, constructing a probability distribution on different final outcomes. In that case, the probability distribution for different final outcomes involving all the particles would show interference effects (akin to how the probability distribution for signal and idler shows interference), but if you just looked at "reduced" final outcomes which only involved paying attention to a particular subsystem of particles in each final measurement, here the probability distributions for different possible outcomes would show virtually no interference (akin to how the probability distribution for signal photon alone, throwing out information about the idler, doesn't show interference).

 

[rodsika]

Decoherence means that the reduced density matrix for a bone, say, would come to be a mixed state where the amplitude is concentrated on position eigenstates and the interference terms were close to zero. But that's not quite the same as saying the particles have become "classical with definite positions", the mixed state would still look like a statistical ensemble of different position eigenstates, each of which would feature the particles in different sets of positions. So for example one position eigenstate might involve a given leg bone being upright because the cat is in a standing position, another might involve it being horizontal because the cat is in a sleeping position, the reduced density matrix would include both. Decoherence doesn't provide a "collapse" that selects one of the many position eigenstates, it just means that it's approximately correct to treat it as a statistical mixture of these different states rather than a superposition where you have to worry about interference effects.

I think it's misleading to say they "become classical", as I said they don't go to any single definite set of positions, and it's only approximately correct to treat them as a regular statistical ensemble if you're just looking at those 50 particles and ignoring the other 380, you would still find interference effects if you looked at all 430 at once. Again think of the delayed choice quantum eraser, where if you just look at the probability distribution for signal photons ignoring correlations with idlers, you find a non-interference pattern on the screen behind the double slit, but if you consider the conditional probability that a signal photon landed at various positions given that the idler was detected at some detector, then here there may be an interference pattern.

Copenhagen is just sort of agnostic about what's "really" going on in the box before measurement, it just says that we model the inside of the box as a big superposition in order to figure out the probabilities of various outcomes when we open it. In terms of Copenhagen, say you could prepare some vast number of boxes with the identical cats measured to be in identical initial pure states at the moment the box was sealed, and then for each one you opened the box after some fixed time and made another exhaustive measurement of all the particles, and then you looked at the statistical patterns in all these different experiments, constructing a probability distribution on different final outcomes. In that case, the probability distribution for different final outcomes involving all the particles would show interference effects (akin to how the probability distribution for signal and idler shows interference), but if you just looked at "reduced" final outcomes which only involved paying attention to a particular subsystem of particles in each final measurement, here the probability distributions for different possible outcomes would show virtually no interference (akin to how the probability distribution for signal photon alone, throwing out information about the idler, doesn't show interference).

About this whole density matrix thing. A week ago I posted a thread called "Predictivity Sieves questions" asking something about it but no one answered so hopefully you can answer it (reproduced in the following) so I can understand better this density matrix as it pertains to decoherence, many thanks.

I wrote in that thread the following:

There is something that escapes my understanding about decoherence and the so called predictivity sieves. In Max Tegmark paper:

http://arxiv.org/abs/quant-ph/0101077"The second unanswered question in the Everett picture was more subtle but equally important: what physical mechanism picks out the classical states — face up and face down for the card — as special? The problem was that from a mathematical point of view, quantum states like "face up plus face down" (let’s call this "state alpha") or "face up minus face down" ("state beta", say) are just as valid as the classical states "face up" or "face down".

So just as our fallen card in state alpha can collapse into the face up or face down states, a card that is definitely face up — which equals (alpha + beta)/2 — should be able to collapse back into the alpha or beta states, or any of an infinity of other states into which "face up" can be decomposed. Why don’t we see this happen?

Decoherence answered this question as well. The calculations showed that classical states could be defined and identified as simply those states that were most robust against decoherence. In other words, decoherence does more than just make off-diagonal matrix elements go away. If fact, if the alpha and beta states of our card were taken as the fundamental basis, the density matrix for our fallen card would be diagonal to start with, of the simple form

density matrix = [1 0]
--------------------[0 0]

since the card is definitely in state alpha. However, decoherence would almost instantaneously change the state to

density matrix = [1/2 0]
--------------------[0 1/2]

so if we could measure whether the card was in the alpha or beta-states, we would get a random outcome. In contrast, if we put the card in the state "face up", it would stay "face up" in spite of decoherence. Decoherence therefore provides what Zurek has termed a "predictability sieve", selecting out those states that display some permanence and in terms of which physics has predictive power."

---------------------------

Inquiries about the above:

1. What does it mean "if the alpha and beta states of our card were taken as the fundamental basis, the density matrix for our fallen card would be diagonal to start with"? What does becoming "diagonal" mean?

2. How come if the card is in alpha state, decoherence would amost instantaneously change the state to the 2nd density matrix?

3. And why if the card was in alpha state, one would get a random outcome? What random outcome is it talking about?

 

[JesseM]

About this whole density matrix thing. A week ago I posted a thread called "Predictivity Sieves questions" asking something about it but no one answered so hopefully you can answer it (reproduced in the following) so I can understand better this density matrix as it pertains to decoherence, many thanks.

As I said I haven't studied the technical details of decoherence and that goes for density matrices as well, what I know about them is from nontechnical explanations from physicists, for example the basic idea that a density matrix is just a statistical ensemble of different possible pure states. Probably it's possible to figure out the basics from the definitions in http://qis.ucalgary.ca/quantech/443/chapter_five.pdf though. For a 2x2 density matrix as in Tegmark's example, they would write the four entries as:

$$\left[ \begin{matrix} \rho_{11} & \rho_{12}\\ \rho_{21} & \rho_{22}\\ \end{matrix} \right]$$

And define the entries as follows:

$$\rho_{11} = \sum_i p_i \langle a_1 \mid \psi_i \rangle \langle \psi_i \mid a_1 \rangle$$
$$\rho_{12} = \sum_i p_i \langle a_1 \mid \psi_i \rangle \langle \psi_i \mid a_2 \rangle$$
$$\rho_{21} = \sum_i p_i \langle a_2 \mid \psi_i \rangle \langle \psi_i \mid a_1 \rangle$$
$$\rho_{11} = \sum_i p_i \langle a_2 \mid \psi_i \rangle \langle \psi_i \mid a_2 \rangle$$

Where here $$\mid a_1 \rangle$$ and $$\mid a_1 \rangle$$ would be the measurement basis, so they'd be the states Tegmark called "alpha" and "beta" respectively since he said that's what was being measured, in terms of bra-ket notation it would be clearer to write them as |alpha> and |beta>. Then $$\mid \psi_1 \rangle$$ and $$\mid \psi_2 \rangle$$ would be the pure states in the statistical ensemble, assigned probabilities p1 and p2, I think Tegmark's point was that decoherence would drive the card into a statistical ensemble of classical states, in this case "face up" which he defined as (|alpha> + |beta>)/2 and "face down" which I guess would be (|alpha> - |beta>)/2 (normally superpositions like this involve a factor of 1/sqrt(2), not 1/2, I wonder if he wrote it incorrectly here). So if there's an equal probability of either, meaning p1=1/2 and p2=1/2, then I think if you plug all this into the above (keeping in mind that products <alpha|beta> and <beta|alpha> are equal to zero, while <alpha|alpha> and <beta|beta> should be 1) you will find that the matrix ends up being

$$\left[ \begin{matrix} \frac{1}{2} & 0\\ 0 & \frac{1}{2}\\ \end{matrix} \right]$$

Seems to work for me if we assume the "face up" and "face down" terms have a factor of 1/sqrt(2), not 1/2...

1. What does it mean "if the alpha and beta states of our card were taken as the fundamental basis, the density matrix for our fallen card would be diagonal to start with"? What does becoming "diagonal" mean?

As seen by the equations in the tutorial it looks like you need a set of basis vector (corresponding to the possible outcomes of whatever you plan to measure, in this case alpha or beta) to define $$\mid a_i \rangle$$ in the density matrix. Diagonal just means that only the elements of the matrix along the line from upper left to lower right are nonzero, in this case only $$\rho_{11}$$ and $$\rho_{22}$$, the other terms are zero. When he said "the density matrix for our fallen card would be diagonal to start with", he was taking as the "starting" condition that the card had just been measured in the |alpha> state, so here I guess you could have a statistical ensemble where $$\mid \psi_1 \rangle$$ = |alpha> and $$\mid \psi_2 \rangle$$ = |beta>, in which case measuring it to be in |alpha> would mean p1=1 and p2=0. If you plug that into the above equations you do get

$$\left[ \begin{matrix} 1 & 0\\ 0 & 0\\ \end{matrix} \right]$$

which qualifies as diagonal.

2. How come if the card is in alpha state, decoherence would amost instantaneously change the state to the 2nd density matrix?

From the above, I think it's because decoherence drives the system into a statistical mixture of position eigenstates, in this case a 1/2 probability of face-up and a 1/2 probability of face-down.

3. And why if the card was in alpha state, one would get a random outcome? What random outcome is it talking about?

In that statement he was talking about what would happen when the card was no longer in the alpha state, but had been driven by decoherence into a statistical mixture of face-up and face-down...in that case if you measured again in the alpha/beta basis, it'd be random whether you got alpha or beta. Here were his words:

...the card is definitely in state alpha. However, decoherence would almost instantaneously change the state to

density matrix = $$\left[ \begin{matrix} \frac{1}{2} & 0\\ 0 & \frac{1}{2}\\ \end{matrix} \right]$$

so if we could measure whether the card was in the alpha or beta-states, we would get a random outcome.

 

[rodsika]

Decoherence means that the reduced density matrix for a bone, say, would come to be a mixed state where the amplitude is concentrated on position eigenstates and the interference terms were close to zero. But that's not quite the same as saying the particles have become "classical with definite positions", the mixed state would still look like a statistical ensemble of different position eigenstates, each of which would feature the particles in different sets of positions. So for example one position eigenstate might involve a given leg bone being upright because the cat is in a standing position, another might involve it being horizontal because the cat is in a sleeping position, the reduced density matrix would include both. Decoherence doesn't provide a "collapse" that selects one of the many position eigenstates, it just means that it's approximately correct to treat it as a statistical mixture of these different states rather than a superposition where you have to worry about interference effects.

You said that "Decoherence doesn't provide a "collapse" that selects one of the many position eigenstates".. but isn't it what the Preferred Basis is all about.. where it selects one of the many position eigenstates. This is why I equate decoherence with classicality. So why do we experience classical world while that 50 atoms in the 430 atom buckyball don't.

I think it's misleading to say they "become classical", as I said they don't go to any single definite set of positions, and it's only approximately correct to treat them as a regular statistical ensemble if you're just looking at those 50 particles and ignoring the other 380, you would still find interference effects if you looked at all 430 at once. Again think of the delayed choice quantum eraser, where if you just look at the probability distribution for signal photons ignoring correlations with idlers, you find a non-interference pattern on the screen behind the double slit, but if you consider the conditional probability that a signal photon landed at various positions given that the idler was detected at some detector, then here there may be an interference pattern.

Let's take the case of this universe. Our universe can be said to be totally isolated in a 100% isolation box. Decoherence has produced classical Earth with definite positions and this classicality is all over the entire universe (which becomes classical) or to put it in another form. How come the universe is classical when it supposedly should experience macroscopic superposition just like the cat as both are 100% isolated. The cat is like the universe enclosed in 100% hypothetical isolation box too. How come the cat is in superposition with no definite classical region like a classical liver organ, while this universe has a classical Earth and galaxy and so on. This was what I meant in my previous message. Again let's ignore Many worlds for now because Many worlds are all classical in the branches.

Copenhagen is just sort of agnostic about what's "really" going on in the box before measurement, it just says that we model the inside of the box as a big superposition in order to figure out the probabilities of various outcomes when we open it. In terms of Copenhagen, say you could prepare some vast number of boxes with the identical cats measured to be in identical initial pure states at the moment the box was sealed,[ and then for each one you opened the box after some fixed time and made another exhaustive measurement of all the particles, and then you looked at the statistical patterns in all these different experiments, constructing a probability distribution on different final outcomes. In that case, the probability distribution for different final outcomes involving all the particles would show interference effects (akin to how the probability distribution for signal and idler shows interference), but if you just looked at "reduced" final outcomes which only involved paying attention to a particular subsystem of particles in each final measurement, here the probability distributions for different possible outcomes would show virtually no interference (akin to how the probability distribution for signal photon alone, throwing out information about the idler, doesn't show interference).