I Exploring Measurements in Quantum Field Theory: From Light Cones to Bell Tests

[Delta Kilo]

I don't think there is a measurement problem. Q(F)T works beautifully.
How do you define what constitutes a "measurement"?

For the QM it is defined in a usual way, as per postulates, as the "R" process of wavefunction reduction whereupon wavefunction collapses into one of the eigenstates of observable operator, as opposed to "U" process of unitary evolution. For example this is from Griffiths:

We say that the wave function collapses upon measurement <...> There are, then, two entirely distinct kinds of physical processes: "ordinary" ones, in which the wave function evolves in a leisurely fashion under the Schrodinger equation, and "measurements", in which $\Psi$ suddenly and discontinuously collapses.

Other people like Ballentine sidestep the issue by refusing to deal with individual systems but only with ensembles of identically prepared systems. Yet others say nothing except macroscopic results is real but just some math tricks that just happen to work and give right predictions.

It's actually hard to define exactly what the measurement problem is because it is manyfold and different interpretations tend to solve some parts while glossing over the others. Some examples:
* Criteria for choosing R process over U process (why some interactions are measurements while others are not).
* Measurement apparatus not fully described by QM, need for quantum/classical cut
* Explaining the apparent single outcome when measuring superposition.
* Explaining/deriving Born rule

I know there's been a great deal of progress in decoherence, einselection etc. and it does answer some questions but apparently not all of them.

I did not include any of the Bell test / spooky-action-at-a-distance issues because I feel a lot of them are due to limitations of QM being non-relativistic and non-local by design. Like a perfectly rigid rod in Minkowsi space. It is my hope that QFT can do better in this regard.

 

[A. Neumaier]

Photons are exactly what travel from a source to a detector. They are "real" in any sense I can think of, and are not merely mathematical abstractions. Whether or not they follow a single specific path is a different question.

A single photon ( an object in an eigenstate of the photon number operator with eigenvalue 1) may be considered to be real in any sense. But in the entangled case, it is not two photons that travel but a single two-photon object (an object in an eigenstate of the photon number operator with eigenvalue 2) travels. This is clear from the way quantum mechanics is argued to violate Bell inequalities.

One cannot separate the two-photon object into two photons. Therefore the photons in entangled photons are quite unreal. What is real is just the detector clicks that are interpreted of having recorded a corresponding number of photons.

 

[Lord Jestocost]

It's actually hard to define exactly what the measurement problem is because it is manyfold and different interpretations tend to solve some parts while glossing over the others.

A so-called measurement problem exists only for those who cling with ferocity to the assumption that the "state vector" is a representation of some reality behind the phenomena.

 

[A. Neumaier]

A so-called measurement problem exists only for those who cling with ferocity to the assumption that the "state vector" is a representation of some reality behind the phenomena.

For those who don't cling to this assumption, a much more severe problem is that of connecting in an objective way the subjective probabilities associated with the vague psychological concept of knowledge to physical reality.

Unless they regard physics as a subdiscipline of psychology, they need to define precisely how and under which objective conditions this subjective knowledge changes.

 

[Delta Kilo]

Thinking about measurements in context of QFT and MWI, I believe there is a way to consistently handle entangled measurements with no spooky actions. Consider typical Bell test. There is a diamond arrangement OABC with 2 particles (say photons) starting at O, measured at spacelike separated A and B, and the results are sent to C, equidistant from A and B, where they are compared.

Let's assume that in a small vicinity of each of the vertices OABC, QFT can be approximated by ordinary QM, but there is a limit of $c$ on propagating speed of any disturbances. So we do get a wave function locally but it can go somewhat out of sync over large distances. Yes, and each wavefunction is assumed to have $|the\ rest\rangle$ appended, which absorbs all other dimensions we don't care about.

So the two photons start at $O$ in a state:
$$|\phi_O\rangle = \sum_{ij} c_{ij} |p_a^i\rangle p_b^j\rangle$$
Photon $p_a$ flies to $A$ where it interacts with $A$:
$$|\phi_A \rangle= |\phi_O\rangle |A\rangle \rightarrow |\phi_A \rangle=\sum_{ijk} a_{ijk} |p_a^i\rangle p_b^j\rangle|A^k\rangle$$
where we get $a_{ijk}$ by converting $p_a$ to $A$'s measurement basis $|m_a^k\rangle$ and back:
$$a_{ijk} = \sum_n c_{nj} \langle p_a^i|m_a^k\rangle \langle m_a^k|p_a^n\rangle $$
and because $|A^k\rangle$ are macroscopic states involving lots of degrees of freedom, the world is split locally, with probabilities of each branch:
$$P(A^k) = \sum_{ij}\left\|a^{ijk}\right\|^2$$
Ditto for $|\phi_B \rangle$:
$$|\phi_B \rangle=\sum_{ijl} b_{ijl} |p_a^i\rangle p_b^j\rangle|B^l\rangle$$
When both results meet at $C$ the results are combined. We can treat is as measurement of $|\phi_A\rangle$ in basis of $|\phi_B^l\rangle$ or $|\phi_B\rangle$ in basis of $|\phi_A^k\rangle$, the result is the same:
$$|\phi_C\rangle = \sum_{ijkl} a_{ijk} b_{ijl} |p_a^i\rangle p_b^j\rangle|A^k\rangle|B^l\rangle$$
with the world split 4-ways:
$$P(A^kB^l) = \sum_{ij}\left\|a_{ijk} b_{ijl}\right\|^2$$
Eventually the split will propagate outwards from C back to A and B and everyone gets a consistent picture of events. If we set photons to be maximally entangled, for example $c_{ij} = \delta_{ij}$ and the values for $\langle m_a^k|p_a^i\rangle$ and $\langle m_b^l|p_b^j\rangle$ appropriate for Bell test angles, we'll get 50% for $P(A^k)$ and $P(B^l)$ and expected probabilities of coincidences $\sum_n P(A^nB^n)$.

 

[Delta Kilo]

A so-called measurement problem exists only for those who cling with ferocity to the assumption that the "state vector" is a representation of some reality behind the phenomena.

Which specific implementation do you have in mind? There are some "ostrich" interpretations where pretty much nothing is real, not sure how it helps.
--
The Great Collapsing Hrung Disaster of Gal./Sid./Year 03758 which wiped out all the old Praxibetel communities on Betelgeuse VII is shrouded in deep mystery; in fact, no one ever knew what a Hrung was nor why it had chosen to collapse on Betelgeuse VII particularly.

 

[DrChinese]

I did not include any of the Bell test / spooky-action-at-a-distance issues because I feel a lot of them are due to limitations of QM being non-relativistic and non-local by design. Like a perfectly rigid rod in Minkowsi space. It is my hope that QFT can do better in this regard.

A detail to add to your "rod" analogy: You can entangle Photon A with Photon B, where: i) A has already been measured and no longer exists; ii) B does not yet exist at the time that A is measured; iii) A and B don't need to share any past light cone; and optionally iv) the entanglement can occur after B is measured.

Entanglement Between Photons that have Never Coexisted (2012)

So that rod can have both space-like and time-like extent, and maybe even some offshoots.

A single photon ( an object in an eigenstate of the photon number operator with eigenvalue 1) may be considered to be real in any sense. But in the entangled case, it is not two photons that travel but a single two-photon object (an object in an eigenstate of the photon number operator with eigenvalue 2) travels. This is clear from the way quantum mechaics is argued to violate Bell inequalities.

One cannot separate the two-photon object into two photons. Therefore the photons in entangled photons are quite unreal. What is real is just the detector clicks that are interpreted of having recorded a corresponding number of photons.

This is a good point. I often refer to systems of entangled pairs as "biphotons" just to drive that home. A couple of minor issues though:

a) Where and when does the biphoton become 2 separate photons (assuming you think they become separate objects by the point in time they are detected)?

b) What do you call an entangled system composed of 2 electrons? I guess you could call it a "bielectron". I would note that the entanglement is ultimately verified by photon detection. But the electrons were entangled. And - at least in principle - you could entangle completely different particle types.

Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km (2015)
"We add one particle, for example an electron, to each box. The spin degree of freedom of the electron forms a two level system with eigenstates |↑> and |↓>. For each trial, the two spins are prepared into the entangled state |ψ−> = (|↑↓> − |↓↑>) /√2."

 

[DrChinese]

Eventually the split will propagate outwards from C back to A and B and everyone gets a consistent picture of events.

You realize, I suppose, that an explanation such a this automatically rules out the possibility of acknowledging the existence of action at a distance (or quantum nonlocality or whatever you want to call it) - even if that actually exists. Basically you are saying: "my uncle is not dead until I learn about it".

In other words: You are denying that events recorded at A and B are "real" until those results are brought together at C (and re-broadcast). That automatically eliminates all proofs of AAD, because the A/B results must be transmitted to C at a maximum of light speed. No form of AAD - assuming that signal locality itself is respected - could ever pass that test! Call it what you will, that's circular reasoning.

A so-called measurement problem exists only for those who cling with ferocity to the assumption that the "state vector" is a representation of some reality behind the phenomena.

Not sure how much "ferocity" is needed to believe there is reality to the "state vector", "quantum state", "wave function", "density matrix" or whatever you want to call it. After all, there is that PBR thing. :smile" Does the quantum state represent some reality (ontic) ? Or does it represent only our knowledge of reality (epistemic) ?

On the reality of the quantum state (2011)
Matthew F. Pusey, Jonathan Barrett, Terry Rudolph
"Here we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory."

Experimental test of the no-go theorem for continuous ψ-epistemic models (2016)
"Here we experimentally test this theorem with high-dimensional single photon quantum states without additional assumptions except for the fair-sampling assumption. Our experimental results reproduce the prediction of quantum theory and support the no-go theorem."

There is controversy about their conclusions (which are denied by those whose viewpoints differ from the results of these papers). But I wouldn't say that someone who concurs with PBR (and its experimental realizations) is swimming against the stream. I follow their basic viewpoint, so you might say I "cling" to the idea: the measurement problem is an issue yet to be fully understood.

 

[WernerQH]

It's actually hard to define exactly what the measurement problem is

Indeed. Although QM is nearly a century old and extremely successful, quite a few physicists are unhappy about it, and the number of proposed interpretations is still rising. There seems to be no issue with the stationary solutions of the time-independent Schrödinger equation. I think everybody agrees that they are time-averaged, statistical representations of something real that's going on in the atom. But the time-dependent Schrödinger equation is a different story. It seems to suggest continuous and deterministic evolution, as if the world fundamentally still were Newtonian. It is obvious that this doesn´t square with the abruptness and randomness that we observe in the real world. A free neutron doesn´t gradually turn into a proton over the timescale of minutes, but in a fraction of less than a microsecond (the time it takes the neutrino to leave the laboratory). So it´s clear, except to MWI-adherents, that the wave function cannot be the whole story. The "R" process and the Born rule were added to make contact with the real world. But a neutron will decay even if it is not "measured", and a lot of hand-waving is invoked that the environment causes decoherence and can thus be seen as performing some kind of measurement. I think it's clearer to accept that QFT is a statistical, stochastic theory.

* Explaining the apparent single outcome when measuring superposition.

Yes, a lot of explaining is necessary if you believe that an individual system must be described by a wave function at all times. But, as I have already indicated, I cannot believe that the time-dependent wave function can be a faithful representation of something that happens in the real world (e.g. in the laboratory or in the interior of the sun). The wave function is a mathematical object accounting for things that could happen, not for the events that actually unfold.

* Explaining/deriving Born rule

I think the Born rule is an independent ingredient of quantum theory, and cannot be derived. (Except, of course, from another, equivalent postulate.) Rather than focusing on the wave function as the key to quantum theory, one should turn to the statistical operator (density matrix) that allows writing down (observable!) correlation functions (expectation values) directly. I can see no need to talk about measurements and collapsing wave functions.

 

[A. Neumaier]

a) Where and when does the biphoton become 2 separate photons (assuming you think they become separate objects by the point in time they are detected)?

When a detector clicks, the photon number is reduced by 1, and a single photon is left. When two detectors click simultaneously (which happens only with probability zero for an exact temporal coincidence) then the photon number is reduced by 2, and nothing is left.

b) What do you call an entangled system composed of 2 electrons? I guess you could call it a "bielectron". I would note that the entanglement is ultimately verified by photon detection. But the electrons were entangled. And - at least in principle - you could entangle completely different particle types.

It is a single bilocal object in an entangled 2-electron state.

 

[Delta Kilo]

You realize, I suppose, that an explanation such a this automatically rules out the possibility of acknowledging the existence of action at a distance (or quantum nonlocality or whatever you want to call it) - even if that actually exists. Basically you are saying: "my uncle is not dead until I learn about it".

Well, yes. Until the wavefront of environmental decoherence reaches you, you remain unaware. That follows from MWI + relativity.

In other words: You are denying that events recorded at A and B are "real" until those results are brought together at C (and re-broadcast). That automatically eliminates all proofs of AAD, because the A/B results must be transmitted to C at a maximum of light speed. No form of AAD - assuming that signal locality itself is respected - could ever pass that test! Call it what you will, that's circular reasoning.

No, it's not like that. Measurement at A is perfectly real for an observer at A from the moment the measurement is done and it remains real and does not change. It is also real for all observers in the future cone of A defined by the speed of decoherence $\leq c$. Same for B. But only the observes in intersection of those future cones are aware of both and able to compare them.
"Rebroadcast" from B,C back to A is not essential to the picture. I only mentioned it to show that once it is complete all observers have the same idea about what is going on, before that the observes at different locations may be out of sync.
And yes, there are no actions at a distance in this picture and I don't see it as a problem. The probabilities $P(A)$, $P(B)$ and $P(AB)$ come out right at the end, violating Bell's inequality as expected.

 

[PeterDonis]

there are no actions at a distance in this picture

Only because you are not counting the change in the wave function as "action at a distance". But the wave function itself is nonlocal in these experiments; it includes entangled degrees of freedom that are spatially separated, and the local interaction that operates on one of those degrees of freedom takes the other entangled degrees of freedom along with it in the wave function.

 

[Delta Kilo]

But the time-dependent Schrödinger equation is a different story. It seems to suggest continuous and deterministic evolution, as if the world fundamentally still were Newtonian. It is obvious that this doesn´t square with the abruptness and randomness that we observe in the real world.

I'm not at all put off by randomness and abruptness. There are plenty of sources of randomness and chaos in classical theory already. And the role of environment should not be underestimated. Penrose gave this example: even if we knew exact coordinates and momenta of all particles in a parcel of air here on Earth, moving 1Kg by 1m on Alpha Centauri (4 years ago) would completly scramble their trajectories in a very short time.

Here we are measuring a single photon using a device made up from great many particles in unknown state. And all these particles are presumably heavily entangled between themselves and with the environment so I can't even begin to imagine the number of individual degrees of freedom involved, it's too scary.

Actually my biggest issue with QM is the amount of information required to completely describe a system. One would need $2^{10^{23}}$ coefficients to describe a mole of qubits
This is why MWI appeals to me: it puts all these numbers to good use

 

[Delta Kilo]

Only because you are not counting the change in the wave function as "action at a distance". But the wave function itself is nonlocal in these experiments; it includes entangled degrees of freedom that are spatially separated, and the local interaction that operates on one of those degrees of freedom takes the other entangled degrees of freedom along with it in the wave function.

Yes it does take another entangled degree for a ride and carry it around but it does not travel faster than $c$.

This is not exactly standard QM. There are two assumptions: first, that relativistic quantum theory can be locally approximated by standard QM in small enough neighbourhoods of points O,A,B,C and second, that for large enough distances nothing travels faster than $c$. Kind of piecewise linear approximation.
So the wavefunctions at points OABC are similar but different. For example wavefunction at B feels the effects of measurement done at A in due time and vice versa.

Yes, I am first to admit there is a lot of handwaving. This is why I'd like to see it done properly in the context of fully relativistic QFT.

 

[PeterDonis]

it does take another entangled degree for a ride and carry it around but it does not travel faster than $c$.

There is no concept of "speed of travel" involved at all. The wave function is nonlocal; it just updates instantly whenever any entangled degree of freedom is acted on. There is no such thing as one degree of freedom "carrying along for the ride" another. The wave function is not a function in ordinary space.

This is not exactly standard QM.

Then you need to give a reference for where you are getting whatever interpretation you are using. You can't just make one up; that's off limits here. I am describing my understanding of the MWI.

So the wavefunctions at points OABC are similar but different. For example wavefunction at B feels the effects of measurement done at A in due time and vice versa.

There is nothing in the math that corresponds to this.

Yes, I am first to admit there is a lot of handwaving.

Not handwaving, just incorrectly describing what the math says.

This is why I'd like to see it done properly in the context of fully relativistic QFT.

I don't know of any accepted fully relativistic treatment of the MWI. If you want to say that's a missing piece of the MWI, that's fine, but it still doesn't mean you can just make things up.

 

[Delta Kilo]

There is no concept of "speed of travel" involved at all. The wave function is nonlocal; it just updates instantly whenever any entangled degree of freedom is acted on. There is no such thing as one degree of freedom "carrying along for the ride" another. The wave function is not a function in ordinary space.

That's only in NRQM, presumably not in QFT, there is no such thing as "instantly". I'm specifically considering relativistic setting here.

Then you need to give a reference for where you are getting whatever interpretation you are using. You can't just make one up; that's off limits here. I am describing my understanding of the MWI.

I'm pretty sure I've seen something along these lines, but I won't be able to provide a reference. I'm sorry for that. If you feel that post was not appropriate, let's disregard it, I won't insist. I can delete it if you want.
However I hope I'm allowed to discuss the two assumptions at the start of that post, or is it going to be considered original research too? Do you think these assumptions are sensible?

There is nothing in the math that corresponds to this.

That was a direct consequence of the two assumptions I stated at the start of that post.

I don't know of any accepted fully relativistic treatment of the MWI. If you want to say that's a missing piece of the MWI, that's fine, but it still doesn't mean you can just make things up.

All I'm saying is I'd love to see a properly done full relativistic treatment of the scenario I described. I haven't seen one yet. That was just my understanding which direction it might be going but I might be wrong and it may not work out. In any case I believe standard non-relativistic QM comes to the limit of its applicability here and that's where all the paradoxes and spooky actions come from.

 

[PeterDonis]

That's only in NRQM, presumably not in QFT

Which is why I made a point of saying that AFAIK there is no accepted relativistic version of the MWI. So if you're going to talk about the MWI, then this...

I'm specifically considering relativistic setting here.

...is really off topic here since there is no valid reference you can use to tell you what the MWI says in such a setting.

I hope I'm allowed to discuss the two assumptions at the start of that post, or is it going to be considered original research too?

Without a valid reference, yes.

I believe standard non-relativistic QM comes to the limit of its applicability here

If by "here" you mean "in experiments where Bell inequality violations in measurements of entangled systems are observed", non-relativistic QM makes the same predictions as QFT in this domain because relativistic effects are negligible, so saying that non-relativistic QM "comes to the limit of its applicability" is quite a stretch. People who favor particular interpretations of QM might be disappointed that relativity and QFT don't seem to play any role in the actual predictions, but that's a problem with those interpretations, not with either NRQM or QFT.

 

[PeterDonis]

I'm interested in how measurement, entanglement, bell test etc are handled in QFT.

As my previous post will show you, the short answer is, "the same way they are handled in NRQM". As I noted, relativistic effects are negligible in these experiments, so NRQM makes the same predictions as QFT. Which means nobody bothers to go to all the trouble of using QFT to make the predictions, since NRQM is a lot easier mathematically.

What you really appear to be after is a relativistic interpretation, i.e., an interpretation that tells you a story about how Bell inequality violations and other related effects are produced that seems to you to not violate the spirit of relativity. (The letter of relativity is already known not to be violated in these experiments: you can't transmit actual information FTL, and the measurements commute so there are no causal ordering issues.) Unfortunately, as I have said, AFAIK no such relativistic interpretation exists.

 

[Delta Kilo]

Well, let's just agree to disagree. I still think that "instantaneous update of wavefunction in spacelike separated regions" is a proverbial perfectly rigid rod in Minkowski space. It does not exist, it is an oxymoron.
And "nothing to see here, move along" attitude is not going to help solve these issues either.

 

[PeterDonis]

let's just agree to disagree.

Which, as the guidelines for this subforum will tell you, is what you should expect to happen in many, if not most (if not all) QM interpretation discussions at our current state of knowledge.

I still think that "instantaneous update of wavefunction in spacelike separated regions" is a proverbial perfectly rigid rod in Minkowski space.

Which means you do not like interpretations that require this. Which is fine. You just need to realize that it's a matter of interpretation and opinion; nobody has a way to resolve such issues by experiment at our current state of knowledge.

"nothing to see here, move along" attitude

Is not the attitude I'm taking. I'm just making clear what we do not currently know. That's because you appear to be looking for something that does not currently exist. That is not at all the same as saying you can't go out and try to build it. The latter effort is off topic here at PF, since PF is not for doing original research. But it is on topic to point out that, at our current state of knowledge, original research is what is required to produce what you appear to be looking for.

 

[DrChinese]

When a detector clicks, the photon number is reduced by 1, and a single photon is left.

Thanks for your straightforward answer. Not trying to debate that answer (which is good), or to be clever or anything. Just wondering about these follow-up questions, if you care to consider.

a) In Bell tests, there are usually polarizing beam splitters before the detector. Is it still a biphoton after the PBS and before the detectors?

[Moderator's note: question b) and the responses it generated have been spun off into a separate thread here.]

(Actually simply looking for your take on this, and nothing more.)

 

[A. Neumaier]

Thanks for your straightforward answer. Not trying to debate that answer (which is good), or to be clever or anything. Just wondering about these follow-up questions, if you care to consider.

a) In Bell tests, there are usually polarizing beam splitters before the detector. Is it still a biphoton after the PBS and before the detectors?

If the state of the biphoton has a nonzero component in the polarizing direction, yes. If it is orthogonal to the polarizing direction, the particle number reduces by 1.

 

[DrChinese]

If the state of the biphoton has a nonzero component in the polarizing direction, yes. If it is orthogonal to the polarizing direction, the particle number reduces by 1.

Thanks, answers my question nicely.

 

[Lord Jestocost]

A so-called measurement problem exists only for those who cling with ferocity to the assumption that the "state vector" is a representation of some reality behind the phenomena.

For those who don't cling to this assumption, a much more severe problem is that of connecting in an objective way the subjective probabilities associated with the vague psychological concept of knowledge to physical reality.

Unless they regard physics as a subdiscipline of psychology, they need to define precisely how and under which objective conditions this subjective knowledge changes.

The entire formalism of QM is a computational recipe for predicting the probabilities of various observable macroscopic outcomes. Unlike classical probability, however, the quantum probability is not the probability of where - let's say for example - an electron is. It is the objective probability of where you (or anyone else) will find it in a particular experimental context. What the heck has this to do with pychology?

 

[A. Neumaier]

The entire formalism of QM is a computational recipe for predicting the probabilities of various observable macroscopic outcomes. Unlike classical probability, however, the quantum probability is not the probability of where - let's say for example - an electron is. It is the objective probability of where you (or anyone else) will find it in a particular experimental context. What the heck has this to do with pychology?

Psychology enters because the wave function is updated when the knowledge of the observer changes. Knowlege is a psychological property of an observer, not a physically well-defined state of the latter.

 

[PeterDonis]

Moderator's note: A subthread about temporal entanglement has been moved to its own thread:

https://www.physicsforums.com/threads/measurements-in-qft.1059303/post-6990834

I have also added a moderator's note in post #51 of this thread to show where the split begins.

 

[PeterDonis]

@DrChinese I have moved your latest post to the new thread on temporal entanglement. See the link in post #56.

 

[DrChinese]

@DrChinese I have moved your latest post to the new thread on temporal entanglement. See the link in post #56.

Thanks!