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

[Auto-Didact]

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

Do you have a source for that? In either case, in Bohmian mechanics there is a clear answer: spin is not a property of the particle but of the guiding wave, which is a spinor; the guiding wave imparts spin onto the particle.

In SED (a semi-Bohmian semiclassical competitor to QED) particles are fundamentally spinless as well; spin is imparted onto particles by the circularly polarized modes of the ground state of the background field via the Lorentz force.

 

[vanhees71]

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

Yes, this is a generalization of idealized measurements to imprecise measurements of real-world detectors, formalized in terms of the POVM formalism. Idealized measurements, i.e., precise measurements are a special case, where the $\hat{E}_m$ are projectors $|m \rangle \langle m|$ with $m$ labelling a complete orthonormal set of eigenvectors of the measured observable. I've no clue, how else to interpret the example used by Peres, if the $\hat{J}_k$ are not the self-adjoint (I don't think hermitian is sufficient though Peres claimes this) operators representing spin.

 

[DarMM]

I've no clue, how else to interpret the example used by Peres, if the $\hat{J}_k$ are not the self-adjoint (I don't think hermitian is sufficient though Peres claimes this) operators representing spin.

He's not saying $\hat{J}_k$ don't represent spin. He's saying that a typical POVM cannot be understood as measurement of $J\cdot n$ for some direction $n$, i.e. a typical POVM cannot be associated with spin in a given direction or in fact with any classical quantity. It seems to be simply an abstract representation of the responses of a given device.

 

[vanhees71]

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.

Of course, the entropy measure depends on the context. That it's strength! It's completely legitimate to define an entropy $H$ and also obviously useful in some investigations is quantum informatics as Peres. To avoid confusion, I'd not call it Shannon entropy.

Let me try again, to make the definition clear (hoping to have understood Peres right).

Peres describes the classical gedanken experiment to introduce mixed states and thus the general notion of quantum state in terms of a statistical operator (which imho should be a self-adjoint positive semidefinite operator with trace 1): Alice (A) prepares particles in pure states $\hat{P}_n=|u_n \rangle \langle u_n|$, each with probability $p_n$. The $|u_n \rangle$ are normalized but not necessarily orthogonal to each other. The statistical operator associated with this situation is
$$\hat{\rho}=\sum_n p_n \hat{P}_n.$$
Now Peres defines an entropy by
$$H=-\sum_n p_n \ln p_n.$$
This can be analyzed using the general scheme by Shannon. Entropy in Shannon's sense is a measure for the missing information given a probability distribution relative to what's considered complete information.

Obviously Peres takes the $p_n$ as the probability distribution. This distribution describes precisely the situation of A's preparation process: It describes the probability that A prepares state $\hat{P}_n$, i.e., an observer Bob (B) uses $H$ as the entropy measure if he knows that A prepares the specific states $\hat{P}_n$, each with probability $p_n$. Now A sends him such a state. For B complete information would be to know which $\hat{P}_n$ this is, but he doesn't know it but only the probability $p_n$. That's why B uses $H$ as the measure for missing information.

Now the mixed state $\hat{\rho}$ defined above describes something different. It provides the probability distribution for any possible measurement on the system. Complete information in QT means that we measure precisely (in the old von Neumann sense) a complete set of compatible observables $O_k$, represented by self-adjoint operators $\hat{O}_k$ with orthonormalized eigenvectors $|\{o_k \} \rangle$. If we are even able to filter the systems according to this measurement we have prepared the system as completely as one can according to QT, namely in the pure state $\hat{\rho}(\{o_k \})=|\{o_k \} \rangle \langle \{o_k \}|$.

The probabilities for the outcome of such a complete measurement are
$$p(\{o_k \})=\langle \{o_k \} |\hat{\rho}|\{o_k \} \rangle.$$
Relative to this definition of "complete knowledge", given A's state preparation described by $\hat{\rho}$ B associates with this situation the entropy
$$S=-\sum_{\{o_k \}} p(\{o_k \}) \ln p(\{o_k \}).$$
Now it is clear that this entropy is independent of which complete set of compatible observables B chooses to define what's complete knowledge in this quantum-theoretical sense means, since obviously this entropy is given by
$$S=-\mathrm{Tr} (\hat{\rho} \ln \hat{\rho}).$$
This is the usual definition of the Shannon-Jaynes entropy in quantum theory, and it's identical with von Neumann's definition by this trace. There's no contradiction between $H$ and $S$ of any kind, it are just entropies in Shannon's information theoretical sense referring to different information about the same preparation procedure.

One has to keep in mind, to which "sense of knowledge" the entropy refers to, and no confusion can occur. As I said before, I'd not call H the Shannon entropy to avoid confusion, but it's fine as a short name for what Peres clearly defines.

 

[A. Neumaier]

(I don't think hermitian is sufficient though Peres claimes this) operators representing spin.

Spin operators are Hermitian and bounded. This implies already that they are selfadjoint.

 

[vanhees71]

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.

We have discussed this a zillion of times. This is the standard treatment in introductory text, and rightfully so, because you have to first define the idealized case of precise measurements. Then you can generalize it to more realistic descriptions of imprecise measurements.

Peres is a bit contradictory when claiming everything is defined by defining some POVM. There are no POVMs in the lab but only real-world preparation and measurement devices.

If, as you claim, precise measurements were not what Peres calls a "quantum test", which sense then would this projection procedure make? Still, I don't think that it's good language to mix the kets representing pure states with eigenstates of the observable operators. At latest at the point if you bring in dynamics using different pictures of time evolution, this leads to confusion. I understood this only quite a while after having learned QT for the first time by reading the book by Fick, which is among the best books on QT I know. The only point which I think is wrong is to envoke the collapse postulate. Obviously one can not have a QT textbook that gets it completely right :-((.

 

[vanhees71]

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:

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?

No. As I tried to explain in #154 there's only one definition of Shannon entropy, which is very general but a very clear concept. It's on purpose context dependent, i.e., that's not a bug but a feature of the whole concept.

The only confusion arises due to the unconventional use of the word Shannon entropy in the context of the probabilities described by quantum states.

 

[A. Neumaier]

No. As I tried to explain in #154 there's only one definition of Shannon entropy, which is very general but a very clear concept. It's on purpose context dependent, i.e., that's not a bug but a feature of the whole concept.

The only confusion arises due to the unconventional use of the word Shannon entropy in the context of the probabilities described by quantum states.

Shannon entropy is a classical concept that applies whenever one has discrete probabilities. Peres, DarMM, and you appply it consistently to three different situations, hence are all fully entitled to call it Shannon entropy. You cannot hijack the name for your case alone.

 

[vanhees71]

Sigh. Shannon entropy is not restricted to classical physics. It's applicable for any situation described with probabilities. I also do not see what's wrong with the usual extension of the Shannon entropy to continuous situations. After all entropy in the context of statistical physics has been defined using continuous phase-space variables.

 

[A. Neumaier]

Sigh. Shannon entropy is not restricted to classical physics. It's applicable for any situation described with probabilities.

This is just what I had asserted. Instead of sighing it might be better to pay attention to what was actually said: You, DarMM and Peres consider three different kinds of quantum situations described with probabilities and hence get three different Shannon entropies. They all fully deserve this name.

I also do not see what's wrong with the usual extension of the Shannon entropy to continuous situations.

The Shannon entropy of a source is defined as the minimal expected number of questions that need to be asked to pin down the classical state of the source (i.e., the exact knowledge of what was transmitted), given the probability distribution for the possibilities. It applies by its nature only to discrete probabilities, since for continuous events no finite amount of questions pins down the state exactly.

After all entropy in the context of statistical physics has been defined using continuous phase-space variables.

In the statistical physics of equilibrium, the Hamiltonian must have a discrete spectrum; otherwise the canonical density operator is not defined: Indeed, if the spectrum is not discrete, $e^{-\beta H}$ is not trace trace class, and the partition function diverges.

On the other hand, Boltzmann's H is, by the above argument, not a Shannon entropy.

 

[Auto-Didact]

So basically, the criterion being given is that for something to be "real" it must be capable of being acted on by other "real" things, whereas the wave function is not acted on by anything. But this doesn't seem right, because the wave function is determined by Schrodinger's Equation, which includes the potential energy, and the potential energy is a function of the particle configuration. So I don't think it's correct to say that the wave function is not acted on by anything.

This is a misunderstanding, but a very intriguing one, namely a category error. The wave function is a solution to the Schrödinger equation, which is specifically determined by positions (among other things which is not relevant for the rest of the argument).

Being a solution to an equation is definitely not the same kind of logical or mathematical relationship as e.g. the relationship between two dynamical objects such as two masses mutually acting upon each other; the former is a relationship between input and output, while in the latter the relationship is between two inputs who together determine an output.

 

[PeterDonis]

This is a misunderstanding, but a very intriguing one, namely a category error.

In the context of, say, the Copenhagen interpretation of QM, it is, yes. But not in the context of Bohmian mechanics, which was the interpretation under discussion in the post of mine you quoted and the subthread it is part of. In Bohmian mechanics the wave function is a real thing; the Schrodinger Equation is simply an equation that governs the dynamics of this real thing.

 

[Morbert]

In Bohmian mechanics the wave function is a real thing

Hmm, I'm not sure that's necessarily the case. BM can frame the wavefunction as nomological rather than ontological. I.e. Instead of being a thing that exists, it is a representation of the behaviour of things that exist.

From "Quantum Physics Without Quantum Philosophy" by Goldstein et al

"It should be clear by now what, from a universal viewpoint, the answer to these objections must be: the wave function of the universe should be regarded as a representation, not of substantial physical reality, but of physical law."

"As such, the wave function plays a role analogous to that of the Hamiltonian function H = H (Q, P ) ≡ H (ξ ) in classical mechanics [...] And few would be tempted to regard the Hamiltonian function H as a real physical field, or expect any back-action of particle configurations on this Hamiltonian function."

 

[Auto-Didact]

In the context of, say, the Copenhagen interpretation of QM, it is, yes. But not in the context of Bohmian mechanics, which was the interpretation under discussion in the post of mine you quoted and the subthread it is part of. In Bohmian mechanics the wave function is a real thing; the Schrodinger Equation is simply an equation that governs the dynamics of this real thing.

Actually, that isn't exactly true: in BM the wavefunction isn't a field in physical space(time), but a vector field in configuration space (NB: the fact that the wavefunction is living in configuration space while acting upon particles in space(time) is also why BM is an explicitly nonlocal theory).

This is quite similar to the Hamiltonian vector field which exists in phase space; the difference is that the Hamiltonian vector field is static, while the wavefunction as a solution to the SE is dynamic; I suspect however that this difference might be a red herring, because we actually know of a static wavefunction, namely the solution to the Wheeler-de Witt equation.

 

[PeterDonis]

in BM the wavefunction isn't a field in physical space(time), but a vector field in configuration space

That's true, but it doesn't change what I said.

 

[PeterDonis]

BM can frame the wavefunction as nomological rather than ontological.

The reference you give, as far as I can tell, isn't talking about BM.

 

[Morbert]

The reference you give, as far as I can tell, isn't talking about BM.

Bohmian Mechanics is the main subject of the book. The quotes are from chapter 11.5: "A Universal Bohmian Theory". Specifically, the wavefunction is described as nomological in response to the objection that the wavefunction in BM doesn't experience any back-action from the existing configuration.

 

[PeterDonis]

the wavefunction is described as nomological in response to the objection that the wavefunction in BM doesn't experience any back-action from the existing configuration

I don't have the book, but the position you describe seems similar to that described in the paper @DarMM linked to in post #50. We discussed that earlier in the thread.

 

[Morbert]

I don't have the book, but the position you describe seems similar to that described in the paper @DarMM linked to in post #50. We discussed that earlier in the thread.

Ah ok, I missed the earlier context of the discussion.

 

[vanhees71]

This is just what I had asserted. Instead of sighing it might be better to pay attention to what was actually said: You, DarMM and Peres consider three different kinds of quantum situations described with probabilities and hence get three different Shannon entropies. They all fully deserve this name.

The Shannon entropy of a source is defined as the minimal expected number of questions that need to be asked to pin down the classical state of the source (i.e., the exact knowledge of what was transmitted), given the probability distribution for the possibilities. It applies by its nature only to discrete probabilities, since for continuous events no finite amount of questions pins down the state exactly.

In the statistical physics of equilibrium, the Hamiltonian must have a discrete spectrum; otherwise the canonical density operator is not defined: Indeed, if the spectrum is not discrete, $e^{-\beta H}$ is not trace trace class, and the partition function diverges.

On the other hand, Boltzmann's H is, by the above argument, not a Shannon entropy.

So you are saying that the classical examples for the application of equilibrium statistics is flawed, i.e., no ideal gases, no Planck black-body radiation, specific heat of solids, and all that? How can it then be that it works so well in physics? That you have to take the "thermodynamic limit" very carefully is clear.

 

[A. Neumaier]

So you are saying that the classical examples for the application of equilibrium statistics is flawed, i.e., no ideal gases, no Planck black-body radiation, specific heat of solids, and all that? How can it then be that it works so well in physics? That you have to take the "thermodynamic limit" very carefully is clear.

The derivation of thermodynamics from statistical mechanics is sound. I give such a derivation in Part II of my online book in terms of the grand canonical density operator, independent of any interpretation in terms of probabilities and of any thermodynamical limit.

I am only claiming that the thermodynamic concept of entropy in general (e.g., in Boltzmann's H-theorem) has nothing to do with Shannon entropy. It is not amenable to an information theoretic analysis. The latter is limited to discrete probabilities.

 

[DarMM]

The latter is limited to discrete probabilities

Do you mean a discrete sample space?

 

[vanhees71]

The derivation of thermodynamics from statistical mechanics is sound. I give such a derivation in Part II of my online book in terms of the grand canonical density operator, independent of any interpretation in terms of probabilities and of any thermodynamical limit.

I am only claiming that the thermodynamic concept of entropy in general (e.g., in Boltzmann's H-theorem) has nothing to do with Shannon entropy. It is not amenable to an information theoretic analysis. The latter is limited to discrete probabilities.

Well, there are two camps in the physics community: The one camp likes the information theoretical approach to statistical physics, the other hates it. I belong to the first camp, because for me the information theoretical approach provides the best understanding what entropy is from a microscopic point of view.

What I mean by "thermodynamical limit" is the limit to take the volume to infinity, keeping densities constant. This is the non-trivial limit you seem to refer to when saying you need descrete probability distributions. Indeed at finite volume with the appropriate (periodic in the case you want proper momentum representations) spatial boundary conditions, momentum gets discrete and you can so all calculations in a (pretty) well-defined way.

 

[A. Neumaier]

Well, there are two camps in the physics community: The one camp likes the information theoretical approach to statistical physics, the other hates it.

My arguments are of a logical nature. They do not depend on emotional feelings associated with an approach. The information theoretical approach is limited by the nature of the questions it asks.

What I mean by "thermodynamical limit" is the limit to take the volume to infinity, keeping densities constant.

In this thermodynamical limit, all statistical uncertainties reduce to zero, and no trace of the statistics remains.

In particular, it is not the limit in which a discrete probabiliy distribution becomes a continuous one. Thus it does not justify to apply information theoretical reasoning to continuos probability distributions.

Neither is Boltzmann's H-theorem phrased in terms of a thermodynamic limit. No information theory is needed to motivate and understand his results. Indeed they were obtained many dozens year before Jynes introduced the thermodynamic interpretation

 

[A. Neumaier]

Do you mean a discrete sample space?

I mean a discrete measure on the sigma algebra with respect to which the random variables associated with the probabilities are defined. This is needed to make sense of the Shannon entropy as a measure of lack of information. For example, Wikipedia says,

The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable [...] is usually referred to as the continuous entropy, or differential entropy. A precursor of the continuous entropy h[f] is the expression for the functional Η in the H-theorem of Boltzmann. [...] Differential entropy lacks a number of properties that the Shannon discrete entropy has – it can even be negative. [...] The differential entropy is not a limit of the Shannon entropy for n → ∞. Rather, it differs from the limit of the Shannon entropy by an infinite offset. [...] It turns out as a result that, unlike the Shannon entropy, the differential entropy is not in general a good measure of uncertainty or information.

for me the information theoretical approach provides the best understanding what entropy is from a microscopic point of view.

What should a negative amount of missing information mean? Which improved understanding does it provide?

 

[vanhees71]

Where in the usual applications in statistical physics does the entropy become negative?

 

[A. Neumaier]

Where in the usual applications in statistical physics does the entropy become negative?

In Boltzmann's H-theorem. since there the energy has a continuous spectrum.

 

[vanhees71]

In the usual definition you start with a finite volume and thus the discrete case. Obviously the entropy is always positive,
$$S=-\mathrm{Tr} \hat{\rho} \ln \hat \rho=-\sum_{i} \rho_i \ln \rho_i,$$
where $\rho_i$ are the eigenvalues of $\hat{\rho}$. Since $\hat{\rho}$ is positive semidefinite and $\mathrm{Tr} \hat{\rho}=1$ you have $\rho_i \in [0,1]$. For $\rho_i=0$ by definition in the entropy formula you have to set $\rho_i \ln \rho_i=0$. Thus $S \geq 0$. Taking the thermodynamic limit keeps $S \geq 0$.

At which point in the derivation of the Boltzmann equation becomes $S<0$ then?

 

[A. Neumaier]

In the usual definition you start with a finite volume and thus the discrete case. Obviously the entropy is always positive,
$$S=-\mathrm{Tr} \hat{\rho} \ln \hat \rho=-\sum_{i} \rho_i \ln \rho_i,$$
where $\rho_i$ are the eigenvalues of $\hat{\rho}$. Since $\hat{\rho}$ is positive semidefinite and $\mathrm{Tr} \hat{\rho}=1$ you have $\rho_i \in [0,1]$. For $\rho_i=0$ by definition in the entropy formula you have to set $\rho_i \ln \rho_i=0$. Thus $S \geq 0$. Taking the thermodynamic limit keeps $S \geq 0$.

At which point in the derivation of the Boltzmann equation becomes $S<0$ then?

In the above, you didn't discuss Boltzmann entropy but von Neumann entropy. It is definable only for trace class operators $\rho$, which necessarily have discrete spectrum. Thus they produce discrete probability densities, which of cource may be interpreted in terms of information theory. (Though it is artiicial as one cannot implement on the quantum level the decision procedure that gives rise to the notion of Shannon entropy. )

On the other hand, Boltzmann entropy is an integral over classical phase space. He didn't yet know quantum physics when he introduced the H-theorem. Boltzmann entropy can be negative and has no interpretation in terms of information theory.

 

[vanhees71]

Well, supposedly that's then the deeper reason for the necessity of starting with some "regularization" such that the statistical operator has a discrete spectrum, and the thermodynamic limit is not that trivial.

That there's trouble with entropy in classical statistics is well-known since Gibbs ;-)).

 

[A. Neumaier]

Well, supposedly that's then the deeper reason for the necessity of starting with some "regularization" such that the statistical operator has a discrete spectrum, and the thermodynamic limit is not that trivial.

This is not a regularization. Real materials to which statistical mechanics applies have bounded volume.

The thermodynamic limit is an idealization in which all uncertainties vanish, and hence all statistical connotations disappear.

 

[A. Neumaier]

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.

We have discussed this a zillion of times. This is the standard treatment in introductory text, and rightfully so, because you have to first define the idealized case of precise measurements. Then you can generalize it to more realistic descriptions of imprecise measurements.

But an idealized case can be no more than a didactical prop. Good foundations must be general enough to support all uses.

Your position on the foundations of quantum mechanics is like calling systems of harmonic oscillators the foundations of classical mechanics, because you have to first define the idealized case of precise oscillations. But the true foundations of classical mchnaics are the Lagrangian and Hamiltonian formalisms!

Similarly, in quantum mechnaics, true foundations must feature open systems and POVMs, since these describe the realistic scenarios.

 

[vanhees71]

We want to do physics. In the way, how POVMs are introduced by Peres, you'd never have discovered QT as a tool to describe what's observed. He also defines a quantum test as a projection operation to a one-dimensional subspace in Hilbert space. As he rightfully stresses, there are no Hilbert spaces nor hermitean operators in the lab but real-world equipment. E.g., where in the quantum-optics literature have you ever needed POVMs rather than the standard formulation of QT to understand all the stringent Bell tests? There you meausure photon detection rates of various kinds and analyze them in terms of an appropriate initial state of your photons and the n-photon correlation functions, sometimes also corrected for non-ideal detectors.

This is also the main obstacle for a physicist to understand what you want to say in your "thermal interpretation". A physical theory needs more than the statement of some axioms. You need an operational meaning, i.e., how to apply the formalism to the operations with real-world equipment in the lab. That's what "interpretation" is all about. There's no necessity for philosophical confusions, and overly abstract axiomatic foundations without relation to real-world experiments may be nice mathematical edifices of pure thought but have not much to do with theoretical physics applicable to phenomenology. A famous example is string theory and its relatives ;-)).

 

[A. Neumaier]

In the way, how POVMs are introduced by Peres, you'd never have discovered QT as a tool to describe what's observed.

Well, it was neither discovered through how Born's rule is introducd by you, but through trying to theoretically understand black body radiation, the photoeffect, and spectra. Good foundations should not follow the way of discovery, which is often erratic and tentative, but should procvide the concepts needed to be able to handle the general situation by sraightforward specialization.

where in the quantum-optics literature have you ever needed POVMs rather than the standard formulation of QT to understand all the stringent Bell tests?

They are needed to characterize the equipment in such a way that one can talk reliably about efficiencies nd close various loopholes. Most of quantum optics works with POVMs rather than von Neumann measurements; these figure only in the simplified accounts.

You need an operational meaning, i.e., how to apply the formalism to the operations with real-world equipment in the lab. That's what "interpretation" is all about.

Yes, and that's why one needs POVMs rather than Bon's rule. Only in introductory courses is the latter sufficient.

 

[vanhees71]

Why then are POVMs so rarely used in practice? I've not seen them used in quantum optics papers dealing with the foundations. Can you point me to one, where they are needed to understand an experiment?

 

[A. Neumaier]

Why then are POVMs so rarely used in practice? I've not seen them used in quantum optics papers dealing with the foundations. Can you point me to one, where they are needed to understand an experiment?

They are used a lot for different purposes. For example, any quantum phase measurement is necessarily a POVM,
https://iopscience.iop.org/article/10.1088/0954-8998/3/1/002/metaso is any joint measurement of position and momentum (or the quadratures in quantum optics)To see the principles at work, one can works with the simple, idealized version. This is the stuff discussed in textbooks, popular articles, and theoretical papers where the idealization simplifies things a lot.

But to see the limits of real equipment one needs the POVMs - no real detector is ideal . Then it gets messy, e.g.,
https://arxiv.org/pdf/1204.1893and specifically in the context of Bell inequalities:
https://arxiv.org/pdf/quant-ph/0007058https://arxiv.org/pdf/1304.7460
https://arxiv.org/pdf/quant-ph/0407181This is why people avoid the details if possible and just use simple efficiency proxys.

Some other papers:
https://arxiv.org/pdf/quant-ph/9809063https://arxiv.org/pdf/quant-ph/0011042https://arxiv.org/pdf/0804.3082https://arxiv.org/pdf/1304.7460https://journals.aps.org/pra/abstract/10.1103/PhysRevA.82.062115https://arxiv.org/pdf/1007.3043https://arxiv.org/pdf/quant-ph/0608128https://arxiv.org/pdf/1111.5874https://arxiv.org/pdf/1206.6054
start looking, and you find a nearly endless collection of work...

 

[vanhees71]

https://arxiv.org/pdf/1204.1893

Well, this is very clearly using the standard quantum-theoretical formalism to construct (!) the POVM description of the measurement device. That rather confirms my view on the POVM formalism than is an argument against it.

 

[A. Neumaier]

Well, this is very clearly using the standard quantum-theoretical formalism to construct (!) the POVM description of the measurement device.

What do you mean by ''construct the POVM description of the measurement device''?

In general, any device has a POVM description that cannot be postulated to be of Born type but needs to be found out by quantum tomography, using the formula (1) for probabilities. This is not Born's riule but a proper extension of it. It cannot be reduced to Born unless one adds nonphysical stuff (ancillas, that have no physical representation) to the description!

 

[vanhees71]

Are we talking about the same paper? Eq. (1) IS Born's rule. What else should it be?

 

[A. Neumaier]

Are we talking about the same paper?

Yes, but you didn't read it carefully enough.

Eq. (1) IS Born's rule. What else should it be?

No. In equation (1) on p.2 of https://arxiv.org/pdf/1204.1893, $\Pi_n$ is an arbitrary positive operator from a POVM. Born's rule in its most general form is only the special case of (1) where all $\Pi$ are orthogonal projectors.
https://arxiv.org/pdf/1204.1893

 

[vanhees71]

Why is (1) not Born's rule? I thought $\Pi_n$ is still a self-adjoint operator. At least that's the case in the treatment of the POVM formalism in Peres's textbook. The only difference is that the $\Pi_n$ are not orthonormal projectors as in the special case of ideal von Neumann filter measurements.

Also, as far as I understand the paper is about, how to determine the POVM for a given apparatus, and the necessary analysis is through the standard formalism of measurements in quantum optics, using a sufficiently large set of input states (in this case they use coherent states, aka Laser light).

I still don't see, in which sense the POVM formalism is an extension of standard QT. To the contrary it's based on standard QT, applied to open systems in contradistinction to the idealized description of measurements on closed systems.

 

[A. Neumaier]

Why is (1) not Born's rule? I thought $\Pi_n$ is still a self-adjoint operator.

$\Pi_n$ is Hermitian and bounded, hence self-adjoint. But in the formula for probabilties in Born's rule only orthogonal projection operators figure;:

The only difference is that the $\Pi_n$ are not orthonormal projectors as in the special case of ideal von Neumann filter measurements.

This is an essential difference. It means that Born's rule is only a very special and often unrealistic (i.e., wrong!) case of the correct rule calculating probabilities for quantum detectors. To write down the correct rule in an introduction to quantum mechnaics would in fact be easier than writing down Born's rule, because one needs no discussion of the spectral theorem. Thus there is no excuse for giving in the foundations a special, highly idealized case in place of the real thing.

Also, as far as I understand the paper is about, how to determine the POVM for a given apparatus, and the necessary analysis is through the standard formalism of measurements in quantum optics

The method is quantum tomography, which is based on POVM's and semidefinite programming only, nothing else. Of course it needs sources with known density operator. Only for these, textbook quantum optics is used.

I still don't see, in which sense the POVM formalism is an extension of standard QT. To the contrary it's based on standard QT, applied to open systems in contradistinction to the idealized description of measurements on closed systems.

On a closed system, one cannot make a measurement at all, not even one satisfying Born's original rule.
Thus your distinction is meaningless.

The POVM formalism applies (quantitatively correctly) in exactly the same circumstances where Born's rule is claimed to apply (qualitatively, quantitatively only in ideal cases): Between the source and the detector, the system under discussion in the paper (and everywhere else) can be as closed as you can make it; it doesn't change at all the POVM properties of the detector. It is only the detection process itself where the system is open for a moment.

 

[vanhees71]

Fine, I'd still not know, how to teach beginners in QT using the POVM concept. I'd not think that a book like Peres's is adequate for this purpose. It's not clear to me, what he considers concretely to be a "quantum test". Interestingly enough he introduces the POVM using the Born rule in the standard way for a closed system tracing out what he calls the "ancilla". I don't think it's possible to introduce POVMs for physicists without using the standard formulation in the usual terms of observables and states.

 

[A. Neumaier]

The only difference is that the $\Pi_n$ are not orthonormal projectors as in the special case of ideal von Neumann filter measurements.

Another important difference is that a POVM measurement makes no claim about which values are measured.

It just says that one of the detectors making up the detection device responds with a probability given by the trace formula. The value assigned to the $k$th detection event is pure convention, and can be any number $a_k$ - whatever has been written on the scale the pointer points to, or whatever has been programmed to be written by an automatic digital recording device. This is reality. Nothing about eigenvalues.

The state dependent formula for the expectation of the observable measured that follows from POVM together with the value assignment is $\langle A\rangle=Tr~\rho A$ with the operator $A=\sum a_k\Pi_k$.
Note that the same operator $A$ in the expectation can be decomposed in many ways into a linear combination of many POVM terms. The spectral decomposition is just the historically first one, but usually not the most realistic one.

This is similar to the classical situation where a detector returns a number measured in inches or measured in cm, depending on how you label the scale. You could also measure length-squared by changing the scale nonlinearly.

 

[A. Neumaier]

he introduces the POVM using the Born rule in the standard way for a closed system tracing out what he calls the "ancilla".

This is not the introduction, but after having already introduced POVMs he shows that the concept is consistent with the traditional setting, but on an (unphysical, just formally constructed) extended Hilbert space.

 

[DarMM]

This is not the introduction, but after having already introduced POVMs he shows that the concept is consistent with the traditional setting, but on an (unphysical, just formally constructed) extended Hilbert space.

This is the whole "church of the smaller/larger Hilbert space" issue in quantum foundations. Whether POVMs are fundamental or if they're always PVMs with ancillas.

 

[A. Neumaier]

This is the whole "church of the smaller/larger Hilbert space" issue in quantum foundations. Whether POVMs are fundamental or if they're always PVMs with ancillas.

Well, at least once you go to QFT, there is no natural way to add the ancillas. It is a purely formal trick to reduce POVMs and related measurement issues to the standard (problematic) foundations.

 

[DarMM]

Well, at least once you go to QFT, there is no natural way to add the ancillas. It is a purely formal trick to reduce POVMs and related measurement issues to the standard (problematic) foundations.

I agree. Just to inform people of the terms should they encounter them. I myself can't my sense of the "always due to an ancilla" view of POVMs.

 

[Elias1960]

How can the wave function be not ontic when its dynamics determines the positions at future times?
Something nonexistent cannot affect the existent.

This happens in the objective Bayesian probability interpretation. There exists some reality, and there exists incomplete but nonetheless objective information about it. It defines a probability distribution - the one which maximizes entropy given the particular information.

If time changes and no new information appears, there will be dynamics - equations which derive that probability distribution for later times from that of the initial time.

There is already a well-developed version of thermodynamics based completely on this interpretation of probability. In this version, the entropy is not something really existing, but a function characterizing our incomplete knowledge of what really exists.

And it has been extended by Caticha to an interpretation of quantum theory too, based on the formulas of Nelsonian stochastics.

Caticha, A. (2011). Entropic Dynamics, Time and Quantum Theory, J Phys A 44:225303, arxiv:1005.2357

It assumes that there are, beyond the configuration q, also some other variables y (which may be simply the configuration of everything else, including the preparation device). Incomplete knowledge of all this is some \rho(q,y) Then one restricts this incomplete knowledge to knowledge about the system itself, integrating over this probability:
$\rho(q) = \int_{y\in Y} \rho(q,y) d y$
$S(q) = \int_{y\in Y} -\ln \rho(q,y) \rho(q,y) d y$
and then uses those two functions to define the wave function.

 

[A. Neumaier]

there exists incomplete but nonetheless objective information about it. It defines a probability distribution - the one which maximizes entropy given the particular information.

This is a very questionable statement.

What is the probability distribution of something of which the information is given that 5 times someone observed 1, twice 3 was observed, and once 6 was observed? One cannot maximize the entroy given this information. But all information one can gather about aspects of the universe is information of this kind.