What is a Pure State and Mixed State?

[vanesch]

Hi,

Being new to this forum, I'm not sure this is the right place to ask my question. I had no time to read all the posts of this very interesting debate, and maybe this subject was already addressed somewhere...

My question is :

B. D'Espagnat states that the reduced density matrix of a subsystem A of a composite system "A+B", obtained through the partial trace operation on B, doesn't necessarily represent a mixture, but what he calls an "improper mixture".

in the following article :

http://arxiv.org/PS_cache/quant-ph/pdf/0109/0109146.pdf

the author, K.A. Kirkpatrick shows that D'Espagnat is wrong somewhere in his reasoning, and that, therefore, this reduced matrix may indeed be considered as representing a "true mixture".

Kirkpatrick's argument is based on indistinguishability, but I don't really get it.

Could someone explain it ? My last question is, finally, who is right and who is wrong ?

Thanks for help,

Bertrand

I only skimmed through the article, but I think Kirkpatrick is somehow wrong when attacking Hughes' argument (with which I think I'm familiar) on top of p 2, because I think this argument is correct.
There's a difference between having independent statistics for the systems S and M, given by the two reduced density matrices, and the correlated statistics that will result when applied to the pure state. However, this correlation will not show, of course, if all measurements are of the form A_S x 1 or 1 x B_M (in other words, when we do not measure correlations between S and M, but do independent measurements on S alone, or on M alone), at least, in the case when the systems S and M are entangled (that means, that the pure state is not a product state). If the systems S and M are not entangled, then the individual reduced density matrices will also give rise to individual pure states. The only case in which the reduced density matrix is not pure, while the "master" state is pure, is when the system is entangled.

So the point of Hughes (which I think is correct), is that he physical state of a global system (S + M) in a pure state is not correctly described by only the mixtures given by the reduced density matrices. These only give the correct result for measurements who do not try to establish correlations between the systems S and M. Only the pure state gives the right correlations, while the local density matrices assume statistical independence and hence erroneous results for these correlations.Kirkpatrick might be right, however, that in absense of explicit projection, there's no such thing as a proper mixture if we didn't start with one. But that's known: it is the entire issue of the measurement problem: how to produce a genuine mixture, by unitary evolution, from a pure state (which doesn't work).

Simple example:

|psi> = a |+> |-> + b |->|+>

This gives, for the first system, a reduced density matrix as mixture:
a^2 |+><+| + b^2 |-><-|

and for the second system:
b^2 |+><+| + a^2 |-><-|

But an overall operator measuring the state:
a |+>|-> + b|->|+> ==> outcome 1
all other orthogonal states ==> outcome 0

would give expectation value = 1 when applied to the pure state,
and an expectation value below 1 when applied assuming statistical
independence.

 

[selfAdjoint]

From the paper

"Consider a composite system in the pure state \rho^{S \oplus M}, of which the component states are the mixed states \rho^S and \rho^M. For the sake of
argument, assume that \rho^S = a1 | u1 &gt;&lt; u1 | +<br /> a2 | u2 &gt;&lt; u2 |, while \rho^M = b1 | v1 &gt;&lt; v1 | +<br /> b2 | v2 &gt;&lt; v2 |, with a1 \neq a2 and b1 \ne b2, so there are no problems of degeneracy. Then, according
to the ignorance interpretation of \rho^S and \rho^M,
system S is really in one of the pure states | u1 > or | u2 >, and system M is really in one of the pure states | v1 > or | v2 >. . . . But this would mean that the composite system is really in one of the four states | u_j v_k &gt;, with probabilities a_jb_k respectively — in other words, that the composite system is in a mixed state. Since this contradicts our original assumption, the ignorance interpretation simply will not do."
This argument is so clearly stated by Hughes that its error stands out: the claim that “the composite system is in a mixed state” is not supportable — nothing external to S ⊕M distinguishes those states from one another. We must add the state vectors (not the projectors): |\rho^{S \oplus M}&gt; = \sum_{jk} \psi_{jk} | u_j v_k &gt; — a pure state.

So the indistinguishable states are the supposed components of the mixed states |u_jv_k&gt;, and they are indistinguishable because their difference cannot be observed.

 

[CarlB]

If you recall my posts # 56 and # 62 in this thread, I am convinced (along with greats like Einstein, compared to whom I am less than nobody, that's understood) that the experiential now and therefore the concept of an objective split between an open future and a "fixed and settled" past is an illegitimate projection into the objective world of physics of our self-experience as agents in a successively experienced world. To my mind, the domain of quantum mechanics is the spatiotemporal whole, in which it correlates measurement outcomes.

Einstein's relativity, and classical mechanics in general, has to do with the rules governing the "fixed and settled" past. Quantum waves have to do with the open future, with the relation between these being given by the probability postulate. At least that is my version of things.

I realize that Einstein was not a fan of this sort of thing, but then again, he wasn't much of a fan of quantum mechanics in general. As long as physics rejects an objective split between the past and the future, it should be no great surprise that the laws of physics do not possesses an explicit arrow of time. The act of measurement give an explicit arrow of time.

Yes, the act of measurement can be reversed (at least in thought), and the laws of physics will work just fine, but that hardly proves anything. It doesn't make sense to apply probabilistic laws to an experiment that has already been made and whose result is known, so applying the laws of physics backwards is not a very useful thing to do. It is the action of measurement that indicates the arrow of time of course this arrow can be pointed in whatever direction one wishes to assume. But in our physical world, there can be no doubt which direction the arrow points.

Carl

 

[vanesch]

From the paper

<quotation>

So the indistinguishable states are the supposed components of the mixed states |u_jv_k&gt;, and they are indistinguishable because their difference cannot be observed.

No, that's the point. It is sufficient to use a measurement on the combined system which has eigenvalue 1 for exactly this pure state, and 0 for all the others, and this is a measurement that can distinguish between the pure state and the mixture. It is only if you lock yourself up in the basis of the original measurement that you can't tell the difference between a mixture and a pure state.

This measurement is of course a correlation measurement, and not a measurement on one of the systems alone.

 

[Bertrand]

Hello Vanesh,

Thank you for this very clear explanation !

I have now another one

You say this :

"Kirkpatrick might be right, however, that in absense of explicit projection, there's no such thing as a proper mixture if we didn't start with one. But that's known: it is the entire issue of the measurement problem: how to produce a genuine mixture, by unitary evolution, from a pure state (which doesn't work)."

In this sentence, I undersatnd that for the moment it is not proved that one may obtain a projection operator from unitary evolution.

However, recently I read an article from Zurek (Decoherence and the Transition from Quantum to Classical-Revisited) who states (page 4) :

" Decoherence leads to the environment-induced superselection (einselection) that justifies the existence of the preferred pointer states. It enables one to draw an effective border between the quantum and the classical in straightforward terms, which do not appeal to the "collapse of the wavepacket" or any other such deus ex machina"

In this sentence, I understand that Zurek claims he has solved this problem of constructing a projection operator from unitary operators. Is it really what he claims ?

Maybe this is not the right place to ask this question. I guess this topic must have already been addressed somewhere else in the forum.

Thanks again for your explanations,

Bertrand

 

[vanesch]

Hello Vanesh,

Thank you for this very clear explanation !

I have now another one

You say this :

"Kirkpatrick might be right, however, that in absense of explicit projection, there's no such thing as a proper mixture if we didn't start with one. But that's known: it is the entire issue of the measurement problem: how to produce a genuine mixture, by unitary evolution, from a pure state (which doesn't work)."

In this sentence, I undersatnd that for the moment it is not proved that one may obtain a projection operator from unitary evolution.

Worse, it is simply demonstrated that this cannot be the case! It takes 5 lines or so to show that a unitary operator can never be a projector. von Neumann knew that already, hence his distinction between "process 1" and "process 2". (process 2 was the unitary evolution, and process 1 was the projection)

In the language of hilbert states, process 1 is a projection (writing out the state in a specific measurement basis (eigenbasis of the measurement operator), and statistically picking out one component). In the language of density operators, process 1 is writing out the density operator in a specific basis (the measurement basis) and putting the non-diagonal elements to 0. This changes a pure state into a mixture.

Now, a density operator, under process 2 (unitary evolution) changes as follows: rho(t) = U(t) rho(0) U+(t)
and the condition for a pure state is: rho^2 = rho

Now, from this, follows simply that, if rho is a pure state at t = 0, then rho remains a pure state under process 2:

rho(t)^2 = rho(t) rho(t) = U(t) rho(0) U+(t) U(t) rho(0) U+(t)
= U(t) rho(0)^2 U+(t) = U(t) rho(0) U+(t) = rho(t)

So, once a pure state, and unitary evolution, always a pure state.

EDIT: I would like to add, that taking the reduced density matrices by partial trace, and putting these reduced density matrices again into an overall density matrix (which makes the measurements on the two systems statistically independent) is half way between process 1 and process 2: it puts the elements in the original overall density matrix to 0, which deal with correlations between the two systems, but doesn't put all non-diagonal elements to 0. But in any case, this cannot be obtained through unitary evolution.

However, recently I read an article from Zurek (Decoherence and the Transition from Quantum to Classical-Revisited) who states (page 4) :

" Decoherence leads to the environment-induced superselection (einselection) that justifies the existence of the preferred pointer states. It enables one to draw an effective border between the quantum and the classical in straightforward terms, which do not appeal to the "collapse of the wavepacket" or any other such deus ex machina"

In this sentence, I understand that Zurek claims he has solved this problem of constructing a projection operator from unitary operators. Is it really what he claims ?

Well, no. Decoherence doesn't solve the measurement problem. However, it can solve the preferred basis problem, and as thus, suggest a set of associated projection operators. The suggested preferred basis is then simply a basis that remains robust under time evolution (that is, each robust term by itself evolves while remaining within its "superselection subspace").

In fact, in the book "decoherence and the appearance of a classical world" by Joos, Zeh et al, it is clearly stated that decoherence does not solve entirely the measurement problem but only illuminates a specific aspect of it. Zeh himself wrote that (in chapter 2) and explained carefully what decoherence does, and what it doesn't. I know some claim that it does, but this is an erroneous claim.
However, *if* you have a way to tackle the measurement problem (by accepting one or other interpretational scheme), then decoherence shines some light on otherwise strange phenomena. The most natural setting to consider decoherence is an Everett-style interpretation of course, because of the fact that the apparatus and environment are provided with quantum states.

 

[Bertrand]

Hello, Vanesh !

Your answer has the merit to be clear ! I had difficulties to find out, from the articles I read, whether or not Zurek's decoherence theory was able or not to explain completely the measurement process. Some people I discussed with told me with certitude that yes indeed it did, however, I remained very doubtfull, some points remaining very unclear for me, especially, how does the theory explain why a given "pointer state" is "really selected" among all the possible states.

So far as I understand it, Everett's theory gives the same degree of reality to all the potential results of a measurement, and therefore requires that somehow the universe splits each time a measurement is performed. If this is really the correct interpretation, well, maybe the theory is interesting for some mathématical aspects, but for the rest, it seems crazy.

Concerning measurement, an other aspect that must be taken into account by a theory supposed to explain the detail of the measurement process, without making use of the projection principle contained in the Copenhague Interpretation, is the time irreversibility.

My understanding of his theory, is that Zurek obtains this time- irreversibility through a mecanism akin to the second principle of thermodynamics, that is, the principle of the increase of the entropy.

He seems to say that the measuring process is irreversible, because the "information" flows into the degrees of freedom of the environment, and the time reversal of this process is highly unprobable.

Do I correctly get his point, and in case of a positive answer, what do you think of this argument ?

Bertrand

 

[vanesch]

Your answer has the merit to be clear ! I had difficulties to find out, from the articles I read, whether or not Zurek's decoherence theory was able or not to explain completely the measurement process. Some people I discussed with told me with certitude that yes indeed it did, however, I remained very doubtfull, some points remaining very unclear for me, especially, how does the theory explain why a given "pointer state" is "really selected" among all the possible states.

Well, if you place yourself in an Everett view, then decoherence does describe the measurement *process* of course, but you're still confronted with the fact that all branches are present in the wavefunction.
Now, in an Everett-like view, you then just say that an observer is not a physical object, but a physical state, so if an object occurs in several entangled states, then that corresponds to so many subjective worlds (with different probabilities to be experienced).

So the "resolution" of the measurement *problem* comes from the interpretation, but at least the *process* is described using decoherence. what decoherence brings in (once one has a scheme to resolve the measurement process), is two things: first of all, the absence of quantum interference between branches. This is important: even if there are parallel branches, in order for us to live our subjective life in one of them, it is important that most of the time, we do not get disturbed by "neighbouring branches". In other words, decoherence helps us understand why the "neighbouring branches" become unobservable.
The second point decoherence brings us, is: why things seem to have definite positions (in a branch). This comes from the structure of the interaction hamiltonian between a system and the environment, which is highly position-sensitive. This means that only coarse-grained states with approximately given positions to objects can be robust against time evolution.

So far as I understand it, Everett's theory gives the same degree of reality to all the potential results of a measurement, and therefore requires that somehow the universe splits each time a measurement is performed. If this is really the correct interpretation, well, maybe the theory is interesting for some mathématical aspects, but for the rest, it seems crazy.

I'm in fact an Everettian. Not that I think that this is ultimately the correct world description (I'm ignorant of that), but I think it is the correct view on the formalism of quantum mechanics as it stands now, simply because it is the only view in which there is no arbitrary distinction between "measurement" and "evolution", and because it is the only view that allows for an analysis of the measurement process itself. Moreover, the Everett view is the only ontological view that is entirely compatible with the standard QM predictions, and locality. All other views which give an ontological interpretation need some form of non-locality.

All other interpretations are in fact cutting away "a piece of quantum mechanics", and do not want to face what this theory tells us. It would be a bit like using the Lorentz transformations, but saying that time dilatation doesn't apply to human beings or so.


Now, I've exposed so many times my views on this, here, that I'm not going to reiterate them (do a search on my name and MWI or something if you're interested). However, there's one common misconception of the Everett view I want to correct: the *universe* doesn't branch, it is only the *observer* that branches.

Concerning measurement, an other aspect that must be taken into account by a theory supposed to explain the detail of the measurement process, without making use of the projection principle contained in the Copenhague Interpretation, is the time irreversibility.

My understanding of his theory, is that Zurek obtains this time- irreversibility through a mecanism akin to the second principle of thermodynamics, that is, the principle of the increase of the entropy.

He seems to say that the measuring process is irreversible, because the "information" flows into the degrees of freedom of the environment, and the time reversal of this process is highly unprobable.

Do I correctly get his point, and in case of a positive answer, what do you think of this argument ?

I think so, yes. This is the way irreversibility can be reconciled with time-reversible microdynamics: you start from a very unprobable initial state, and then it evolves most probably into more probable states. This is "irreversibility". It's only statistical, but if the probabilities are small enough to go backward (which they are), you will not observe it.

 

[Bertrand]

"Now, I've exposed so many times my views on this, here, that I'm not going to reiterate them (do a search on my name and MWI or something if you're interested)"

OK Vanesh, I'll do this when I come back in 3 weeks.

Thank you again for all your explanations,

Bertrand