Issues on notation and concept of entanglement

[entropy1]

As a follow up from my other thread, where I consider popular media describe entanglement sort of as:

IF particle A is found to be spin-up, "we know that" particle B "has" spin-down.

and I think this may be wrong.

As a follow up question I want to put forward this: A singlet state of entangled particles is notated in a superposition of product states as: $|up, down \rangle - |down, up \rangle$. It is not clear to me if this singlet state describes measurement outcomes or wavefunctions. Nevertheless, when the measurement is done, it represents both measurement outcome and wavefunction.

So I think this could cause some confusion to less informed people, like me.

Suppose the first half of the product state applies to Alice's measurement, and the second half to Bob's measurement. Then IF Alice measures spin up THEN Bob measures spin down, and IF Bob measures spin down THEN Alice measures spin up. But the singlet state is a general state that describes more than just measurements that are made in an identical angle of measurement. If the angles of measurement of Alice and Bob are NOT aligned (e.g. not parallel), THEN IF Alice measures spin up, THEN Bob may measure spin down OR spin up. But if we assume that IF Alice measures spin up, the particle that Bob measures instantly takes on spin value "down", THEN the math gives the correct probability of Bob measuring spin up of down.

There are problems with this interpretation. First of all, if Alice and Bob's measurements are spacelike separated, there is no "instant" interaction, AND second, there is no FIRST or SECOND measurement! Furthermore, and I feel perhaps the most significant issue, it is not clear if the spin states ('up' or 'down') refer to measurement outcomes or wavefunctions. It can't be measurement outcomes. However, the product state notation suggests that IF Alice measures spin up, Bob will measure a spin as if his particle of the entangled pair has value spin down (before measurement). Vice versa, IF Bob measures spin down, Alice will measure a spin as if her particle has value spin up (before measurement). So this does not al all make clear who is determining the ontological state of the particle of who.

However, assuming that Bob's particle takes on a spin value before he measures it gives the right answers. But it can't be really happening. And this appears to me as a source of confusion often made in popular media on entanglement, and understandably so, because physisists are using this product state notation exactly this way!

So I am curious who disagrees with something I wrote here, and if someone could layout how, in this Dirac product state notation (or other), there can be made sense of entanglement, in popular media, and also in science.

 

[Morbert]

IF particle A is found to be spin-up, "we know that" particle B "has" spin-down.

A weaker more interpretation-neutral statement would be "If Alice records spin-up for particle A, then we know that Bob, if he makes a measurement, will record (or will have recorded) spin-down for particle B". This statement does not assert any actually-existing microscopic properties of the particle; only what they might imprint on macroscopic apparatus. This avoids the pitfall you bring up later re/spacelike-separated variables, and a stronger statement about the particle rather than the observation brings you into the weeds of interpretation.

As a follow up question I want to put forward this: A singlet state of entangled particles is notated in a superposition of product states as: . It is not clear to me if this singlet state describes measurement outcomes or wavefunctions. Nevertheless, when the measurement is done, it represents both measurement outcome and wavefunction.

The actually existing outcome $i$ is e.g. the reading on a dial. A quantum theory at the very least makes contact with an actually existing outcome with a probability that the outcome occurs $p(i)$. Our expectations for what outcomes occur and their likelihoods are informed by other outcomes, and the quantum state codifies this information (see e.g. Born's rule, Gleason's theorem etc). This is the minimum commitment demanded by orthodox QM. You could therefore say something like "The quantum state of a system can be expressed as the trace-class operator that gets you the right probabilities and expectation values for outcomes of measurements that can be performed on the system".

 

[entropy1]

"If Alice records spin-up for particle A, then we know that Bob will record (or will have recorded) spin-down for particle B"

I would not instantly agree because of:

If the angles of measurement of Alice and Bob are NOT aligned (e.g. not parallel), THEN IF Alice measures spin up, THEN Bob may measure spin down OR spin up.

The actually existing outcome $i$ is e.g. the reading on a dial. A quantum theory at the very least makes contact with an actually existing outcome with a probability that the outcome occurs $p(i)$. Our expectations for what outcomes occur and their likelihoods are informed by other outcomes, and the quantum state codifies this information (see e.g. Born's rule, Gleason's theorem etc). This is the minimum commitment demanded by orthodox QM.

I fully embrace the notion that it all agrees with the math, but I see ambiguity in the interpretation of that math concerning entanglement in Dirac notation, namely what are we talking about: measurement outcomes or (ontological) wave functions. I am not sure if "the math says so" is enough to accept those ambiguities, because we don't know what we are talking about.

 

[phinds]

A weaker more interpretation-neutral statement would be "If Alice records spin-up for particle A, then we know that Bob, if he makes a measurement, will record (or will have recorded) spin-down for particle B".

A statement that I find less confusing is "whichever of Alice and Bob is the second to make a measurement will get the opposite result of that gotten in the first measurement by the other".

 

[Morbert]

For posterity:

I would not instantly agree because of: "If the angles of measurement of Alice and Bob are NOT aligned (e.g. not parallel), THEN IF Alice measures spin up, THEN Bob may measure spin down OR spin up. "

Yes I was referring to a measurement where the spins are aligned.

 

[entropy1]

For posterity:

Yes I was referring to a measurement where the spins are aligned.

As far as I know the singlet notation is more general than that.

 

[Morbert]

As far as I know the singlet notation is more general than that.

Yes. The state permits you to make those kinds of statements (and assign them probabilities). But it is not reducible to anyone statement of that kind.

 

[entropy1]

Yes. The state permits you to make those kinds of statements (and assign them probabilities). But it is not reducible to anyone statement of that kind.

Still, if Alice measures her particle, Bob can measure either spin up or spin down if measuring along a different axis than Alice? So the suggestion that $|up, down \rangle$ applies solely to measurement, does not hold in my view.

 

[vanhees71]

A statement that I find less confusing is "whichever of Alice and Bob is the second to make a measurement will get the opposite result of that gotten in the first measurement by the other".

The temporal order of the measurements doesn't play a role. The measurement events can be space-like separated and still you have the 100% correlation between the outcomes of measuring the spin components (in the same direction of course).

 

[phinds]

The temporal order of the measurements doesn't play a role.

Which is exactly what I said.

 

[Demystifier]

First of all, if Alice and Bob's measurements are spacelike separated, there is no "instant" interaction,

Are you sure about that? The Bell theorem seems to say the opposite. Experts do not agree on what exactly the Bell theorem says, but the conclusion of Bell himself was that there is some kind of instant interaction.

 

[vanhees71]

Bell's model is not Q(F)T. In relativistic local QFTs by construction there are no causal connections between space-like separated events.

 

[martinbn]

Are you sure about that? The Bell theorem seems to say the opposite. Experts do not agree on what exactly the Bell theorem says, but the conclusion of Bell himself was that there is some kind of instant interaction.

What is your definition of interaction?

 

[entropy1]

Are you sure about that? The Bell theorem seems to say the opposite. Experts do not agree on what exactly the Bell theorem says, but the conclusion of Bell himself was that there is some kind of instant interaction.

It needn't be really happening, but if you assume it does, you get the statistically correct outcome.

I think that because it is only statistically right, individual measurements are not required to be thought of as having instant interaction.

 

[Nugatory]

Still, if Alice measures her particle, Bob can measure either spin up or spin down if measuring along a different axis than Alice? So the suggestion that $|up, down \rangle$ applies solely to measurement, does not hold in my view.

The notation inside the ket is just an arbitrary label, chosen by whoever is writing the ket because they find it convenient. If you find up, down as the label for a particular state to be confusing then you can always choose a different label.

 

[entropy1]

The notation inside the ket is just an arbitrary label, chosen by whoever is writing the ket because they find it convenient. If you find up, down as the label for a particular state to be confusing then you can always choose a different label.

The question is, where do they apply to. It can't be measurement outcomes alone.

 

[Nugatory]

The question is, where do they apply to. It can't be measurement outcomes alone.

They don't apply to anything, they're just labels. If seeing the words "up" and "down" inside the kets confuses you then you can use different labels... write the entangled state from your first post as $|bob,sue\rangle-|sue,bob\rangle$ instead of $|up,down\rangle-|down,up\rangle$ and the math will work just as well.

 

[Demystifier]

I think that because it is only statistically right, individual measurements are not required to be thought of as having instant interaction.

So how do you explain statistical correlation if not by interaction?

 

[entropy1]

So how do you explain statistical correlation if not by interaction?

I don't know how to explain the correlation, like I don't know how to explain an interaction.

 

[Morbert]

So how do you explain statistical correlation if not by interaction?

Probably treading old ground but: Correlations in quantum experiments are explained by the preparation, even though correlations in quantum theories are richer than in classical theories. Why is nature more correctly accounted for by quantum rather than classical theories? That might be like asking why nature is more correctly accounted for by one Hamiltonian over the other.

 

[Demystifier]

What is your definition of interaction?

I don't have a strict definition. Do you? But intuitively, by interaction I mean correlation caused by something.

 

[Demystifier]

Correlations in quantum experiments are explained by the preparation

Preparation of what? Orthodox QM defines preparation as a procedure in a laboratory, without saying what are the things that are being prepared. Kochen-Specker theorem, in my view, says that if there are actual things that are being prepared, then correlations cannot be explained by preparation alone.

 

[martinbn]

I don't have a strict definition. Do you? But intuitively, by interaction I mean correlation caused by something.

Well, in the standard terminology there are four fundamental interactions. The rest reduce to those four. In this case which interactions are involved? Electromagnetic, weak, strong, gravity? Or do you use the word differently when you say electromagnetic interaction and the interaction in the entanglement case?

 

[Morbert]

Kochen-Specker theorem, in my view, says that if there are actual things that are being prepared, then correlations cannot be explained by preparation alone.

I don't think Kochen-Specker forces the dichotomy that either the microscopic world does not exist or the correlations we see in experiments on the microscopic world warrant an explanation beyond preparation. Instead the Kochen-Specker theorem says the representation of the microscopic world in our theory cannot be naively made independent from measurement context.

We can accept that the there is a microscopic world. But the manner in which it is made intelligible is through apparatus in the macroscopic world.

 

[Demystifier]

Well, in the standard terminology there are four fundamental interactions. The rest reduce to those four. In this case which interactions are involved? Electromagnetic, weak, strong, gravity? Or do you use the word differently when you say electromagnetic interaction and the interaction in the entanglement case?

Even in this standard terminology, there is also a fifth interaction, the one mediated by the Higgs field. But of course, in the context of explaining correlations associated with entanglement, by interaction I mean something different. I mean interaction without mediator, not unlike the interaction in Newtonian gravity.

 

[vanhees71]

Preparation of what? Orthodox QM defines preparation as a procedure in a laboratory, without saying what are the things that are being prepared. Kochen-Specker theorem, in my view, says that if there are actual things that are being prepared, then correlations cannot be explained by preparation alone.

The correlations are indeed properties described by the state and are thus due to the preparation of the two-photon system in an entangled state. Let's assume the two-photon state is prepared by parametric downconversion. Then indeed the entanglement is due to the interaction of a photon absorbed by the crystal, which then emits the two spin-entangled photons.

The fact that the correlations are detectable when the photons are measured at two distant locations with measurement events space-like separated together with the description of this entire process within standard QED, which excludes faster-than-light causal effects, indicates that the correlations are indeed a property of the prepared two-photon state and not due to the causal influence of one of the measurements on the other.

 

[AndreiB]

Bell's model is not Q(F)T. In relativistic local QFTs by construction there are no causal connections between space-like separated events.

Can you prove that in QFT the following statement must be false:

Bob's measurement result caused (determined) Alice's measurement result?

 

[martinbn]

Even in this standard terminology, there is also a fifth interaction, the one mediated by the Higgs field. But of course, in the context of explaining correlations associated with entanglement, by interaction I mean something different. I mean interaction without mediator, not unlike the interaction in Newtonian gravity.

There is a difference, in the Newtonian case Bob can tell imidiately if Alice is doing something or not. With entanglement that is not possible even in principle.

If you consider interaction without a mediator, which cannot be detected in principle, you are adopting a very strong form of antirealism. Wasn't that one of the things you were not happy about CI?

 

[vanhees71]

Can you prove that in QFT the following statement must be false:

Bob's measurement result caused (determined) Alice's measurement result?

If A's and B's measurement events (clicks of photodetectors for a typical Bell experiment with entangled photon pairs) are space-like separated and accepting standard relativistic local QFT implies that there cannot be any causal impact of one of the measurements on the other. The temporal order of space-like separated events is frame-dependent and thus they cannot be causally connected, and relativistic local QFT excludes such a posibility by fulfilling the microcausality property of local observables.

 

[AndreiB]

standard relativistic local QFT implies that there cannot be any causal impact of one of the measurements on the other.

Can this claim be rigorously proven inside QFT? Can you, by assuming that A caused B arrive at a mathematical/logical contradiction?

The temporal order of space-like separated events is frame-dependent and thus they cannot be causally connected, and relativistic local QFT excludes such a posibility by fulfilling the microcausality property of local observables.

Can you show, in a rigorous way, how the assumption the A caused B leads to a contradiction with microcausality?

 

[vanhees71]

What do you have to prove? If all operators representing local observables (which inculdes the Hamilton density of the theory) commute at space-like separation of their arguments there cannot be any faster-than-light influence between events. That's why one envokes this "microcausality" property in the first place, implying the existence of antimatter, the spin-statistics relation, and CPT symmetry, all of which are in accordance with (very accurate) observations.

 

[AndreiB]

What do you have to prove? If all operators representing local observables (which inculdes the Hamilton density of the theory) commute at space-like separation of their arguments there cannot be any faster-than-light influence between events.

I do not understand this argument. You have:

P1. all operators representing local observables commute at space-like separation.
P2. The measurement at A caused B.

Where is the contradiction?

 

[vanhees71]

If local relativistic QFT describes the experiment, then P2 is excluded due to P1.

 

[AndreiB]

If local relativistic QFT describes the experiment, then P2 is excluded due to P1.

You assert that, you did not prove it. What is the relationship between commutativity and causation?

 

[vanhees71]

I still don't understand what you need to prove.