# Preferred Basis and Superposition

## Main Question or Discussion Point

Hi Guys, I have a question about observable's and superposition's that I haven't been able to find a definitive answer to (purely for the fact that it doesn't seem to be addressed), and would greatly appreciate an answer.

When in a superposition state of an observable, you are also in a definite state of another observable. For example, suppose you have two incompatible observable's from the 1/2 spin states: X-spins and Z-spins. If you are in a definite X-up position, then you are in a superposition of Z-spin down and Z spin up. So mathematically we could express this as:

$$(1.0)\mid S_x=+ \frac{1}{2}> = a\mid S_z=+\frac{1}{2}> + b\mid S_x=-\frac{1}{2}>$$
Where: $$\mid a \mid ^2 +\mid b \mid ^2 =1$$

In the MWH (Many Worlds Hypothesis), we have a problem known as the 'basis problem'. Essentially the crux of the problem is due to the fact that when we perceive the 'collapse of the wavefunction' and thus the splitting of the superposition component states into different branches (or worlds), it all is all dependent on the observable we choose to use, so is the mathematical representation of the splitting a real ontological split or just part of the mathematical eigenbasis representation we chose? That is the 'basis problem' in the MWH.

Now that makes sense to me, but it made me think about superpositions and choosing an observable to measure in general and what it actually means (regardless if we adhere to the MWH or not).

So my question going back to equation (1.0) is: if the system is in a definite state of X spin-up, does that mean X spin-up is a superposition of the Z-spin? So when we have a definite state of X-spin up, are we are also observing a superposition of the Z-spins (a superposition of Z spins is the definite state of X spin up) or is it just Z-spins are in a superposition, when we have a definite X spin-up?

I know the distinction can be hard to grasp so I'm trying to make an effort to spell it out. What I'm trying to say is, because the mathematical Hilbert space representation says X spin-up equals the superposition of z-spin down and z-spin up, does that mean they are identical things? Being in an X-spin up state is a superposition of z-spin down and z-spin up? Or is it just the case that when a particle is in a definite state of X spin-up, the other observable Z-spins are in a superposition, but they are not identical things?

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Simon Bridge
Homework Helper
If you are in a definite X-up position, then you are in a superposition of Z-spin down and Z spin up.
Correction: If you are in a definite spin z state, you are also in a superposition of x and y states.

so is the mathematical representation of the splitting a real ontological split or just part of the mathematical eigenbasis representation we chose?
That is the issue - but it makes no difference in fact.

Note: The process of doing the math is not the same thing as the process of doing a measurement.

The basis, irl, depends on the experimental setup - the rest is just maths.
i.e. the reason we care about a particular spin component is that there is some apparatus imposing some directionality wrt the spin.

More generally - just because we can expand out a state vector in a basis does not mean anything about the physical situation ... just like we may have 4lbs of mass... we also know that 2+2=4, but that does not mean that it makes sense in every situation to say that out 4lbs of mass is literally the same as having two separate 2lb masses. It just tells us what we may be able to have should the situation warrant (the mass passes through a cutting machine).

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Correction: If you are in a definite spin z state, you are also in a superposition of x and y states.
Yes my mista'ke, I should've stated I was assuming the partial representation of spin states in R^2 for simplicity of my illustration (since y spin states are represented in C^2).

My question however was not about the basis problem as it pertains to the MWH, but rather my query further in the post.

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Simon Bridge
Homework Helper
Note: a specific z state for spin is not "the same thing" as the superposition of x and y states.
But as you note - that just means you pulled a poor example out of your hat.

It is possible to represent a state vector as a superposition of eigenstates of some observable.
There are many such superpositions possible.

It's not real - it's just maths. The physical setup of the experiment determines the favored basis.
i.e. in the Stern-Gerlach experiment, the apparatus decides on which direction counts as z. Without the magnets, all spin components are uncertain.

Note: I edited the previous post while you were replying.

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More generally - just because we can expand out a state vector in a basis does not mean anything about the physical situation ... just like we may have 4lbs of mass... we also know that 2+2=4, but that does not mean that it makes sense in every situation to say that out 4lbs of mass is literally the same as having two separate 2lb masses. It just tells us what we may be able to have should the situation warrant (the mass passes through a cutting machine).
The reason we 2+2=4 is not 'literally' the same as having two separate 2lb masses is because the equality sign in that respect is not in relation to entities but rather 'mass'. It is the case that 2lb+2lb is identical to 4lb with 'respect to the mass'. The equality sign in that equation (=) represents identity with respect to mass not anything else.

If I had apples which were all 2lb's each and banana's which were also 2lb's each per unit. Then i can easily construct the mathematical equation 2apples (lb) = 2 banana's (lb), since they are 'identical' in respects to mass. But it would be absurd to say 2 apples = 2 banana's without specification because those two things are not identical with respect to colour, appearance, taste, geometrical shape etc.

So in regards to superposition, I am asking, what is the equality sign referring with respect to? Is it saying a definite X Spin-up state is the superposition of Z states (and Y states) or that whenever we have a definite X Spin-up state, we will also have superposition's of Z and Y states?

If we take the former to be true, then it is possible for us to say that being in a definite X state is what it means to be in a superposition of Z and Y states. To imagine a superposition of Y and Z states is to see an definite X-state, since they are identical.

It's not real - it's just maths.
Well that is the question. Alot of physicists and philosophers would say a superposition is represented in physical reality rather than just being a mathematical construct. That is after all how the many worlds hypothesis came to be. Its adherents believes that the wavefunction is a true physcial construct which obeys the dynamical evolution as postulated by the Schrodinger equation and superposition is a real feature of reality, its just the constituents states of superposition's occupy different worlds which cannot be directly measured within 'our' world, hence why they had the 'basis problem' in the first place. If the splitting of worlds is predicated on the wave function which was to be regarded as real - which 'basis' should it take? since there is an infinite amount of ways of representing observable's. A good book talking about the ontology of the wavefunction is The Wave Function: Essays on the Metaphysics of Quantum Mechanics edited by Alyssa Ney and David Albert. But that's correct, I was trying to ask what is the true physical meaning of a superposition. If you take the statistical view of QM and see it rather as an instrumental tool than actually representing physical reality, then of course my questions would seem quite irrelevant.

But thanks anyway in showing me why it seems I don't see this question directly addressed within textbooks, considering i assume many physicists would follow your interpretation of the mathematics, and wouldn't be interested in the interpretational issues of QM but the more 'practical' aspects of the theory.

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Simon Bridge
Homework Helper
So in regards to superposition, I am asking, what is the equality sign referring with respect to?
Maths.

But thanks anyway in showing me why it seems I don't see this question directly addressed within textbooks, considering i assume many physicists would follow your interpretation of the mathematics, and wouldn't be interested in the interpretational issues of QM but the more 'practical' aspects of the theory.
That topic is banned here. You want a quantum philosophy forum.

I was asking what is the equality sign (=) referring to? The equality sign is a mathematical symbol so I dont know what you mean by its referring to maths, I'm asking what is it referring to as being equal between the LHS and RHS of the equation. In the (2 + 2)lb = 4lb example, the equality sign (this '=' sign) is referring to equality of masses. The mass of the LHS equals (is in equality, is identical, is the same as - they all mean the same thing they all mean this '=' symbol) the mass of the RHS of the equation.

All I'm asking is what is the equality sign (=) is referring to being equal to in the superposition equation in (1.0)? What's the physical interpretation of the mathematics. That is not a philosophical question, we interpret the mathematics to be different things all the time. Calculus for example can be used to represent economic, physics or social science phenomena, they're all interpretations of the mathematics into physical reality - the maths abstractly doesn't mean anything.

So once again, I'm just asking, what is the LHS of equation (1.0) equal to, in the RHS of the equation? As I showed earlier the 2+2=4 example you showed, the equality sign just tells us the masses of the LHS and the RHS are equal, not anything else. Simply I am asking what is equal with respect to the LHS and RHS of the wavefunction equation in (1.0).

Simon Bridge
Homework Helper
When we say that |z> = a|x> + b|y> we mean exactly that state vector |z> is mathematically the same thing as a|x> + b|y> ... if you were to work out the full mathematical structure behind the symbols you'd end up with the same numbers for both sides. It's just maths.

A wavefunction can be expanded in a wide range of different bases.
This has no meaning by itself. It's just different ways of writing it down using different symbols. It's just maths.

The wavefunction does not mean anything until you operate on it - then the outcome of the operation may be assigned a meaning.

 I have a feeling I'm talking at a tangent to what you are asking though.

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bhobba
Mentor
If the splitting of worlds is predicated on the wave function which was to be regarded as real - which 'basis' should it take?
This is the so called preferred basis problem.

You need to consult tomes on decoherence for exactly how that happens - eg a bit of a technical argument shows that for radial type interactions (which are the majority) a position basis is usually singled out.

If you want to pursue it further the book to get is Schlosshauer's:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20&tag=pfamazon01-20

Regarding what superposition means its simply an artifact of the math of QM. Technically the reason has to do with having continuous transformations between so called pure states. If you want to pursue this check out:
http://www.scottaaronson.com/democritus/lec9.html

At an even deeper level it has to do with entanglement:
http://arxiv.org/abs/0911.0695

Thanks
Bill

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Yeah, I was just spelling out what the preferred basis problem was in the context of the Many Worlds Hypothesis, I knew what it was hence the title of the thread. Thank you very much though for the further readings, this will help me alot with my honours thesis that i'm getting prepared for. The first book looks like a great introduction to decoherence.

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meBigGuy
Gold Member
Note that when you say 4lb = 4lb, you are saying the number of pounds are equal. You have assigned units, which makes it "more understandable" in a way. When you say 2 = 2 with no units its difficult to say what a standalone 2, with no units, represents other than the value or state 2. So how does one talk about the units or value of the state vector |x> + |y>, or assign it meaning when it stands alone? Or even |z>? It's a notation that represents numbers or values describing the possible states of a system. The state of the system is the sum of the states of 2 sub-systems?

This is the so called preferred basis problem.

You need to consult tomes on decoherence for exactly how that happens - eg a bit of a technical argument shows that for radial type interactions (which are the majority) a position basis is usually singled out.

If you want to pursue it further the book to get is Schlosshauer's:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20&tag=pfamazon01-20

Regarding what superposition means its simply an artifact of the math of QM. Technically the reason has to do with having continuous transformations between so called pure states. If you want to pursue this check out:
http://www.scottaaronson.com/democritus/lec9.html

At an even deeper level it has to do with entanglement:
http://arxiv.org/abs/0911.0695

Thanks
Bill
Did I understand you to say that talking about quantum interpretations is banned on this forum? Because invoking 'decoherence' as a solution to the preferred basis problem is an interpretational approach that is not immune to problems. E.g. there is circular argumentation involved there. The original poster is asking very good questions about what determines the basis of measurement. These are best answered by taking absorption into account, i.e., in the transactional picture.

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bhobba
Mentor
Yeah, I was just spelling out what the preferred basis problem was in the context of the Many Worlds Hypothesis, I knew what it was hence the title of the thread. Thank you very much though for the further readings, this will help me alot with my honours thesis that i'm getting prepared for. The first book looks like a great introduction to decoherence.
It's THE book on decoherence.

I always recommend it to people after they have a good foundation in QM and an interest in interpretations - it explains the issues clearly - as well as the problems. And even aside from its value wrt interpretations its a very important and interesting phenomena.

I don't know your background in QM but I think its very important when doing anything related to this area to understand its exact axiomatic basis.

Its really based on two axioms and in fact the superposition principle is really a consequence of one of those axioms called the Born Rule. Schrodinger's equation etc actually follow from symmetry, and are not really separate axioms - which is also a VERY important foundational point - even aside the fact its a pretty amazing result in and of itself.

You will find this approach in what I (and others) think is THE book on QM - Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20&tag=pfamazon01-20

The other interesting thing is the Born Rule is not entirely independent of the first axiom (it's that the possible outcomes of an observation are the eigenvalues of its observable). There is this marvelous theorem called Gleason's Theorem that shows it follows from that if you assume basis independence:
http://kof.physto.se/theses/helena-master.pdf

It seems almost like magic that QM can be derived from just one axiom. Of course it can't - its just that in this approach each further assumption is simply very reasonable eg in Gleason's Theorem it's very reasonable to assume your measure is basis independent - but it's an assumption - and careful analysis shows its quite a strong one at that leading to whats called non-contextuality.

Anyway I do strongly suggest you at least scan through the first 3 chapters of Ballentine. Once you do that you will see, for example, the superposition principle is a pretty trivial requirement following from the Born rule for so called pure states. But I wont go into it further - its better if you nut it out.

Post here if you have any problems.

Thanks
Bill

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Note that when you say 4lb = 4lb, you are saying the number of pounds are equal. You have assigned units, which makes it "more understandable" in a way. When you say 2 = 2 with no units its difficult to say what a standalone 2, with no units, represents other than the value or state 2. So how does one talk about the units or value of the state vector |x> + |y>, or assign it meaning when it stands alone? Or even |z>? It's a notation that represents numbers or values describing the possible states of a system. The state of the system is the sum of the states of 2 sub-systems?
That's exactly what I was trying to get at before. What property is the LHS and RHS equal with respect to in equation 1.0? Like I said earlier, you can say 2 Banana's (4lb) = 2 Apple's (4lb) because the equality sign is with respect to the mass of the LHS and RHS as being equal even though 2 Bananas do not equal 2 Apples with respect to other properties. I could say 1 Banana is 2lb and 10 oranges are 2lbs thus 1 bannana (lb) = 10 oranges (lb) with respect to mass, however without that interpretation 1 = 10 does not make any sense.

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bhobba
Mentor
Did I understand you to say that talking about quantum interpretations is banned on this forum? Because invoking 'decoherence' as a solution to the preferred basis problem is an interpretational approach that is not immune to problems. E.g. there is circular argumentation involved there. The original poster is asking very good questions about what determines the basis of measurement. These are best answered by taking absorption into account, i.e., in the transactional picture.
It's not banned - just kept on a tight reign to ensure it doesn't slip on over to philosophy which is off-topic for this forum.

And decoherence does have issues in explaining all aspects of the preferred basis problem (in particular the problem of why we get any outcomes at all - which is a doodle in your interpretation, MW, Bohm etc - but by itself decoherence can't do it) - but the standard text I linked to explains carefully what it does and does not do.

Thanks
Bill

Thanks -- although from what I've seen, Schlosshauer does not really acknowledge the extent of the circularity facing the decoherence account of the preferred basis.

bhobba
Mentor
What property is the LHS and RHS equal with respect to in equation 1.0? Like I said earlier, you can say 2 Banana's (4lb) = 2 Apple's (4lb) because the equality sign is with respect to the mass of the LHS and RHS as being equal. I could say 1 Banana is 2lb and 10 oranges are 2lbs thus 1 bannana (lb) = 10 oranges (lb) with respect to mass, however without that interpretation 1 = 10 does not make any sense.
OK. I was hoping you would nut it out for yourself from Ballentine, but I can see its something that's perplexing you a bit so I will make a stab at it now.

A key foundational axiom of QM is the so called Born rule.

It says given an observable O then there exists a positive operator of unit trace P such that the expected value of the observation associated with O is Trace (PO). By definition P is called the state of the system.

By definition states of the form |x><x| are called pure. States that are the convex sum of pure states are called mixed ie of the form ∑ pi |xi><xi| pi positive, ∑ pi = 1. It can be shown all states are either mixed or pure.

The usual states you read about in beginning and intermediate texts on QM are pure states - but in fact its a lot wider than that including mixed states. This is explained carefully in Ballentine. The other thing is they are not elements of a vector space - they are operators. But for pure states one can associate the |x> in |x><x| with elements of the underlying vector space - but not uniquely - multiplying by a phase factor gives exactly the same operator. Its this association that leads to the superposition property, principle, or whatever you want to call it - it's simply that it forms a vector space - which is hardly surprising since you have mapped it to a vector space. What such an equality 'means' is determined by the Born rule - you can derive all sorts of connections - but I will leave you to play around with that.

Thanks
Bill

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bhobba
Mentor
Thanks -- although from what I've seen, Schlosshauer does not really acknowledge the extent of the circularity facing the decoherence account of the preferred basis.
He is pretty careful in looking at its problems, and explaining you need further interpretative assumptions to make use of it.

I don't think the issue is one of circularity but, for things like why we get any outcomes at all, decoherence is silent about.

Although it must be said that tracing over the environment can only be justified by first assuming the Born Rule, so later you cant use the improper mixed state that comes out of that to justify the Born rule - that is circular. One must accept tracing over the environment as a separate process and interpret that to avoid circularity. Care is certainly required to prevent circularity.

Thanks
Bill

bhobba
Mentor
Interesting related paper on the basis problem in MWI: http://arxiv.org/abs/1210.8447
This is the so called factoring problem.

I originally thought Schlosshauer didn't cover it in his text - but my memory was faulty - he does.

Thanks
Bill

The circularity I'm concerned about is the same sort of problem facing Boltzmann's "H-theorem" allegedly proving that entropy always increases. Boltzmann assumed 'molecular disorder' to get the result, which amounted to assuming what he was trying to prove. Similarly, 'derivations' of decoherence assume that you can define already-decoherent environmental subsystems - i.e. they divide everything into observed system + environment where the environmental subsystems have random phases wrt each other and the system. Thus they have to presuppose a crucial degree of already-existing decoherence in order to 'demonstrate' the decoherence of the system's relevant degrees of freedom.

bhobba
Mentor
Thus they have to presuppose a crucial degree of already-existing decoherence in order to 'demonstrate' the decoherence of the system's relevant degrees of freedom.

Indeed key theorems are missing showing regardless of how you factor it exactly the same basis results. I did come across a paper a while ago that, for a simple model, showed just that - but it needs to be greatly extended.

Thanks
Bill

This is the so called factoring problem.

I originally thought Schlosshauer didn't cover it in his text - but my memory was faulty - he does.

Thanks
Bill
Could you please quote? Because this is new and interesting to me.
Recently I saw Max Tegmark had a knack at it, aswell as Lev Vaidman recognizing it, so if it's already "solved" I would imagine these guys would've been in the know.

bhobba
Mentor
Could you please quote? Because this is new and interesting to me.
Recently I saw Max Tegmark had a knack at it, aswell as Lev Vaidman recognizing it, so if it's already "solved" I would imagine these guys would've been in the know.
Sorry mate - I didn't take note of where it was and quoting from texts is a bit painstaking.

I will have to leave it to you to get the book and check it out for yourself.

But its pretty simple really. Its simply that current derivations use a particular factoring into system, environment, and observational apparatus. A particular basis is then singled out. Theorems are necessary to prove it happens regardless of how its done.

Thanks
Bill

Thanks Bill --

But there would still be circularity even if such a theorem were found.

Showing that presupposing an already-decohered division can yield 'decoherence,' regardless of which division is presupposed, doesn't remove the circularity problem. What would have to be shown is that decoherence occurs when there is no presupposed decoherence to begin with, since that's the actual circumstance that's alleged in the program of using decoherence to solve the measurement problem --i.e., 'Quantum Darwinism", that the macroscopic world of appearance emerges naturally from a quantum foundation. In my view, this is why the program is overrated and does not really work as claimed. You need absorption to define the measurement process, and without that, you end up going in circles.