Does a measurement setup determine the reality of spin measurement outcomes?

[A. Neumaier]

I've to read the paper again, but that the moon is there when nobody looks is for sure not necessarily leading to Bohmian mechanics. The usual conservation laws are sufficient.

Are the usual conservation laws valid in each single case or only statistically in the mean? From your minimal foundations you can conclude only the latter, hence cannot claim anything for the single instance of the moon that we have.

 

[vanhees71]

A spin component is an operator. In classical and semiclassical (large quantum number limit) physics, the value of a spin component is a real number. It is a matter of interpretation what a spin component means in terms of measurement; the quantum formalism doesn't say anything about it, except through Born's rule which is about measurement.

The thermal interpetation negates Born's rule as a fundamental principle and degrades it to an approximate law with limitations like most law of physics. Thus Born's rule cannot be invoked directly in the thermal interpretation. Instead, the thermal interpretation preserves the classical and semiclassical continuous nature of the spin, which is much more intuitive. The discrete response is therefore due to the experimental setup.But it is neither a measurement in the sense of Born's rule, nor is it discrete.

A spin component is an observable, formally described by an operator. Spin has no classical counterpart. It's generically quantum. Classical mechanics/field theory is an approximation of QT not the other way around. It's not the response that's discrete but the observable. It's true that through the SGA the measurement of the discrete spin observable is through entanglement with a continuous position observable, but with the properly set up SGA the determination of this continuous observable is sufficiently accurate to resolve the discrete spin values.

 

[Demystifier]

but that the moon is there when nobody looks is for sure not necessarily leading to Bohmian mechanics. The usual conservation laws are sufficient.

No, as I explained to you several times, the conservation laws are not sufficient. The conservation laws cannot prohibit the transformation of Moon into a completely different object (perhaps a giant pink elephant) carrying the same total energy, momentum and charge. With conservation laws you can be sure that something is there when you don't look, but you cannot be sure that it is the Moon and not something else. The Moon has some fine structure that makes it different from the giant pink elephant of the same mass. Bohmian mechanics has something to do with this fine structure. The claim that the Moon is there when you don't look means that its fine structure is there when you don't look, and this fine structure is positions of small parts of which the Moon is made. The simplest extrapolation of the notion of "positions of small parts" to smaller and smaller distances naturally leads to Bohmian mechanics. Different extrapolations are imaginable too, but Bohmian mechanics is the simplest.

 

[A. Neumaier]

A spin component is an observable, formally described by an operator. Spin has no classical counterpart. It's generically quantum.

Though you seem to be ignorant about it, spin has a classical counterpart. It is not specifically quantum.

Classical spin is given by the coadjoint representation of SO(3) on the Bloch sphere.
Its (geometric) quantization naturally produces the spin s representations for half-integral s.

See, e.g., Section 2 of the paper

• Müller, L., Stolze, J., Leschke, H., & Nagel, P., https://www.researchgate.net/profile/Hajo_Leschke/publication/13383184_Classical_and_quantum_phase-space_behavior_of_a_spin-boson_system/links/00b49536cef105ea27000000.pdf, Physical Review A, 44 (1991), 1022.

 

[vanhees71]

Sect. 2 is a nice mathematical game, but what's the physics of their $\vec{s}$?

 

[Demystifier]

Though you seem to be ignorant about it, spin has a classical counterpart. It is not specifically quantum.

Classical spin is given by the coadjoint representation of SO(3) on the Bloch sphere.

It's interesting to note that it is rarely mentioned in high-energy QFT textbooks, but often discussed in condensed-matter QFT textbooks.

 

[vanhees71]

This is no surprise since in non-relativistic physics you have rigid bodies that can intrinsically rotate, and you may call this a classical analogue of "spin" though it's of course not more than that. If it were the same as quantum-mechanical spin, any gyro-factor different from 1 would be a mystery!

 

[Demystifier]

and you may call this a classical analogue of "spin" though it's of course not more than that.

It is more than that. When you quantize it by path integrals, you get exactly nonrelativistic QM of spin.

 

[A. Neumaier]

Sect. 2 is a nice mathematical game, but what's the physics of their $\vec{s}$?

Its the classical spin vector. Its geometric quantization by the standard recipes produces standard quantum spin, of any spin 1/2, 1, ..., corresponding to the irreducible representations of SU(3).

 

[PeterDonis]

corresponding to the irreducible representations of SU(3)

Do you mean SU(2)?

 

[A. Neumaier]

Do you mean SU(2)?

I meant the irreducible projective representations of SO(3), which are the same as the irreducible linear representations of SU(2). The relevant group is the rotation group as a subgroup of the Poincare group ISO(1,3).

 

[DarMM]

Irredicible projective reps of $SO(3)$?

 

[A. Neumaier]

Irredicible projective reps of $SO(3)$?

Yes, typo corrected.

 

[A. Neumaier]

Sure. Peres is only much more eloquent in expressing these ideas. There's always verbatim what I also always state: There are no Hilbert spaces and operators in the lab but real-world equipment like accelerators ("preparation devices") and detectors ("meausrement apparati") of various kinds, all of which function according to the generally valid physical laws, including QT (as far as we know today).

Of course, if you want to describe inaccurate measurements (i.e., 99.9% of measurements done in daily live) you must use the POVM formalism, but I don't see, where this would contradict any part of the standard minimal interpretation.

Well, read his paper from which I took the quote. He says explicitly,

Traditional concepts such as “measuring Hermitian operators,” that were borrowed or adapted from classical physics, are not appropriate in the quantum world.

 

[vanhees71]

Interesting, I've to read the paper in more detail. In his book, he refers to the standard formulation of the theory to begin with. He also gives a detailed account on the more modern theory of measurement in terms of POVMs too, and I don't see, where this contradicts the foundations in any way. I rather understood the POVM formalism as derived from the foundational standard.

In which sense contradicts the POVM formalism the standard formalism? Is this a proper revision of QT and if so, what are differences in the physical predictions? If it's an alternative theory to QT, is now QT obsolete and has to be substituted by the POVM formalism and if so, how can it be formulated in a self-consistent way, i.e., without reference to the standard formalism?

I also never claimed that one "meaures Hermitian operators". As I said above, there are no Hermitian operators in the lab (which statement also Peres makes almost verbatim).

 

[DarMM]

In which sense contradicts the POVM formalism the standard formalism?

Just extends it. Peres's point is that measurements in general cannot be considered as measurements of quantities known from the classical theory.

He gives the example of a POVM measurement on the spin degree of freedom of a particle that can't be understood as measuring $\textbf{J}\cdot \textbf{n}$ for any direction $\textbf{n}$.

 

[A. Neumaier]

In his book, he refers to the standard formulation of the theory to begin with.

No. In his book, he is careful to talk only about quantum tests (for which Born's rule is on safe grounds) rather than measuring observables (for which Born's rule is questionable). On p.14 he says,

We can now define the scope of quantum theory: In a strict sense, quantum theory is a set of rules allowing the computation of probabilities for the outcomes of tests which follow specified preparations.

And he acknowledges (on p.11) that the statistical interpretation ...

is not entirely satisfactory. The use of a specific language for describing a class of physical phenomena is a tacit acknowledgment that the theory underlying that language is valid, to a good approximation. This raises thorny issues

... which he discusses in Chapter 12 without resolving them.

In the formal Chapters 2 and 3 he uses over 50 pages to define his general setup in a careful way.
On p.63 he reemphasizes (on the formal level)

“quantum measurement” is nothing more than its original definition: It is a quantum test whose outcomes are labelled by real numbers.

and mentions that many kinds of measurements have nothing to do with (the caricature notion called) measurement in Born's rule.

In which sense contradicts the POVM formalism the standard formalism?

The standard formalism is an idealization. To reduce POVM to this idealization one needs to introduce a bigger, unphysical Hilbert space encoding a suitable ancilla (p.282+288), and pretend (via Neumark's theorem, p.285f) that the idealization holds for an artificially constructed dynamics in this extended Hilbert space.

In the preface of his book, Peres compares (p.xiii) his approach with that of von Neumann, which can be taken to represent the standard view:

[This book differs from von Neumann’s classic treatise in many respects. [...]
This approach is not only conceptually different, but it also is more general than von Neumann’s. The measuring process is not represented by a complete set of orthogonal projection operators, but by a non-orthogonal positive operator valued measure (POVM). This improved technique allows to extract more information from a physical system than von Neumann’s restricted measurements.
/QUOTE]
On p.292 he says that the Shannon entropy of an unpolarized state (given by a multiple of the identity) is zero; hence he doesn't equate (as you seem to do) $S=-Tr~\rho\log\rho$ with the Shannon entropy.

I also never claimed that one "measures Hermitian operators".

This is a shorthand for measuring observables described by means of Hermitian operators - his arguments go against the latter! Indeed, in his book, he sometimes calls (e.g. in formula (3.53)) Hermitian operators observables.

 

[DarMM]

On p.292 he says that the Shannon entropy of an unpolarized state (given by a multiple of the identity) is zero; hence he doesn't equate (as you seem to do) $S = -Tr\rho\log\rho$ with the Shannon entropy.

I think he does, if you read the earlier section of that chapter. He defines the Von Neumann entropy as the minimal Shannon entropy taken over all contexts.

 

[A. Neumaier]

I think he does, if you read the earlier section of that chapter. He defines the Von Neumann entropy as the minimal Shannon entropy taken over all contexts.

I referred to what Peres wrote on p.292:

Within this logical framework, a statement that the initial state is a random mixture, ρ ~ 1, leaves no room for a priori probabilities. It is a complete specification of the preparation: the Shannon entropy is zero

 

[DarMM]

Yes, but immediately proceeding that on p.290 he says such a state has non-zero Shannon entropy.

 

[A. Neumaier]

Yes, but immediately proceeding that on p.290 he says such a state has non-zero Shannon entropy.

I have the 2002 edition, and don't find it on that page; so please quote some text.

On p.281 he says, '' it is necessary to distinguish'' between Shannon entropy H (of the subjective knowledge that the state $\rho_i$ has probability $p_i$) and Von Neumann entropy S (of the mean state $\rho=\sum p_i\rho_i$, see (3.78)), which is consistent with what he says on p.292. What I find on p.290 is the statement that if it is known that the state is an eigenstate of $\sigma_z$ then the Shannon entropy is log 2. This does not contradict what he says on p.292, since the situation is a different one where the state is 1/2 identity and the Shannon entropy is 0. The mean state and hence the von Neumann entropy is the same in both cases.

 

[A. Neumaier]

He defines the Von Neumann entropy as the minimal Shannon entropy taken over all contexts.

Where does he do this? it would contradict p.292.

 

[DarMM]

Where does he do this? it would contradict p.292.

p. 263 "Entropy of a preparation"

Note I should say I don't think you're wrong, I'm just not sure of how his statements all fit together.

This and his discussion of superobservers and reversibility always confused me a bit. The latter because the ability of superobservers to reverse measurements isn't the main problem superobservers pose to a Copenhagen like view such as Peres's.

 

[PeterDonis]

Moderator's note: Thread level changed to "I" based on how the discussion has developed.

 

[A. Neumaier]

p. 263 "Entropy of a preparation"

This is not his Shannon entropy H used later, and he indeed uses a different name for it. The $p_i$ don't mean the same on p.263 (here they are computed probabilities for outcomes of tests for pure states) and on p.281-292 (here they are assumed subjective probabilities for the state being $\rho_i$) .

Note I should say I don't think you're wrong, I'm just not sure of how his statements all fit together.

The statements fit perfectly. Note that Peres is a true quantum Bayesian, who thinks that the subjective knowledge consists of an assumed probability distribution on the densities. His Shannon entropy refers to the lack of knowledge of the true density. For a Bayesian , the latter behaves like a beable (defined as what the subjective knowledge is about). We had started a discussion about this in another thread, but then you disappeared without having reached consensus.

his discussion of superobservers

I am not interested in discussing superobservers. They are fiction. In reality, different observers can in principle observe each other, and there is no hierarchy.

 

[DarMM]

This is not his Shannon entropy H used later, and he indeed uses a different name for it. The $p_i$ don't mean the same on p.263 (here they are computed probabilities for outcomes of tests for pure states) and on p.281-292 (here they are assumed subjective probabilities for the state being $\rho_i$) .

I see. You see in Quantum Information today the Shannon entropy usually refers to the Shannon entropy of the induced classical statistical model in a given context. I'm not sure what the standard name is for what Peres is calling the Shannon entropy here.

The statements fit perfectly. Note that Peres is a true quantum Bayesian, who thinks that the subjective knowledge consists of an assumed probability distribution on the densities. His Shannon entropy refers to the lack of knowledge of the true density. For a Bayesian , the latter behaves like a beable (defined as what the subjective knowledge is about). We had started a discussion about this in another thread, but then you disappeared without having reached consensus.

So he reserves "Shannon entropy" for the distribution over densities.

Regarding the other thread, I realized I wasn't informed enough and have since being reading over this point but have not reached a conclusion myself.

I am not interested in discussing superobservers. They are fiction. In reality, different observers can in principle observe each other, and there is no hierarchy.

"No hierarchy" meaning it is not possible for one observer to observe another to the subatomic scale, because if they can you do get an apparent paradox that Copenhagen interpretations have to explain.

 

[A. Neumaier]

it is not possible for one observer to observe another to the subatomic scale

It is not possible in Nature, hence fiction, hence no physics.

 

[DarMM]

It is not possible in Nature, hence fiction, hence no physics.

I agree with this. It's interesting because it means in Copenhagen views there is no quantum state in a certain sense for a table considered as a collection of atoms.

 

[A. Neumaier]

I agree with this. It's interesting because it means in Copenhagen views there is no quantum state in a certain sense for a table considered as a collection of atoms.

In the Copenhagen interpretation (dated end of1927), the table is a classical object and has no quantum state.

For von Neumann (not quite Copenhagen), the table is a quantum object, but all true states are pure. The grand canonical ensemble for the table is a proper mixture representing uncertainty about the true pure state.

 

[DarMM]

In the Copenhagen interpretation, the table is a classical object and has no quantum state.

In Bohr's treatment whether something was "classical" was an issue of framing the cut, he didn't divide the world into two different classes of systems. Niels Bohr and Complementarity: An Introduction by Arkady Plotnitsky as well as the following by Schlosshauer say more on this:
https://arxiv.org/abs/1009.4072
Most allow a table to have a quantum state, e.g. Omnes does initially before showing that it has no operational meaning.

For von Neumann (not quite Copenhagen), the table is a quantum object, but all states are pure

This would cause superobserver difficulties.

 

[A. Neumaier]

In Bohr's treatment whether something was "classical" was an issue of framing the cut, he didn't divide the world into two different classes of systems. Niels Bohr and Complementarity: An Introduction by Arkady Plotnitsky as well as the following by Schlosshauer say more on this:
https://arxiv.org/abs/1009.4072

I added the qualification (dated end of 1927), as otherwise the notion of ''Copenhagen interpretation'' is too interpretation dependent. I think it was von neumann in his book who first claimed that the cut is arbitrary. Bohr 1927 only claimed,

for every particular case it is a question of convenience at what point the concept of observation involving the quantum postulate with its inherent ‘irrationality’ is brought in

and convenience dictates that the cut is between microscopic and macroscopic, possibly already between microscopic and microscopic.

Most allow a table to have a quantum state, e.g. Omnes does initially before showing that it has no operational meaning.

But Omnes and ''Most" are much later than Copenhagen.

 

[DarMM]

I added the qualification (dated end of 1927)...Bohr 1927 only claimed

Bohr's view changes a good bit between this and his final view of the 1940s. With this qualification I agree. Later Bohr did permit a table to have a quantum state.

But Omnes and ''Most" are much later than Copenhagen.

Typically they are counted as Copenhagen. Sometimes given the qualification "Neo" to distinguish them.

Having a fundamental ontic quantum-classical cut avoids Wigner friend problems. Not having one, i.e. the cut being a matter of application and framing of the experiment and thus allowing a table to have a full quantum description, does run into a seeming paradox if superobserver's exist. Later Copenhagenists like Peres and Omnes for this reason tend to argue that they don't. Convincingly I think. However I think Peres's treatment is not as good as Omnes's for focusing too much on the side issue of revesability which isn't the true problem.

 

[A. Neumaier]

Bohr's view changes a good bit between this and his final view of the 1940s. With this qualification I agree.

End of 1927 is the natural date for fixing the content of the Copenhagen interpretation, since in this year Born and Heisenberg claimed that ''quantum mechanics is a complete theory for which the fundamental physical and mathematical hypotheses are no longer susceptible of modification'' (my English rendering of their French). Apparently nobody made a later, similar claim for a modification.

 

[DarMM]

End of 1927 is the natural date

Natural date for defining the Copenhagen interpretation you mean? I think in the history of physics this is now becoming the consensus and Bohr's later views of the 1940s are coming to be called "The Complementarity Interpretation". So your terminology is becoming the standard one.

I think the confusion stems from the fact that for a long time "Copenhagen" meant any roughly Operationalist take on QM.

 

[A. Neumaier]

Natural date for defining the Copenhagen interpretation you mean?

Yes.