Does the Observer have anything to do with conscious choice

[micromass]

It's OK. One should first define "convex combination of states" starting with the axioms and only with them. Otherwise why would we need axioms?

OK, but don't you think that the concept of "convex combination" is well known to everybody reading this paper. I mean, convex combinations don't even have anything to do with QM, they are rather a mathematical concept.

 

[bhobba]

This is what boobba wrote:

"It means a state that can't be expressed as the convex sum of other states."

"Its a definition - no proof required."

But I do not see the term "convex" in the definition.

Its dead simple. Look the outcomes with dead certainty are unit vectors. These are the pure states of probability theory. A vector with positive entries that adds up to one is a probability vector. Only the vectors with unit entries can't be expressed as a convex sum of other vectors. I suppose no proof required might be a little off the mark - but not by much - it's rather trivial.

Both those papers are a bit like that - they are somewhat of a slog to go through in that if you haven't come across these terms before it may take a bit of getting used to.

Thanks
Bill

 

[bhobba]

OK, but don't you think that the concept of "convex combination" is well known to everybody reading this paper. I mean, convex combinations don't even have anything to do with QM, they are rather a mathematical concept.

My bad. Like I say my background is in math so I guess stuff that's common in math I take more or less as given.

Both those papers are of the mathematical physics variety so that sort of terminology tends to be rampant.

Thanks
Bill

 

[arkajad]

OK, but don't you think that the concept of "convex combination" is well known to everybody reading this paper.

Convex combination of "what"? Of two real numbers? I know. Of two trees? I do not know, because I do not know how to add trees and multiply them by numbers to get other trees. The same with "states"? State has been introduced in the axioms. But no operations on state have been discussed. Neither the meaning of these operations.

 

[bhobba]

Convex combination of "what"? Of two real numbers? I know. Of two trees?

Scratching head - of probability vectors.

Thanks
Bil

 

[micromass]

Convex combination of "what"? Of two real numbers? I know. Of two trees? I do not know, because I do not know how to add trees and multiply them by numbers to get other trees. The same with "states"? State has been introduced in the axioms. But no operations on state have been discussed. Neither the meaning of these operations.

Well, since a state is defined as a certain operator, I think it's not difficult to infer that we take the sum and scalar multiplication of operators as defined in mathematics. If somebody doesn't know how to add two operators, then I don't think he should be reading this paper.

I mean, it's clear this paper takes for granted the required math such as linear algebra/functional analysis. So we should read the paper with that in mind. Otherwise, the paper would be far too long, as they would also need to explain what hermitian is and what the trace is.

 

[arkajad]

Well, since a state is defined as a certain operator.

Definition of "State" from Hardy's paper:

"The state associated with a particular preparation is defined to be (that thing represented by) any
mathematical object that can be used to determine the probability associated with the outcomes of any measurement that may be performed on a system prepared by the given preparation."

So it is not defined as an operator.

And then we have

"Axiom 5 Continuity. There exists a continuous reversible transformation on a system between any
two pure states of that system.

The first four axioms are consistent with classical probability theory but the fifth is not (unless the
word “continuous” is dropped). "

Nowhere in the axioms we have operators.

 

[bhobba]

Nowhere in the axioms we have operators.

Its a definitional thing. Given a measurement setup with outcomes yi like they have in the paper one defines the operator associated with the setup as Ʃ yi |bi><bi| for some |bi> not yet determined (they are the states that are the possible outcomes). This is the first axiom of the treatment you find in Ballentine. The second axiom in Ballentine, Born's rule, is what the paper is concerned about - it really is the key axiom. It's a very interesting thing that QM basically depends on just these two axioms. And also a very interesting thing that you can start with the first axiom as a definition (of course you need to justify why you would want such a cockeyed definition in the first place - but that's another story) and derive the second via Gleason's theorem with the assumption of non-contextuality. When people ask me what's the simplest way of viewing QM so it doesn't look like its pulled out of a hat its what I tell them. At its foundations, when viewed the right way, QM is quite natural - but only when viewed the right way.

Although I haven't seen it, and need to check them out when I get a bit of time, I am sure a lot of this stuff is in the video lectures on a course in Quantum Foundations the Perimeter Institute have archived:
http://www.pirsa.org/C12002

Because its something that really interests me I sometimes forget it not as well known as perhaps it should be. This is one of the reasons I am such a big fan of Ballentine - he develops QM this way from the start and examines closely exactly what it means.

Thanks
Bill

 

[meBigGuy]

For some reason you don't seem to get an observation is something that leaves a mark here in the macro world.

I totally get what you are saying, but it is a relative thing (relational). My "here in the macro world" is different than a particle's "here in the macro world". When two particles interact, what happens? Did one particle leave its mark on the other, or not? Yet to us they are still represented by wave functions.

Actually, I have a problem with the phrase "here in the macro world" as if there is some other world. Where is "there in the non-macro world"? Seems there is one world that things exist in, and our knowledge about them is limited, so we express what we know as a probability: the probability of what we might measure when we really pin it down. Relational interpretations are saying that different entities can see different things about an objects state or potential measurable state. One might see it as a superposition, and the other might not. ("for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may be in a superposition of two or more states."

And this isn't limited to the micro world:
"However, it is held by relational quantum mechanics that this applies to all physical objects, whether or not they are conscious or macroscopic."

I just want to understand two interacting particles that I have not observed but know things about as a pair. How does each particle view the other particle. How is that view different than how I view the pair. They obviously have to be consistant, but not neccessarily the same.

Sometimes you seem to take a pure instrumentalist view, that these things are beyond QM, but then other times you acknowledge there is something more.

 

[bhobba]

When two particles interact, what happens? Did one particle leave its mark on the other, or not?

A mark is something visible here in the macro world. Obviously particles interacting do not do that. When they interact you consider their tensor product and via their interaction Hamiltonian they become entangled - that's the very essence of entanglement.

Basically for particles to leave a mark they need to interact with a macro object. It becomes entangled with the macro object and leaves a mark. QM is a theory about such marks. What its doing or any other visualizations like particles leaving marks on other particles are not part of the theory. Sure it can become entangled with it but that is entirely different to being measured, leaving a mark, observed or the other words bandied about.

Can I suggest you study a book like Ballentine so you know the detail of this sort of stuff?

Or at least have a look at Susskinds lectures on it:
http://theoreticalminimum.com/courses/quantum-entanglement/2006/fall

Thanks
Bill

 

[bhobba]

Actually, I have a problem with the phrase "here in the macro world" as if there is some other world. Where is "there in the non-macro world"? Seems there is one world that things exist in, and our knowledge about them is limited, so we express what we know as a probability: the probability of what we might measure when we really pin it down.

It's by definition the world of everyday experience. There is obviously another level - namely the world of atoms and their constituent particles. We know of their existence by the outcomes of measurements we make that leave traces in the everyday world. QM is a theory about those traces - that's it, that's all.

We have the issue of why the world of everyday experience behaves the way it does since its composed of that quantum stuff, and that is a deep issue - but a lot of progress has been made.

I am an applied mathematician - these kind of word games like where is there a non macro world sound suspiciously like philosophy and if that's what interests you then it might be better looking elsewhere for answers. Guys like me tend to take literal commonsense views of words without getting caught up in these types of semantics.

Sometimes you seem to take a pure instrumentalist view, that these things are beyond QM, but then other times you acknowledge there is something more.

I think you misunderstand because you are confused with a state; thinking its something more than a codification of the outcomes of observations. People talk of a quantum state in a literal sort of way and because of that you think its the reality - but its not implying its anything more than what I have said it is. When two particles interact the description is about its quantum state (specifically the tensor product of the states of the particles) which codifies the outcomes of observations - but its not being observed so its not telling us anything more.

How does each particle view the other particle. How is that view different than how I view the pair.

I have carefully explained what an observation is, its clearly and obviously not applicable to two particles interacting yet you continue to use words like one particle viewing another. View, observe measure etc etc are in this context not applicable to states. Why do you do that?

Imagine speaking to an actuary that talks pretty freely about mortality rates, contingencies etc etc. Then ask them - how does a contingency view a mortality rate - you are likely to get strange looks.

I am not into philosophy but am reminded of Wittgenstein: 'Whereof one cannot speak, thereof one must be silent'

Thanks
Bill

 

[meBigGuy]

Which is a pure instrumentalist view with no room for any interpretations.

I understand what QM is as much as I can without understanding QM. Interpretations exist above (or below depending on your semantics) hard instrumentalist QM.

You seem to take issue with relational interpretions,
"for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may be in a superposition of two or more states." and "However, it is held by relational quantum mechanics that this applies to all physical objects, whether or not they are conscious or macroscopic." which are both from the wikipedia paragraph.

I didn't realize that when a proton and an electron interacted as a hydrogen atom that they became entangled. Or two hydrogen atoms interacting become entangled.

 

[arkajad]

Its a definitional thing. Given a measurement setup with outcomes yi like they have in the paper one defines the operator associated with the setup as Ʃ yi |bi><bi|/QUOTE]

I am not talking about Ballentine's axioms. I am talking about axioms 1-5 or 1-3 in the quoted papers. The operation Ʃ yi |bi><bi| is undefined in these axiom systems. So what is the point of using "axiomatic approach" if you are not following the axiomatic way?

Of course I know what is a pure state in the usual standard approach. Do you want to say: we are assuming, from the very beginning the standard formalism of QM where "states" are in one to one correspondence with density matrices and pure states by one-dimensional projection operators? Or: states are represented by normalized positive functionals on a von Neumann (or, perhaps, C*) algebra? But if so, why it is not written in the preamble to these papers? I really would like to know the point of taking this axiomatic approach.

 

[bhobba]

I am talking about axioms 1-5 or 1-3 in the quoted papers. The operation Ʃ yi |bi><bi| is undefined in these axiom systems.

No its not. They prove the existence of a Quantum state as the element of a complex Hilbert space from which the definition is well defined.

Also see section 8.7 of Hardy's paper where operators are introduced.

'Hence, we obtain pmeas = trace(Aˆρˆ) (90) which is shown to be equivalent to pmeas = r · p in section 5. We now need to prove that the restrictions from quantum theory on Aˆ and ρˆ follow from the axioms. Both ρˆ and Aˆ must be Hermitean since r is real.'

Thanks
Bill

 

[arkajad]

No its not.

The term should be defined before it is introduced in the axioms. Not after. Otherwise axioms do not make any sense. Unless it is a primitive term. But that option does not make sense in this context either.

And it would be not too difficult to do it. Except that the question whether we can distinguish operationally between mixed and pure states leads to a rather nontrivial discussion. Compare the related conclusion in Belinfante's "Can One Detect the State of an Individual System?", Foundations of Physics, Vol. 22, No. 3, 1992.

Jauch and Piron understood the problems. But Hardy apparently is not paying sufficient attention to the logical consistency of his "axiomatic approach".

 

[bhobba]

The term should be defined before it is introduced in the axioms.

It is.

I don't know why but for some reason you are having a lot of trouble understanding what is in plain sight. In section 8.7, A, the operator associated with the observation, is defined before the standard trace formula is deduced.

This paper has been around for quite a while and Hardy is a very well respected physicist working on the foundations of QM. Its highly unlikely he would make fundamental errors of that nature.

Thanks
Bill

 

[arkajad]

Its highly unlikely he would make fundamental errors of that nature.

I am pointing out the error. Whether someone is highly respected or not is not relevant. I am interested in the consistency of the axiomatic system, and not whether some person is respected or not. Highly respected (and for the right reasons) people were making errors. It is not unusual.

Of course one may ask whether the error is an important one or is it just inessential detail. I would not be making such an issue if I would think it is just "an easily correctable omission". In my view fixing this error would require an essential rewrite of the whole paper, including the conclusions.

 

[bhobba]

I am pointing out the error.

The point is there is no error. The operator A associated with the observation is defined BEFORE use in the key trace formula.

My comment was not an appeal to authority, I was merely pointing out it has been gone over countless times - fundamental errors like that would be picked up early. It's more likely you missed something, which I think you have.

Thanks
Bill

 

[arkajad]

The point is there is no error.

The point is that there is an error. Because the term "pure state" is being used in Axiom 5, before it has been defined. It does not matter that it is defined "somewhere else". Axiomatic systems are being used for definite purposes and have their rules. Do we want to abandon logic in mathematical physics? I would not vote for this. It would be a shame.

fundamental errors like that would be picked up early.

I know of fundamental errors that have not been detected quickly.

 

[bhobba]

The point is that there is an error. Because the term "pure state" is being used in Axiom 5,

What a pure state is defined in the framework of generalized probabilistic theories of which bog standard probability theory and QM are just examples eg:
http://www.cgogolin.de/downloads/gpt.beamer.pdf
'Definition: States that can not be written as a convex combination of states are called pure or extremal.'

Its just you haven't come across it before. I agree it's not the stuff that's generally taught to mathematicians and physicists but people into the area of quantum foundations know of it. Axiom 5 differentiates the pure states of normal probability theory from those of Quantum Theory. Both are theories under the heading of generalized probabilistic theories for which the concept of pure state is applicable. But the class of theories for which the concept applies is much wider eg QM like theories based on real vector spaces and quaternion spaces.

If you don't get this then its really only going to continue going around in circles.

Thanks
Bill

 

[bhobba]

Which is a pure instrumentalist view with no room for any interpretations.

I have stated right from the outset this is the usual standard Copenhagen or Ensemble interpretation. If you want to go beyond that - fine - but please say that's what you are doing.

Why you think I have an issue with the relational view is beyond me. I have clearly stated I have always more or less assumed its like that anyway. My departure from it is it's not really of much value because once decoherence has occurred, which happens very very quickly, all observers agree.

Thanks
Bill

 

[bhobba]

I didn't realize that when a proton and an electron interacted as a hydrogen atom that they became entangled. Or two hydrogen atoms interacting become entangled.

It depends on the type of interaction - in that case they are - but not always.

Its not usually thought of that way though.

Thanks
Bill

 

[arkajad]

What a pure state is defined in the framework of generalized probabilistic theories of which bog standard probability theory and QM are just examples eg:
http://www.cgogolin.de/downloads/gpt.beamer.pdf
'Definition: States that can not be written as a convex combination of states are called pure or extremal.'

Let me recall Axiom 1 from Hardy's paper:

The state associated with a particular preparation is defined to be (that thing represented by) any
mathematical object that can be used to determine the probability associated with the outcomes of any measurement that may be performed on a system prepared by the given preparation.

Now, how do you take "convex combination" of "any mathematical object"?

It does not make sense. Evidently the author was not thinking deep enough at this point. So, how would you correct this axiom so that "convex combination" would make sense? Please, replace this axiom by some other so that what you are talking about would make sense.

 

[bhobba]

Now, how do you take "convex combination" of "any mathematical object"?
It does not make sense.

One more time then that's it for me - you obviously don't get it. And you will probably say I don't so there is really no point in continuing and going around in circles.

Of course you can't do it for any mathematical object. Generalized probabilistic theories are defined as theories you can. Such theories exist eg both standard probability theory and QM allow it but they are not the only ones - eg real vector space QM. That's all you need to know for the definition to not be vacuos.

Its used all the time in math - eg the abstract definition of a vector space you can add vectors - how do you add any mathematical object. The obvious answer is you can't - vector spaces are defined as ones you can. Generalized Probabilistic theories are defined as ones where convex sum makes sense. This implies the operation of addition and multiplication by a positive number makes sense.

Thanks
Bill

 

[arkajad]

Of course you can't do it for any mathematical object.

So, we have some progress. As it stands the axiom does not make sense. We agree on that.
So, how would you change so that it does make sense? Would you replace "any mathematical object" by, say "an element of a convex set in some vector space"?

 

[bhobba]

So, we have some progress. As it stands the axiom does not make sense.

It makes perfect sense. Axiom 5 applies to theories where convex sum makes sense like vector space applies to objects where sum makes sense.

Generalized probabilistic theories are defined as theories that contain objects called states such that convex sum makes sense ie if ui are states then Ʃ pi ui is also a state where Ʃ pi = 1 and pi positive. A state is pure if you can't find other states such it can be expressed as a convex sum of them.

We are just going around in circles and getting nowhere.

Thanks
Bill

 

[arkajad]

We are just going around in circles and getting nowhere.

We are going somewhere, though with unnecessary friction. Because that is (convex structure) what I was asking at the very beginning.

So, we are associating Particular preparations" with elements of some convex set (in a real vector space). We define pure states as extremal points of this set. That is ok mathematically. But we are discussing physics. "Preparations" is physics term. So my question is: are we able to distinguish between pure and mixed states experimentally? How? Does it make sense? Which "preparations" correspond to pure states and which to "nontrivially mixed states"? Can we do that analyzing preparation procedures? Or only through measurements? How many measurements are necessary? Can we do it in a finite set of measurements? Infinite? How? And when we talk about "state", do we mean "state of an individual system" or "state of an ensemble of identically prepared systems"?

 

[bhobba]

That is ok mathematically. But we are discussing physics.

We are discussing mathematical models. Probabilistic models can be applied to physics, finance, economics and probably tons of other stuff I can't think of off the top of my head. Generalized probability models have certain well defined mathematical properties such as states that have a defined convex sum.

I was not trained in physics, but in applied math and mathematical modelling uses this sort of thing all the time.

The answer to your questions is contained in the model. The assumption of a mathematical model is it makes sense. Are you denying that probability models do not exist that makes sense? Do you deny that bog standard probability theory, which is a probabilistic model makes sense? Many of those questions you are asking are the general crap you hear about when discussing say the frequency interpretation of probability. To be blunt I have been through that one before and, again to be blunt, its philosophical waffle of highly dubious value. What did my statistical modelling professor say about it - it's like discussing Nietzsche - rather pointless really. That got a big laugh in the class - but you know what - he was right.

But like I say - I am an applied mathematician, you have been trained in philosophy, we are worlds apart in approach I suspect.

Thanks
Bill

 

[arkajad]

I was not trained in physics, but in applied math and mathematical modelling uses this sort of thing all the time.

Thanks a lot. And I apologize.

 

[meBigGuy]

Why you think I have an issue with the relational view is beyond me. I have clearly stated I have always more or less assumed its like that anyway. My departure from it is it's not really of much value because once decoherence has occurred, which happens very very quickly, all observers agree.

I don't see how that fits Wikipedia's description of relational QM Interpretation. I quoted two sentences, twice now, that had no inking of rapid decoherence. I'm saying that unobserved (by me) unentangled particles can interact. That one party sees a superposition, another sees a collapsed state. You say that difference is fleeting. I think Relational Interpretations say otherwise. Am I totally wrong?

From the Wikipedia page on RQM
"RQM argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). " <snip>
"Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above."

This seems totally different from other interpretations.

 

[meBigGuy]

From Wikipedia:
RQM is, in essence, quite similar to the Copenhagen interpretation, but with an important difference. In the Copenhagen interpretation, the macroscopic world is assumed to be intrinsically classical in nature, and wave function collapse occurs when a quantum system interacts with macroscopic apparatus. In RQM, any interaction, be it micro or macroscopic, causes the linearity of Schrödinger evolution to break down. RQM could recover a Copenhagen-like view of the world by assigning a privileged status (not dissimilar to a preferred frame in relativity) to the classical world. However, by doing this one would lose sight of the key features that RQM brings to our view of the quantum world.

 

[bhobba]

I don't see how that fits Wikipedia's description of relational QM Interpretation. I quoted two sentences, twice now, that had no inking of rapid decoherence

That's the exact problem with it - it hasn't taken on the lessons of decoherence.

Once you do that you see its really to required - not wrong - simply not required.

Thanks
Bill

 

[meBigGuy]

Now, that's finally a real point. I'll watch Susskind and get back to you.