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

[vanhees71]

This sounds like you'd agree to the quote from Asher Peres in another thread!

There are also no observables described by linear operators, but only positive operators describing the response of the detectors to the preparation devices.

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. The POVM formalism is just a generalization of the idealized treatment of von Neumann filter measurements discussed in introductory textbooks.

Peres's rightfully famous textbook as well starts with the usual formalism of self-adjoint operators in Hilbert spaces. How else should he be able to formulate the theory?

 

[Demystifier]

Can you translate this into understandable language?

It's explained in the paper, which you said you read. In ordinary language, the axiom says that macroscopic things that we can directly perceive (e.g. the Moon) exist even when we don't observe them. This is a rather innocent axiom, yet in the paper I explain how this axiom naturally leads to Bohmian mechanics.

 

[A. Neumaier]

both mathematical in nature.

but one refers to the other, hence the semantic relation is the same. Just with Platonic reality in place of physical reality.

 

[vanhees71]

It's explained in the paper, which you said you read. In ordinary language, the axiom says that macroscopic things that we can directly perceive (e.g. the Moon) exist even when we don't observe them. This is a rather innocent axiom, yet in the paper I explain how this axiom naturally leads to Bohmian mechanics.

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.

 

[A. Neumaier]

A spin component is discrete.

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.

The measurement of the magnetic moment of the electron is among the most accurate fundamental quantities ever measured

But it is neither a measurement in the sense of Born's rule, nor is it discrete.

 

[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.

 

[vanhees71]

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 $\textbf{n}$.

Indeed, I checked his textbook again. There's nothing at all different in the foundations than in any other textbook. It's also a clear statement for the orthodox (minimal) probabilistic interpretation.

 

[ftr]

Indeed, I checked his textbook again. There's nothing at all different in the foundations than in any other textbook. It's also a clear statement for the orthodox (minimal) probabilistic interpretation.

So in what sense it is called spin. Moreover, isn't true that the electron of Dirac equation has no charge or spin unless it interacts with an EM. why is that?

 

[vanhees71]

If you start to discuss about Bohr and von Neumann and about the status of 1927, there's no chance to get to anything. The whole trouble with "interpretation" has its origin in this enigmatic mumblings of Bohr and particularly Heisenberg. It's completely irrelevant nearly 100 years later. QT is successfully used to describe everything hitherto observed and where it is applicable, and that's a lot too.

In standard texts the Shannon entropy is a measure for missing information given a probability distribution relative to complete information. In QT it's the same as von Neumann's definition,
$$S=-k_{\text{B}} \mathrm{Tr}(\hat{\rho} \ln \hat{\rho}).$$
This is consistent with the fact that in QT a pure state refers to complete possible information about the system. Indeed, then $\hat{\rho}=|\psi \rangle \langle \psi|$ with some normalized ket $|\psi \rangle$, and then $S=0$, i.e., there's no "missing information".

In his textbook Peres does not follow the usual definition, which seems to lead to a confusion in some postings above. If I interpret pages 280 and 281 right, it's the following situation (I make up a simple concrete example to make the very dense remarks of Peres more concrete).

Suppose we have Alice preparing pure spin states of a spin-1/2-particle. Let's denote with $|\sigma_{\vec{n}} \rangle$ the eigenvectors of the spin component $\hat{s}_{\vec{n}}=\vec{n} \cdot \hat{\vec{s}}$, where $\vec{n} in \mathbb{R}^3$ is an arbitrary unit vector. Of course, the eigenvalues are $\sigma_{\vec{n}} \in \{1/2,-1/2 \}$ (in the following, using natural units with $\hbar=k_{\text{B}}=1$).

To make a concrete example take three arbitrary unit vectors $\vec{n}_j$ and suppose Alice prepares randomly always particles in the pure states $\hat{P}_j=|\sigma \vec{n}_j =+1/2 \rangle \langle \sigma_{\vec{n}_j=+1/2}|$ with propbalities $p_j$. Don't ask me why, but Peres defines the "Shannon entropy" for this case as
$$S_{\text{Peres/Shannon}} \equiv H=-\sum_{j=1}^3 p_j \ln p_j.$$
The corresponding quantum state is, however, given by
$$\hat{\rho}=\sum_{j=1}^3 p_j \hat{P}_j,$$
and the usual entropy a la von Neumann (and Jaynes) is
$$S=-\mathrm{Tr} \hat{\rho} \ln \hat{\rho}.$$
This is of course usually different from $H$, and indeed the entropies refer to different probablities. $H$ describes the entropy as the lack of information for an observer who knows that Alice prepares with probabilities $p_j$ the states $\hat{P}_j$. In contradistinction to that $S$ answers the question, what's the missing information given $\hat{\rho}$ relative to complete possible information. Complete possible information for a spin would be to have prepared an arbitrary spin component $\hat{s}_{\vec{n}}$ in an arbitrary direction $n$. The probabilities for the possible outcomes are of course
$$p_{\vec{n}}(\sigma_{\vec{n}})=\langle \sigma_{\vec{n}} |\hat{\rho} |\sigma_{\vec{n}} \rangle=\sum_{j=1}^3 p_j \frac{1}{2} (1+2 \sigma_{\vec{n}} \vec{n} \cdot \vec{n}_j)$$
and
$$S=-\sum_{\sigma_{\vec{n}} =\pm 1/2} p_{\vec{n}}(\sigma_{\vec{n}}) \ln (p_{\vec{n}}(\sigma_{\vec{n}}))=-\mathrm{Tr} \hat{\rho} \ln \hat{\rho}.$$
It's of course nothing wrong with Peres's definitions, but it's a bit confusing to deviate from standard terminology. Usually one defines the Shannon-Jaynes entropy of QT as the von Neumann entropy.

Peres uses the Shannon entropy to a different situation and calls this then Shannon entropy. As usual, in any textbook one must read the definitions carefully since the author may deviate from standard terminology or definitions in other textbooks or papers.

 

[vanhees71]

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,

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

... 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)

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

In which sense isn't the von Neumann measurement (projectors) a special case of the POVM formalism. Maybe, I have to read the chapters in Peres's book I skipped over because I found them overly complicated and sometimes confusing, e.g., calling something "Shannon entropy" which is different from the usual use in other QT textbooks (see my previous posting). Of course, I guess within his book he is consistent.

Maybe I also misunderstood, what he means by "quantum test". I understood it such that he means what's usually called "measurement" with the extended unerstanding of the word using POVMs instead of ideal filter measurements.

He has also a formulation sometimes, which I carefully abandon for some time from my language: He says that given a preparation in a state $|\psi_1 \rangle$, the probability for the system of being in state $|\psi_2 \rangle$ is $|\langle \psi_2|\psi_1 \rangle|^2$. This can lead to confusion in connection with the dynamics, and I consider it important to refer to measurements of observables (in the standard sense), i.e., the probabilities are about outcomes of eigenvalues when measuring observables accurately, i.e., in Born's rule one vector in the scalar product must be an eigenvector of an observable operator and the other a state vector, which in general time evolve differently (e.g., in the Schrödinger picture the state vectors propagate with the complete Hamiltonian and the eigenvectors of observables are constant in time).

If it comes to the details, I find Peres not always satisfactory, but concerning interpretation he seems to be on the no-nonsense (i.e., the probabilistic standard interpretation) side ;-)).

 

[vanhees71]

So in what sense it is called spin. Moreover, isn't true that the electron of Dirac equation has no charge or spin unless it interacts with an EM. why is that?

I don't know. Where did you get this from?

The free Dirac field is a quantum field describing particles and antiparticles with spin 1/2. It's an irreducible representation of the orthochronous Lorentz group (but not the proper orthochronous Lorentz group, for which it is an reducible representation). As all the "physical representations" it is characterized by several quantum numbers, which are the mass and the spin. Invariance of the corresponding Lagrangian under phase transformations leads to a conserved charge. The particles and anti-particles have the same mass and spin and opposite charges.

Now you can gauge the global symmetry with its associated conserved charge and interpret the Abelian gauge field as the electromagnetic field. Then you have a model for charges and the electromagnetic field and then the conserved charge of this gauged symmetry you call the "electric charge".

 

[DarMM]

Indeed, I checked his textbook again. There's nothing at all different in the foundations than in any other textbook. It's also a clear statement for the orthodox (minimal) probabilistic interpretation.

I'm not sure which part of his textbook you looked at, but see this paper:
https://arxiv.org/abs/quant-ph/0207020

 

[DarMM]

In his textbook Peres does not follow the usual definition, which seems to lead to a confusion in some postings above

Yeah, as I mentioned above he uses "Shannon Entropy" in a non-standard way as the entropy of distribution over density matrices.

 

[A. Neumaier]

In which sense isn't the von Neumann measurement (projectors) a special case of the POVM formalism.

In no sense. But it is a highly idealized one, and one cannot derive from it the general one. The 'derivation' given proceeds by embedding the physical Hilbert space into a fictitious tensor product with an ancilla space, and has only the status of a consistency check.

In real life, POVMs are not necessarily implemented by the algorithm of Eq. (9.98). There is an infinity of other ways of materializing a given POVM. The importance of Neumark’s theorem lies in the proof that any arbitrary POVM with a finite number of elements can in principle, without violating the rules of quantum theory, be converted into a maximal test, by introducing an auxiliary, independently prepared quantum system (the ancilla).

Maybe I also misunderstood, what he means by "quantum test". I understood it such that he means what's usually called "measurement" with the extended understanding of the word using POVMs instead of ideal filter measurements.

No. A quantum test is a test for ''being in the state $\phi$,'' corresponding to a von Neuman measurement of a projector to the 1-dimensional space spanned by $\phi$. His axiomatization (in Chapters 2 and 3) only concerns these quantum tests, which are indeed the measurements for which Born's rule (saying here that a positive test is achieved with probability $\phi^*\rho\phi$, postulated on p.56) is impeccable.

Peres has no need for claiming having measured eigenvalues of a given operator representing a given observable. Instead he derives this under certain idealizations (not present in a general POVM) by constructing the operator on p.63 given a collection of quantum tests forming a ''maximal test'', for which he postulates realizability on p.54. The maximal test tests for each state in a given complete orthonormal basis. Since only finitely many quantum tests can be realized, this amounts to an assumption of a finite-dimensional Hilbert space, showing the idealization involved in his derivation.

Later he forgets that his definition of an observable is constructed from a given maximal test, hence has no meaning in any other basis: Carried away by the power of the formal calculus (and since he needs it to make contact with tradition), he asks on p.64 for ''the transformation law of the components of these observable matrices, when we refer them to another basis''. So I don't find his point of view convincing.

He has also a formulation sometimes, which I carefully abandon for some time from my language: He says that given a preparation in a state $|\psi_1 \rangle$, the probability for the system of being in state $|\psi_2 \rangle$ is $|\langle \psi_2|\psi_1 \rangle|^2$. This can lead to confusion in connection with the dynamics, and I consider it important to refer to measurements of observables (in the standard sense), i.e., the probabilities are about outcomes of eigenvalues when measuring observables accurately

At least Peres is more careful and consistent than you.

You are using Born's rule claiming, in (2.1.3) in your lecture notes, that measured are exact eigenvalues - although these are never measured exactly -, to derive on p.21 the standard formula for the q-expectation (what you there call the mean value) of known observables (e.g., the mean energy $\langle H\rangle$ in equilibrium statistical mechanics) with unknown (most likely irrational) spectra. But you claim that the resulting q-expectation is not a theoretical construct but is ''in agreement with the fundamental definition of the expectation value
of a stochastic variable in dependence of the given probabilities for the outcome of a measurement of this variable.'' This would hold only if your outcomes match the eigenvalues exactly - ''accurately'' is not enough.

 

[A. Neumaier]

Yeah, as I mentioned above he uses "Shannon Entropy" in a non-standard way as the entropy of distribution over density matrices.

But your usage makes the value of the Shannon entropy dependent on a context (the choice of an orthonormal basis), hence is also not the same as the one vanhees71 would like to use:

Usually one defines the Shannon-Jaynes entropy of QT as the von Neumann entropy.

Thus we now have three different definition, and it is far from clear which one is standard.

On the other hand, why should one give two different names to the same concept?

 

[DarMM]

But your usage makes the value of the Shannon entropy dependent on a context

Thus we now have three different definition, and it is far from clear which one is standard.

Yes, in Quantum Information theory the Shannon entropy is the entropy of the classical model induced by a context. So it naturally depends on the context. I don't see why this is a problem, it's a property of a context. There are many information theoretic properties that are context dependent in Quantum Information.

Von Neumann entropy is a separate quantity and is a property of the state, sometimes called Quantum Entropy and is equal to the minimum Shannon entropy taken over all contexts.

I don't see that what @vanhees71 and I are saying is that different. He's just saying that the von Neumann entropy is the quantum generalization of Shannon entropy. That's correct. Shannon entropy is generalized to the von Neumann entropy, but classical Shannon entropy remains as the entropy of a context.

It's only Peres's use, referring to the entropy of the distribution over densities, that seems nonstandard to me.