Insights Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

[vanhees71]

I do not conflate the two. I'm talking about the meaning of the formalism, and that's probabilistic via Born's rule. All concepts related with the statistical meaning of the formalism are derived from Born's rule, including the trace rule for expectation values of observables. It even implies the probabilities for measurement outcomes, as is well known from standard probability theory.

Of course in measurements there is no Hilbert space, no operators, no trace rule, no Born's rule. You just measure observables and evaluate the statistics of their outcomes, take into account the specifics of the apparatus etc. There is no generally valid formalism for this but it has to be analyzed for any experimental setup. That's not what I'm discussing and it's not related to the interpretation of QT.

 

[A. Neumaier]

What's new is the order of presentation, i.e., it is starting from the most general case of "weak measurements" (described by POVMs)

Could you please point out to which paper (and which page) you refer here? I found no mention of weak measurements or POVMs in the geonium paper by Brown and Gabrielse that you mentioned earlier. The latter is quite interesting but very long, so it takes a lot of time to digest the details. I'll comment on it in due time in a new thread.

Maybe it would help, when a concrete measurement is discussed, e.g., the nowadays standard experiment with single ("heralded") photons (e.g., produced with parametric down conversion using a laser and a BBO crystal, using the idler photon as the "herald" and then doing experiments with the signal photon).

I discussed a different single photon scenario, that of ''photons on demand'', in a lecture given some time ago:

• A. Neumaier, Classical models for quantum light, Slides of a lecture given on April 7, 2016 at the University of Linz.

 

[vanhees71]

I was talking about the textbook by Peres, quoted in the posting you quote. Of course, Brown and Gabrielse use just standard quantum theory to discuss the physics, and obviously it works. There's no need for alternative interpretations than standard QT.

 

[A. Neumaier]

I do not conflate the two. I'm talking about the meaning of the formalism, and that's probabilistic via Born's rule. All concepts related with the statistical meaning of the formalism are derived from Born's rule, including the trace rule for expectation values of observables.

I am also talking about the meaning of the formalism, but using more careful language. I do this without invoking Born's rule, which you take to be a blanket phrase covering everything probabilistic, independent of its origin. This blurs the conceptual distinctions and makes it impossible to discuss details with you.

Of course in measurements there is no Hilbert space, no operators, no trace rule, no Born's rule.

In the mathematical formalism there is also no Born's rule, but only the trace rule defining quantum expectations. Born's rule only relates the trace rule to measurements, and it does so only in special cases - namely when measurements are made on independent and identically prepared ensembles.

As long as there are no measurements - and this includes everything in books on quantum mechanics or quantum field theory when they derive formulas for scattering amplitudes or N-point functions -, everything is independent of Born's rule. The formula $\langle A\rangle:=$Tr$\rho A$ is just a definition of the meaning of the string on the left in terms of that on the right. It has a priori nothing to do with measurement, and hence with Born's rule.

But it seems to me that you simply equate Born's rule with the trace rule, independent of its relation to measurement. Equating this makes trivially everything dependent on Born's rule. But this makes Born's rule vacuous, and its application to measurements invalid in contexts where no ensemble of independent and identically prepared ensembles. exist.

 

[A. Neumaier]

I was talking about the textbook by Peres, quoted in the posting you quote.

Please give a page number. If I remember correctly, Peres never mentions the notion of weak measurement. A search in scholar.google.com for

• author:Peres "weak measurement"

gives no hits at all.

 

[vanhees71]

It's simply not true! As shown in the book by Peres in a very clear way the Born rule is underlying also the more general cases of POVMs. All the experiments, leading to several Nobel prizes (Dehmelt, Wineland, Haroche,...), with single electrons, atoms, ions, etc. in traps has been analyzed in the standard way of quantum theory. The trace formula to calculate expectation values is a direct consequence of the probabilities predicted in the formalism of QT using Born's rule.

Once more the citation of Peres's book:

A. Peres, Quantum Theory: Concepts and Methods, Kluwer
Academic Publishers, New York, Boston, Dordrecht, London,
Moscow (2002).

I don't know, whether he uses the phrase "weak measurement", but he discusses POVMs and gives a very concise description of what's predicted by QT. It seems to be very much along the lines you propose in your paper (as far as I think I understand it).

 

[A. Neumaier]

I discussed a different single photon scenario, that of ''photons on demand'', in a lecture given some time ago.

I forgot to give the link:

• A. Neumaier, Classical models for quantum light, Slides of a lecture given on April 7, 2016 at the University of Linz.

 

[A. Neumaier]

It's simply not true! As shown in the book by Peres in a very clear way the Born rule is underlying also the more general cases of POVMs.

Yes, but he assumes everywhere stationary sources, i.e., ensembles of identically prepared systems. Moreover, he assumes unphysical mathematical constructs called ancillas to reduce POVM measurements on these ensembles to Born's rule.

All the experiments, leading to several Nobel prizes (Dehmelt, Wineland, Haroche,...), with single electrons, atoms, ions, etc. in traps have been analyzed in the standard way of quantum theory.

They use for their analysis a pragmatic approach (i.e., whatever gives agreement with experiments serves as interpretation), not one strictly based on Born's rule. The latter has essential restrictions to apply!

Once more the citation of Peres's book:

I don't know, whether he uses the phrase "weak measurement", but he discusses POVMs and gives a very concise description of what's predicted by QT. It seems to be very much along the lines you propose in your paper (as far as I think I understand it).

Yes, he discusses POVM in the usual, very abstract terms. But everywhere he assumes stationary sources, i.e., ensembles of identically prepared systems. Under this condition he gets the same as what I propose (with different assumptions, not assuming Born's rule).

Peres never discusses single quantum systems and does not use the term "weak measurement". In the Wikipedia reference I cited, the (standard) derivation of the quantum trajectories describing weak measurements only tells what the state is after a sequence of POVM measurements and what is the probability distribution for getting the whole sequence of results. To give meaning to this probability distribution via Born's rule one needs an ensemble of identically prepared systems giving an ensemble of sequences of measurement results! Otherwise one has only a single sequence of measurement results and the probability of getting this single one is 100%!

As we had discussed some years ago, Peres noticed (and does not resolve) this conflict when he enters philosophical discussions in the last chapter of his book (if I recall correctly, don't have the book at hand).

 

[vanhees71]

I forgot to give the link:

• A. Neumaier, Classical models for quantum light, Slides of a lecture given on April 7, 2016 at the University of Linz.

Yes, and I don't see anything contradicting the standard way to relate the formalism of QED to observations. I also think that the idea that $|\psi(t,\vec{x})|^2$ refers to some kind of intensity in Schrödingers first interpretation of the wave function was in analogy to the intensity of light, where it was known to be measured in terms of the energy density. This was however very soon be realized not to be in accordance with the detection of particles (particularly electrons) which indeed leave a single point on a photo plate and not a smeared distribution, and this brought Born to his probabilistic interpretation (in a footnote of his paper on scattering theory of 1926). Today we can use QED to derive that for the em. field the detection probability is indeed proportional to the expectation value of the energy density: It's just following from the first-order perturbation theory and the dipole approximation to describe the photo effect. The formula to evaluate these expectation values is of course based on Born's rule (or postulate). That's all in the standard textbooks about quantum optics and used also in the papers referring to experiments with single photons and/or entangled photon pairs, including all kinds of Bell tests, entanglement swapping, teleportation, and all that.

I still also don't see, why you think that collecting statistics by coupling a single quantum in a trap to a electrical circuit or repeated excitation-dexcitation events via the emitted photons, etc. cannot be understood with standard quantum theory although that's done for decades. Indeed, the many excitation-relaxation processes via an external laser field is defining the ensemble in this example. How else should you get statistics with a single quantum?

The realization of "weak measurements" and the description with the more general concept of POVMs is pretty recent, and as far as I can see, it's not something contradicting the fundamental Born postulate, how QT probabilities and expectation values are related to the formalism (statistical operators to represent the state and self-adjoint (or unitary) operators for observables).

 

[A. Neumaier]

All the experiments, leading to several Nobel prizes (Dehmelt, Wineland, Haroche,...), with single electrons, atoms, ions, etc. in traps have been analyzed in the standard way of quantum theory.

There is no standard way beyond pragmatism (anything successful goes) to do the matching of formalism to complex experiments.

From

• K. Gottfried, Does quantum mechanics carry the seeds of its own destruction? Physics World 4 (1991), 34--40.

My 'orthodoxy' is not identical to that of Bohr, nor to that of Peierls, to mention two especially eminent examples. Hence I must state my definition of 'orthodoxy'.

From

• D. Wallace, What is orthodox quantum mechanics? (2016).

Orthodox QM, I am suggesting, consists of shifting between two different ways of understanding the quantum state according to context: interpreting quantum mechanics realistically in contexts where interference matters, and probabilistically in contexts where it does not. Obviously this is conceptually unsatisfactory (at least on any remotely realist construal of QM) -- it is more a description of a practice than it is a stable interpretation. [...] The ad hoc, opportunistic approach that physics takes to the interpretation of the quantum state, and the lack, in physical practice, of a clear and unequivocal understanding of the state -- this is the quantum measurement problem.

The realization of "weak measurements" and the description with the more general concept of POVMs is pretty recent, and as far as I can see, it's not something contradicting the fundamental Born postulate

I never claimed a contradiction, just a non-applicability. One cannot derive from a postulate that only applies to large ensembles of independent and identically prepared systems any statement about a single system!

the many excitation-relaxation processes via an external laser field is defining the ensemble in this example.

If the processes are carried out identically, this indeed gives an ensemble of identically prepared photons. But if one only sends a handful of photons on demand to transmit a message (the primary reason why one would want to produce them on demand), one only gets an ensemble of not-identically prepared photons!

How else should you get statistics with a single quantum?

Through repeated measurements, with stochasticity induced by the unmodelled interaction with the environment. Just like in classical stochastic processes!

 

[vanhees71]

I don't know what you mean by sending a handful of photons on demand to transmit a message.

Usually one uses heralded photons to prepare single-photon states, which are in fact not so easy to produce (in contradistinction to "dimmed down coherent states", which however are not equivalent to true single-photon states but consist largely of the vacuum state). One way, nowadays kind of standard, is to shine with a laser on a BBO crystal and use the entangled photon pairs from parametric down conversion. Then you use one photon ("idler") to "herald" the other photon ("signal"), which you then use for experiments. This gives an ensemble of identically prepared single photons.

In the experiments with single atoms in a trap you usually use an external em. field to excite these atoms many times an measure the emitted photons. Another example with a single electron in a Penning trap is to measure the currents of the "mirror charges" in response to the motion of the electron in the trap, which also provides the statistics you need (see Dehmelt's or Brown's review papers quoted above).

 

[A. Neumaier]

Then you use one photon ("idler") to "herald" the other photon ("signal"), which you then use for experiments. This gives an ensemble of identically prepared single photons.

I agree. In this version nothing needs to be explained.

I was thinking of potential applications in quantum information processing, where the situation is different.

Another example with a single electron in a Penning trap is to measure the currents of the "mirror charges" in response to the motion of the electron in the trap

Which quantum observables of the electrons are measured by these currents? If Born's rule were involved, you should be able to point to the operators to which Born's rule is applied in this case.

 

[vanhees71]

You measure the energy differences via repeated spin flips of the electron in the trap via repeated two-photon excitations through a coherent rf field + thermal excitation (Fig. 5 +6 of Dehmelt's paper).

 

[A. Neumaier]

You measure the energy differences via repeated spin flips of the electron in the trap via repeated two-photon excitations through a coherent rf field + thermal excitation (Fig. 5 +6 of Dehmelt's paper).

This measurement recipe is not covered by Born's rule since there is no operator on the electron Hilbert space whose eigenvalues are the energy differences.

So how do you think Born's rule applies in this case?

 

[vanhees71]

Well, perhaps you should read the paper more carefully (or the relevant original papers quoted in that review). Here it's Ref. [18]:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.38.310

 

[A. Neumaier]

All the experiments, leading to several Nobel prizes (Dehmelt, Wineland, Haroche,...), with single electrons, atoms, ions, etc. in traps has been analyzed in the standard way of quantum theory. The trace formula to calculate expectation values is a direct consequence of the probabilities predicted in the formalism of QT using Born's rule.

I still also don't see, why you think that collecting statistics by coupling a single quantum in a trap to a electrical circuit or repeated excitation-dexcitation events via the emitted photons, etc. cannot be understood with standard quantum theory although that's done for decades. Indeed, the many excitation-relaxation processes via an external laser field is defining the ensemble in this example.

Another example with a single electron in a Penning trap is to measure the currents of the "mirror charges" in response to the motion of the electron in the trap

Which quantum observables of the electrons are measured by these currents? If Born's rule were involved, you should be able to point to the operators to which Born's rule is applied in this case.

You measure the energy differences via repeated spin flips of the electron in the trap via repeated two-photon excitations through a coherent rf field + thermal excitation (Fig. 5 +6 of Dehmelt's paper).

This measurement recipe is not covered by Born's rule since there is no operator on the electron Hilbert space whose eigenvalues are the energy differences.

So how do you think Born's rule applies in this case?

Well, perhaps you should read the paper more carefully (or the relevant original papers quoted in that review). Here it's Ref. [18]:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.38.310

I have my own interpretation of what is going on, and it does not involve Born's rule.

But you claimed that the experiment is (like all experiments with ion traps) explained by Born's rule. For your convenience and those of the other readers I collected the whole train of your arguments.

I am challenging you to provide a proof of your claim in this particular instance. If you can't do it in the simple case of measuring energy differences, your claim is without any substance!

 

[vanhees71]

It's on you to show that your bold claim that standard QT cannot be used to understand these experimental results explained usually by standard QT. Spectoscopy, i.e. the measurement of energy differences is the topic since day 1 of modern QT, described by standard QT (prediction of the frequencies and intensities of the em. radiation through transitions between atomic energy levels).

 

[Fra]

Through repeated measurements, with stochasticity induced by the unmodelled interaction with the environment. Just like in classical stochastic processes!

This is similar to other "inverse problems". For example i neuroscience, how can you get statistics from a single neuron, when you put an electrode into neural tissue? Then one basic method is clustering, where while the collected signal is influenced by many nearby single neurons, each single neurons (like each quantum state) has it's own "signature", which by clustering one can classify spikes that originate from the same neuron. But that's not an exact mathematical inverse though, it's always theoretically possible that you fail to resolve two neurons that just happened to have very similar signatures. But once clustereted, one collects statstics for single neurons.

/Fredrik

 

[A. Neumaier]

It's on you to show that your bold claim that standard QT cannot be used to understand these experimental results explained usually by standard QT. Spectoscopy, i.e. the measurement of energy differences is the topic since day 1 of modern QT, described by standard QT (prediction of the frequencies and intensities of the em. radiation through transitions between atomic energy levels).

So you finally agree that standard quantum theory involves more than Born's rule in order to relate the mathematical formalism to experiment!.

Indeed, standard quantum physics has a most pragmatic approach to the interpretation of the formalism: Anything goes that gives agreement with experiment, and Born's rule is just a tool applicable in some situations, whereas other tools (such as resonance observations or POVMs) apply in other situations.

Can we agree on that?

 

[A. Neumaier]

This is similar to other "inverse problems". For example i neuroscience, how can you get statistics from a single neuron, when you put an electrode into neural tissue? Then one basic method is clustering, where while the collected signal is influenced by many nearby single neurons, each single neurons (like each quantum state) has it's own "signature", which by clustering one can classify spikes that originate from the same neuron. But that's not an exact mathematical inverse though, it's always theoretically possible that you fail to resolve two neurons that just happened to have very similar signatures. But once clustered, one collects statistics for single neurons.

Yes. The estimation of constants in mathematical models of reality (whether a growth parameter in a biological model or a gyrofactor in a model of a Penning trap) from noisy measurements is always an inverse problem.

 

[vanhees71]

So you finally agree that standard quantum theory involves more than Born's rule in order to relate the mathematical formalsim to experiment!.

Indeed, standard quantum physics has a most pragmatic approach to the interpretation of the formalism: Anything goes that gives agreement with experiment, and Born's rule is just a tool applicable in some situations, whereas other tools (such as resonance observations or POVMs) apply in other situations.

Can we agree on that?

Sure, but the Born rule is one of the basic postulates that's behind all these pragmatic approaches. In Newtonian mechanics there's also much more then the "three laws" to apply it to the analysis of the pgenomena, but they are behind all the corresponding methods.

 

[A. Neumaier]

Sure, but the Born rule is one of the basic postulates that's behind all these pragmatic approaches.

behind all these??

How is Born's rule behind the measurement of the energy difference of two levels of a quantum system?

 

[vanhees71]

You meassure it, e.g., by burning hydrogen and measure the wave lengths of the light using a grating. For that you accumulate a lot of photons. The intensity of the different lines is predicted using Born's rule.

 

[A. Neumaier]

You measure it, e.g., by burning hydrogen and measure the wave lengths of the light using a grating.

But nothing is burnt in a Penning trap, which is the example under discussion.

For that you accumulate a lot of photons. The intensity of the different lines is predicted using Born's rule.

But the intensity of the lines gives no information at all about the energy differences.

The frequency of the photons emitted by the trapped electron, and hence the determination of the position of the resonance peaks from which the high precision gyrofactor is computed is independent of Born's rule.

 

[vanhees71]

The evaluation of experimental data never uses the quantum formalism. The interpretation of these data of an electron in a Penning trap, however, and thus the "mapping" of measured "beat frequencies" and their mapping to the value of $(g-2)$ is based on the QT formalism. This is of course not directly related to Born's rule but to the evaluation of the energy eigenvalues of the electron in the trap (the "geonium"). Even the noisy signal should be predictable by QT and that's then again based on Born's rule.

 

[A. Neumaier]

The evaluation of experimental data never uses the quantum formalism. The interpretation of these data of an electron in a Penning trap, however, and thus the "mapping" of measured "beat frequencies" and their mapping to the value of $(g-2)$ is based on the QT formalism.

More precisely, on the quantum formalism without Born's rule.

This is of course not directly related to Born's rule

It is of course not related at all to it.

but to the evaluation of the energy eigenvalues of the electron in the trap (the "geonium"). Even the noisy signal should be predictable by QT

It is, again by the quantum formalism without Born's rule.

and that's then again based on Born's rule.

Only because you apply again your magic wand that turns every quantum calculation into an instance of Born's rule.

 

[vanhees71]

I think, we won't find an agreement on the status of Born's rule, which I still consider as one of the foundational postulates of QT, without which there's no interpretation at all. Of course, the frequencies in spectroscopy are differences of energy levels and not directly related with Born's rule but rather to the quantum dynamics, usually derived by 1st-order time-dependent PT in the dipole approximation for (spontaneous and induced) photon emission, i.e., based on the dynamical laws and the meaning of the Hamiltonian. The transition probabilities, also obtained in this same calculation, are of course based on Born's rule.

I stop this discussion at this point.

 

[A. Neumaier]

Of course, the frequencies in spectroscopy are differences of energy levels and not directly related with Born's rule but rather to the quantum dynamics, usually derived by 1st-order time-dependent PT in the dipole approximation for (spontaneous and induced) photon emission, i.e., based on the dynamical laws and the meaning of the Hamiltonian.

I agree. No interpretation is needed for this; it predates Born's rule by at least a year.

But it shows that Born's rule doesn't explain quantum measurements of a spectroscopic nature. The latter includes all high precision determinations of constants of Nature such as gyrofactors, mass ratios, etc.

The transition probabilities, also obtained in this same calculation, are of course based on Born's rule.

I agree. But this is independent of the value of the frequencies, and only the latter are measured in the experiment under discussion.

 

[vanhees71]

Of course it's not only Born's rule but all the other postulates of QT too. Of course the predictions of "old QM" a la Bohr and Sommerfeld concerning atomic spectra predate "new QM" and thus also Born's rule by more than a decade (though they are all wrong except for hydrogen ;-)).

 

[bhobba]

Do I miss something?

Probably not. Less formally a Von-Neumann observation is represented by disjoint positive valued operators Ei such that sum Ei = 1. A POVM is simply a generalisation that removes the need to be disjoint. It turns out Gleason's Theorem is much easier to prove for POVM's. In practice, they occur when for example you observe a system with a probe then observe the probe. See for example:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

Thanks
Bill