Are the 2nd and 3rd axioms of QM incompatible?

[Zafa Pi]

Nielsen & Chuang list three axioms for QM. I paraphrase them as follows:
1. States are unit vectors.
2. The evolution of a state is unitary and given by the Schrodinger equation.
3. The measurement of a state yields a value from a probability distribution. The state just before the measurement may be different than just after and also random.

So 2. says the evolution is deterministic and 3. says it's stochastic.
I've seen many treads that dance around this issue, but no clear resolution.
Seeking enlightenment.

 

[Bystander]

"State" is "deterministic." Disturbed by a measurement process that determinism is reset.

 

[bhobba]

You will likely understand the situation better when you see a more advanced treatment such as found in Ballentine - Chapter 3. Axiom 2 is somewhat redundant in that its derivable by symmetry considerations. By definition classical mechanics is the region where the Galilean transformations hold to good approximation. That all by itself determines axiom 2. So its sort of tautological (not really - but sort of) - the dynamics is determined by the definition of the realm we are considering. If you have the Lorentz transformations you get Quantum Field Theory.

That's part of the incredible beauty of QM once you understand it at a deeper level.

If that excites you (and it should) before delving into it first study Landau's beautiful text:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages.' 'Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics'

Ballentine, Landau, and Noethers theorem had a deep effect on me completely changing my view of physics. A staff mentor has posted when he teaches Noether's there there is stunned silence as its implications are grasped. Deep, wonderful, and powerful stuff. Well worth your attention.

Thanks
Bill

 

[Strilanc]

Pragmatically speaking: you can write a computer program that implements all three postulates, therefore they have a model, therefore they are compatible.

You're probably seeing a contradiction because you are strengthening the meaning of the second postulate to "everything that can happen to the state is unitary". The third postulate is defining another kind of thing that can happen to quantum states, besides unitary evolution. The second postulate doesn't apply to that other kind of thing that can happen.

It's also worth knowing that measurement can be implemented in terms of unitary operations. That's what the many worlds interpretation does. The trick is to replace measurement with entanglement against the environment. Take each measurement operation and replace it with a controlled-not operation that toggles some qubit(s) in a large well-mixed pool of neg-entropy. That's measurement in many-worlds.

 

[vanhees71]

Of course postulate 1. is inaccurate. The correct version is:

1. The state is represented by a self-adjoint positive semidefinite trace-class operator with $\mathrm{Tr} \hat{\rho}=1$, the Statistical Operator of the system.

Definition: The system is in a pure state, if its quantum state is described by a projection operator, i.e., if the Statistical operator fulfills the constraint $\hat{\rho}=\hat{\rho}^2$. Then it follows that there is a normalized vector $|\psi \rangle$ in Hilbert space so that $\hat{\rho}=|\psi \rangle \langle \psi |$. For pure states, it's equivalent to say that it is determined completely by a ray in Hilbert space, i.e., a normalized vector modulo a phase.

It is of prime importance to understand this subtlety, because otherwise quite basic facts of the quantum world are ununderstandable, e.g., the existence of half-integer angular momenta (or spins).

 

[Demystifier]

Nielsen & Chuang list three axioms for QM. I paraphrase them as follows:
1. States are unit vectors.
2. The evolution of a state is unitary and given by the Schrodinger equation.
3. The measurement of a state yields a value from a probability distribution. The state just before the measurement may be different than just after and also random.

So 2. says the evolution is deterministic and 3. says it's stochastic.
I've seen many treads that dance around this issue, but no clear resolution.
Seeking enlightenment.

In Nielsen and Chuang, axiom 2. refers to closed systems, while axiom 3. refers to open systems. In this way they avoid logical contradiction. It does not mean that this completely resolves the problem at the deep fundamental conceptual level, but at least they have a consistent set of rules which can be used in all cases (known so far) of practical interest. Nielsen and Chuang is a practical book; it is not a book on quantum foundations. For the latter, I would recommend
F Laloe, Do We Really Understand Quantum Mechanics?
https://www.amazon.com/dp/110702501X/?tag=pfamazon01-20

 

[Zafa Pi]

In Nielsen and Chuang, axiom 2. refers to closed systems, while axiom 3. refers to open systems. In this way they avoid logical contradiction. It does not mean that this completely resolves the problem at the deep fundamental conceptual level, but at least they have a consistent set of rules which can be used in all cases (known so far) of practical interest. Nielsen and Chuang is a practical book; it is not a book on quantum foundations. For the latter, I would recommend
F Laloe, Do We Really Understand Quantum Mechanics?
https://www.amazon.com/dp/110702501X/?tag=pfamazon01-20

Say I want to measure a polarized photon in state |x> with a polarization analyzer (PA) set at 0 degrees (i.e. with measurement operator Z) then |x> is not a closed system as soon as the PA is introduced. A closed system might consist of the photon, the PA, and the kitchen sink (the state of which is not |x>) which does evolve by Schrodinger's equation though the state and Hamiltonian would be horrendous.
Am I making sense?

I am a retired mathematician having never taken a physics course (I was intending to be a physics major but on day 3 the prof bit my head off for a question I asked. He apologized three years later saying my question was good). The mysteries of QM seemed intriguing, yet the lay literature was obviously filled with flapdoodle. I found Nielsen and Chuang fun to read because it required no physics. Between you and bhobba (post 3) its been suggested that I read 2000 pages of text. That is not going to happen. Though I'm happy to say that so far some of you Physics Forums gurus have come out of the clouds and have been helpful. Thanks.

 

[Demystifier]

Say I want to measure a polarized photon in state |x> with a polarization analyzer (PA) set at 0 degrees (i.e. with measurement operator Z) then |x> is not a closed system as soon as the PA is introduced. A closed system might consist of the photon, the PA, and the kitchen sink (the state of which is not |x>) which does evolve by Schrodinger's equation though the state and Hamiltonian would be horrendous.
Am I making sense?

Yes you do. The point is that you can't measure such a horrendous quantum state. You can only measure the state of a small subsystem, which is open.

Between you and bhobba (post 3) its been suggested that I read 2000 pages of text. That is not going to happen.

Fair enough. As an alternative, you may want to read only 31 pages of
http://arxiv.org/abs/quant-ph/0505070
The presentation is simple, pedagogic, short and mathematically precise - perhaps just what you want.

 

[drvrm]

Say I want to measure a polarized photon in state |x> with a polarization analyzer (PA) set at 0 degrees (i.e. with measurement operator Z) then |x> is not a closed system as soon as the PA is introduced. A closed system might consist of the photon, the PA, and the kitchen sink (the state of which is not |x>) which does evolve by Schrodinger's equation though the state and Hamiltonian would be horrendous.
Am I making sense?

.
All physical observables (defined by the prescription of experiment or measurement ) are represented by a linear
operator that operates in linear inner product space (an Hilbert space in case of finite dimensional spaces). States of
the system are represented by the direction/ray (not a vector) in the linear inner product space (again Hilbert space
in the finite dimensional case).

A physical experiment can be divided into two steps: preparation and measurement. The first step determines the
possible outcomes of the experiment, while the measurement retrieves the value of the outcome. In QM the situation
is slightly different: the first step determines the probabilities of the various possible outcomes, while the measurement
retrieve the value of a particular outcome, in a statistic manner. This separation of the experiment is reflected into
the two types of mathematical objects we find in QM. The first step corresponds to the concept of a state of the
system, while the second to observables.
The state gives a complete description of the set of probabilities for all observables, while these last ones are all
dynamical variables that in principle can be measured. All the information is contained in the state, irrespectively on
how I got the state, of its previous history. For the moment we will identify the state with the vectors of an Hilbert
space |ψ)
.

All physical observables (defined by the prescription of experiment or measurement ) are represented by a linear
operator that operates in linear inner product space (an Hilbert space in case of finite dimensional spaces).
States ofthe system are represented by the direction/ray (not a vector) in the linear inner product space
(again Hilbert spacein the finite dimensional case).
The value of the measurement of an observable is one of the observable eigenvalues. The probability of obtaining one
particular eigenvalue is given by the modulus square of the inner product of the state vector of the system with the
corresponding eigenvector. The state of the system immediately after the measurement is the normalized projection
of the state prior to the measurement onto the eigenvector subspace.
In conclusion, our picture of QM is a mathematical framework in which the system is completely described by
its state, which undergoes a deterministic evolution (and invertible evolution). The measurement process, which
connects the mathematical theory to the observed experiments, is probabilistic
Strong Measurements(statement)
Although the result of a single measurement is probabilistic, we are usually interested in the average outcome, which
gives us more information about the system and observable. The average or
expectation value of an observable for a system in state |ψ) is given by ( A) =(ψ|AIψ)
see<http://ocw.mit.edu/courses/nuclear-engineering/22-51-quantum-theory-of-radiation-interactions-fall-2012/lecture-notes/MIT22_51F12_Ch3.pdf> for details

 

[Collin237]

A closed system might consist of the photon, the PA, and the kitchen sink (the state of which is not |x>) which does evolve by Schrodinger's equation though the state and Hamiltonian would be horrendous.
Am I making sense?

What is the kitchen sink?

Also, are you saying that in the QM definition of open and closed, an open system can be a subset of a closed system?

 

[A. Neumaier]

I am a retired mathematician having never taken a physics course (I was intending to be a physics major but on day 3 the prof bit my head off for a question I asked. He apologized three years later saying my question was good). The mysteries of QM seemed intriguing, yet the lay literature was obviously filled with flapdoodle. I found Nielsen and Chuang fun to read because it required no physics. Between you and bhobba (post 3) its been suggested that I read 2000 pages of text.

I wrote an online book giving an introduction to quantum physics for mathematics students. Only nice stuff with elegant math, no physics is assumed (but it is explained in math terms). For you the first half (at least up to Chapter 10) is probably a pleasure to read.

 

[Zafa Pi]

What is the kitchen sink?

Also, are you saying that in the QM definition of open and closed, an open system can be a subset of a closed system?

The kitchen sink is the set theoretic complement of "everything but the kitchen sink". You can google the latter.

An open system can be a subset of a closed system and vise versa. Nothing special about QM.
A cannon ball shot up on the moon has a trajectory that obeys the classical/Newtonian equation of motion and is a closed system. If an obstruction is introduced (to perhaps measure the time of impact) then this open system interferes with the previous evolution/equation. If on the other hand we take into consideration the coefficient of restitution of the cannon ball, along with the properties of the obstruction (heat dissipation etc.) as well as the moon (and the kitchen sink), we could (in principle) apply the classical laws of this larger system so that it is now a closed system.

 

[meopemuk]

2. The evolution of a state is unitary and given by the Schrodinger equation.
3. The measurement of a state yields a value from a probability distribution.

The two statements are not contradictory. Let me give you an example from the classical world, which is a good model of quantum behavior.

Take a roulette table with a wheel and a ball. Each time you spin the wheel you'll get a random outcome: a number from 1 through 36 or a "zero". This is a good model of how physical measurements are performed on a quantum system. The result of a measurement is unpredictable. You can only describe it by a probability distribution (or a wave function, if you like). In the case of the roulette, the measured observable is discrete, its spectrum has 37 points and the probability distribution is uniform 1/37 for each point. Each time you spin the wheel and the ball drops into a certain slot, the probability distribution "collapses" in a random fashion. Quite similarly, when you do a measurement on a quantum system (electron, atom, etc.), its probability distribution (= wave function) collapses randomly each time. This is your Postulate 3.

You can affect the roulette, for example, by slightly tilting the table. By doing that, you will change the probability distribution: now it is not simply 1/37 for each number, but a more complex function, depending on the tilt. You can model the "time evolution" of the roulette's "state" by slowly moving and tilting the table. If you keep spinning the wheel (doing measurements) on the moving/tilting table, you'll still have your random collapses, but the probability distribution will experience continuous time evolution. Perhaps you could write a kind of Schroedinger equation that would describe this time evolution of the probabilities on a moving/tilting roulette table. This is your Postulate 2.

You see, the random collapse and continuous time evolution can coexist quite happily. Quantum systems work in a similar manner. Results of individual experiments are unpredictable. They can be described only by probability distributions. But these probability distributions (= wave functions) change with time in a continuous (unitary) way. There is absolutely no contradiction.

The only difference between the classical roulette table and the quantum electron is that in the former case we can "in principle" calculate the outcome of each spin (though it is very difficult); in the latter case such a calculation/prediction is impossible even "in principle". Results of measurements performed on the electron are genuinely random.

Eugene.

 

[bhobba]

The two statements are not contradictory.

Its obvious they are not. It can change by process 1 OR process 2.

It's very simple logic.

Thanks
Bill

 

[Zafa Pi]

The two statements are not contradictory. Let me give you an example from the classical world, which is a good model of quantum behavior.

Take a roulette table with a wheel and a ball. Each time you spin the wheel you'll get a random outcome: a number from 1 through 36 or a "zero". This is a good model of how physical measurements are performed on a quantum system. The result of a measurement is unpredictable. You can only describe it by a probability distribution (or a wave function, if you like). In the case of the roulette, the measured observable is discrete, its spectrum has 37 points and the probability distribution is uniform 1/37 for each point. Each time you spin the wheel and the ball drops into a certain slot, the probability distribution "collapses" in a random fashion. Quite similarly, when you do a measurement on a quantum system (electron, atom, etc.), its probability distribution (= wave function) collapses randomly each time. This is your Postulate 3.

You can affect the roulette, for example, by slightly tilting the table. By doing that, you will change the probability distribution: now it is not simply 1/37 for each number, but a more complex function, depending on the tilt. You can model the "time evolution" of the roulette's "state" by slowly moving and tilting the table. If you keep spinning the wheel (doing measurements) on the moving/tilting table, you'll still have your random collapses, but the probability distribution will experience continuous time evolution. Perhaps you could write a kind of Schroedinger equation that would describe this time evolution of the probabilities on a moving/tilting roulette table. This is your Postulate 2.

You see, the random collapse and continuous time evolution can coexist quite happily. Quantum systems work in a similar manner. Results of individual experiments are unpredictable. They can be described only by probability distributions. But these probability distributions (= wave functions) change with time in a continuous (unitary) way. There is absolutely no contradiction.

The only difference between the classical roulette table and the quantum electron is that in the former case we can "in principle" calculate the outcome of each spin (though it is very difficult); in the latter case such a calculation/prediction is impossible even "in principle". Results of measurements performed on the electron are genuinely random.

Eugene.

Though I left it out, Postulate 3 includes the statement "if you make a second measurement right after the first one you will get the same value". How do you account for that in your interesting analogy? Do you suddenly tilt the table vertically? If so then there is no continuous and deterministic evolution of tilting, i.e. we are back to collapse. Perhaps the measurement of the position of a particle undergoing continuous Brownian motion provides a more natural analogy, though we still get collapse when the flattening gaussian reverts to a delta function.
I think one still needs what Demystifier said in post #6.

 

[Zafa Pi]

Its obvious they are not. It can change by process 1 OR process 2.

It's very simple logic.

Thanks
Bill

It?, process 1?, OR?, process 2? Sir, you are so cryptic. Talk down to me, I can take it.

 

[bhobba]

It?, process 1?, OR?, process 2? Sir, you are so cryptic. Talk down to me, I can take it.

Process 1 is unitary evolution. Process 2 is collapse from observation.

It can change by either - simple logic - there is no contradiction.

It's like you have money in a bank account. It grows continuously via interest. But it can change discontinuously if you add or take money from the account.

To be honest I have zero idea why anyone wants to argue it - it's utterly obvious.

Note - collapse is an interpretive thing so only applies to interpretations that have collapse. If you want to see a correct axiomatic development see Ballentine. It really only requires two axioms - actually it can be reduced to one but that is another story for another thread.

Thanks
Bill

 

[meopemuk]

Though I left it out, Postulate 3 includes the statement "if you make a second measurement right after the first one you will get the same value". How do you account for that in your interesting analogy?

You are right that my model describes only single acts of measurements. I have no idea how to describe repetitive measurements performed on the same copy of a quantum system. I am not sure that such a description is really necessary in a successful theory. In 99% of experiments only single measurements are done. The used copy of the physical system is discarded after the measurement and the next measurement is done with a fresh copy from the ensemble.

It is true that in some cases we do repetitive measurements on the same copy. For example, when we see a track left by a particle in photo emulsion. But I suspect that in order to describe such a thing, one must have a full quantum description of both the particle and the measuring apparatus (the photographic plate).

Consider for example a 2-slit experiment in which the interference picture is recorded by two parallel photographic plates made of thin glass lightly covered by photo emulsion. In this case, photons (or electrons) will make their marks on the first plate, pass through it and make marks on the second plate as well. Thus we have the case of a repetitive position measurement, where one can expect that both plates will record very similar interference pictures.

But if we would use a denser layer of photo emulsion on the 1st plate, then the image on the 2nd plate will be distorted. So, the interference picture on the second plate will not be describable by simple square of the photon's (electron's) wave function. It will depend on the density and composition of the first layer. Thus, in the case of repetitive measurements, the description of the simple 2-slit experiment would transform into intractable exercise of a particle interacting with complex matter.

Eugene.

 

[Zafa Pi]

Process 1 is unitary evolution. Process 2 is collapse from observation.

It can change by either - simple logic - there is no contradiction.

It's like you have money in a bank account. It grows continuously via interest. But it can change discontinuously if you add or take money from the account.

To be honest I have zero idea why anyone wants to argue it - it's utterly obvious.

Note - collapse is an interpretive thing so only applies to interpretations that have collapse. If you want to see a correct axiomatic development see Ballentine. It really only requires two axioms - actually it can be reduced to one but that is another story for another thread.

Thanks
Bill

OK, thanks I get it. But what bank do you use?

 

[bhobba]

OK, thanks I get it. But what bank do you use?

ING Direct in Australia - but of course its irrelevant.

Thanks
Bill

 

[Zafa Pi]

ING Direct in Australia - but of course its irrelevant.

Thanks
Bill

Not if you're getting interest > h.