# How can quantum entanglement not be non-local?

I know this kind of question has been brought up many times.I have read many posts here regarding this but I still have a problem with a certain aspect of it so please bear with me.

Lets consider the standard case where one of two singlet state electrons is send to Alice and the other is send to spacelike separated Bob.This is repeated many times.

I know that its impossible for Alice to transmit information by measuring her electrons.This is because the density operator for the state of Bob is invariant under a rotaion of Alices axes.To show that, we note that the state prior to Alices measurement is also rotation invariant (using the well known rotation matrices for spin 1/2 objects):

$$\frac{|\uparrow_z\downarrow_z\rangle-|\downarrow_z\uparrow_z\rangle}{\sqrt{2}}=\frac{|\uparrow_n\downarrow_n\rangle-|\downarrow_n\uparrow_n\rangle}{\sqrt{2}}$$

The state has the same components, no matter in which base it is expressed.That means, that no matter which axis Alice chooses, she will always get half of her results spin up and the other half spin down. Now if we use the "model" that Alices measurement collapses the composite state wavefunction, it follows that the particles Bob recieves will be a statistical mixture of one half spin up, one half spin down.This state is also independent of the basis,as can be shown using the same rotation matricies. $$\rho=1/2*|\uparrow_z\rangle\langle\uparrow_z|+1/2*|\downarrow_z\rangle\langle\downarrow_z|=1/2*|\uparrow_m\rangle\langle\uparrow_m|+1/2*|\downarrow_m\rangle\langle\downarrow_m|$$

This means that the statistical distribution of ups and downs that bob measures is completely independent of Alices choice of axis, or even her choice to measure her electron at all. Bob will statistically always get a fifty fifty result.

But nevertheless, and this is my question, Alices measurement makes Bobs wavefunction collapse to a pure state, and it seems to me this will leave an imprint on Bobs measurement in a non local fashion. Lets say Alice only measures along the z axis, and Bob chooses a random axis for every measurment.If Alice tells Bob the outcome of her measurements, Bob will be able to divide his outcomes in two groups: One where the corresponding spin of Alice was measured up, and the other one for the case where Alice has gotten down.And these two groups will show a perfect correlation between Bobs axis and Alices z-axis.Everytime Alice had found spin up, and Bob has chosen the z-axis aswell, he will find that his measurement was down, and the probability distribution will depend on the angle of his axis relative to Alices z-axis by the well known formula

$$|\langle \chi_z|\chi_n\rangle|^2=\cos\left(\frac{\theta}{2}\right)^2$$

Where theta is the angle that Bob measured the spin along, relative to the z-axis.And Alice can change the axis that Bobs measurements are biased to.If she chooses another axis, then Bobs outcomes will show the correlation with respect to this new axis.This correlation will be "burned" into Bobs list of outcomes one by one, everytime Alice measures an electron.

Of course, Bob will only see this correlation when he recieves Alices results, which can only be transmitted via a classical channel. But I mean, "something" has to change in the very moment Alice makes her measurement, because the correlation between the spins is already there, even before Alices information has reached Bob.

How can we rescue locality here?

(I guess this is deeply connected to the question whether the wavefunction actually collapses or something else is going on.But how can we say that there is not actually a collapse if the model of the collapse gives us the right probability distribution for the angles?Maybe you can adress this)

Edit:

I think these answers don't answer my question because I explained that I know that Alice cant send information to Bob this way.Nevertheless it seems to me there is some kind of non locality involved.Like a nonlocal event thats just useless for sending information.I wonder whether there is a way of looking at it that doesnt involve any nonlocal event.

• Possible duplicate of Does entanglement not immediately contradict the theory of special relativity? Commented Oct 15, 2017 at 16:37
• Yes, entanglement is (or can be construed to be) nonlocal, but what's wrong with that? You only require nature to be causal and that's not at risk here. Commented Oct 15, 2017 at 17:32
• I guess my concern still arises because I understand entanglement in the way that Alices measurement "does" something to Bobs electron.It actually changes to state of the electron.My understanding is that an electron actually has a definite state (in a quantum sense of course), some pure state or even some mixed state.And Alices measurement makes the electron change its state into something else.(That looks exactly the same to Bob, but only because hes missing Information). Is this approach flawed? Commented Oct 15, 2017 at 17:53
• If the electron actually changed its state there would have to be some kind of interaction which would of course bring many problems wouldn't it? Commented Oct 15, 2017 at 17:58
• Argument over this contradiction goes back all the way Einstein vs. Bohr. See Einstein/Podolsky/Rosen (en.wikipedia.org/wiki/EPR_paradox) and Bell's Theorem plato.stanford.edu/entries/bell-theorem. Bell's theorem is subject to experimental verification (the original EPR isn't). The experimental evidence at this time is quite clear: Bohr was right and Einstein was wrong: the universe appears to be non-local. Commented Oct 15, 2017 at 18:27

You have reached the realm of interpretation of the quantum formalism. No one doubts that the formalism is correct (at least at the level of the entanglement effects you are referencing). But what is actually happening is a matter of interpretation. Most physicists would agree with you that the results of the various Bell test experiments doom local interpretations of QM. But if you dig, you may find some which preserve locality at the expense of something else. My personal favorite is the transactional interpretation of quantum mechanics, which (in a sense) preserves local interactions but avoids conflict with the predictions of the apparently non-local QM formalism by postulating time-symmetric (including backward-in-time) signals.

You see, locality can indeed be saved. But at what cost?

• I agree with your cautious remark "what is actually happening is a matter of interpretation" (the realist Everett one denying the collapse or the positivist one for instance). Concerning the locality despite Bells inequalities violation, §Bell inequalities violation in clearing up mysteries is rather clear in my opinion. Moreover Each instant of time a new universe provides a two-state vector time-symmetric formulation where Alice measurement causes a change of state only on his side.
– ABC
Commented Jul 12 at 8:57

There is another explanation. Hidden variables. If at their creation entangled electrons are already in pure state due to unknown physics this reproduces all experiments. Of course due to Bell's inequality the hidden variables need to be non-local. This is not very polpular today, not that popularity means anything.

In the EPRB experiments, in the standard one-state vector quantum formalism, a polarization measurement on Alice's side changes the quantum state on both sides. So, from a formal point of view, Alice’s measurements cause instantaneous “collapses-at-a-distance” of Bob’s photon state, in apparent violation of relativistic causality (despite the impossibility to use this alleged relativistic causality violation to send information between space-like separated events).

Let’s go a little bit further. Actually, a polarization measurement on Alice’s side causes an irreversible record of information, hence a creation of entropy on her side (e.g. a loss of so-called irrelevant information). On the contrary, as far as Bob doesn’t perform any polarization measurement, Bob’s alleged photon’s state “collapse-at-a-distance” doesn’t give rise to any observable effect. Indeed, Bob’s photon “collapse-at-a-distance” doesn’t cause any entropy creation on his side...

...But an entropy creation would be necessary to get an observable record of this supposed phenomenon. From a positivist point of view, according to Occam’s razor, the absence of any observable effect of Bob’s photon collapse is enough to assert that this alleged objective Bob’s photon collapse doesn’t actually happen on Bob’s side when only Alice performs a polarization measurement on his side.

Nevertheless, such an answer may be considered doubtful. Indeed, the here above positivist answer needs adhering to a positivist interpretation of quantum state and quantum collapse (the ireversible quantum measurement process breaking the unitary quantum evolution of the quantum state. A process modeled by the projection of the quantum state on an eigenspace of the observable). Bohmian mechanics and its explicit non local effects, a more realist view of quantum physics than the standard quantum formalism, has proven to be a consistent mathematical model, compatible with observable quantum effects. So, the here above positivist assertion doesn’t and can’t discard the possibility of an objective, explicitly non local interpretation of the EPR effect, violating the relativistic causality at an interpretive level.

Let us dig a little bit more. Actually, in the one-state vector standard quantum formulation, there is a lack of information, a subtle hidden variable. This is the second state vector accounted for in the 2-state vector T-symmetric formulation. Actually, as Einstein had guessed, the one-state vector quantum formalism provides an incomplete description of the quantum state, hence an incomplete description of quantum phenomena. That lacking information is hidden in the future, cf. Can a future choice affect a past measurement’s outcome, §6 Summary, b).

The apparent instantaneous “collapse-at-a-distance” of Bob’s photon doesn’t show up in the more complete two-state vector formulation. Indeed, according to Each moment of time a new universe Aharonov, Cohen and Elitzur article, §5 EPR measurement, fig7, a measurement on Alice’s side causes a change of the two-state vector of her photon but no change of the two-state vector of Bob’s photon. There is even more to say about this absence of any real Bob’s photon collapse. Let us see why.

The article Each instant of time a new universe of Aharonov, Popescu and Tollaksen, presents:

• a T-symmetric formulation of the temporal “evolution” of a quantum system which does not evolve (H=0)
• a very important consequence predicted thanks to this formulation concerning the interpretation of the EPRB experiment.

This very interesting 8 pages article is quickly read and a video is presented by Popescu. Thanks to their 2-state vector T-symmetric formalism, Aharonov, Popescu and Tollaksen notably highlight the following facts:

• as long as no quantum measurement is carried out on a given quantum system (undergoing a H=0 Hamiltonian evolution) the 2-time measurement O(t2) - O(t1) between instants t1 and t2 vanishes whatever the observable O. This proves the existence of a time correlation between successive states of a quantum system as long as it does not undergo any quantum measurement.

• On the other hand, the correlation O(t2) - O(t1) = 0 is broken between instants t1 and t2 respectively preceding and following a quantum measurement of the considered system (except in the specific cases when the measurement result is an eigenstate of O).

Hence, concerning EPRB type experiments, this document indicates (§5 Measurements on EPR state):

• The break, on Alice's side, of the 2-time correlations between instants t1 and t2 preceding and following a quantum measurement by Alice. Indeed, except in a particular case when the measurement result is an eigenvalue of O, the 2-time correlation O(t2) - O(t1) = 0 is lost.
• The conservation, on Bob's side, of the 2-time correlations O(t2) - O(t1) = 0. This means that nothing changes on Bob’s side as long as he doesn’t make any measurements on his side.

Now, let us recall 2 hereunder EPR non-locality interpretations at stake:

When only Alice carries out measurements on her side, the prediction of the conservation of the 2-time correlation on Bob's side decides in favor of the positivist interpretation of the EPR effect. This interpretation thus becomes a falsifiable physical assertion instead of a pure philosophical question.

So, such an experimental verification seems solving a 40 years debate between positivist and realist interpretations of Bell's inequalities violation. Hence, this experimental validation seems deserving to be carried out if it has not already been done. I don’t know if this verification has been performed. That’s why I have asked that question on physics stack exchange today.

Now, there is an additional remark I think useful adding to the here above answer. It concerns the EPR non-locality interpretation by Aharonov, Cohen and Elitzur. Despite my using of their 2-time correlation conservation to prove the validity of the positivist interpretation of EPR non locality (when no measurements are performed on Bob’s side) it’s important to note that these authors are, on the contrary, committed to a realist (e.g. observer independent) interpretation of physics.

In particular, they are committed to an observer independent interpretation of causal relations. They interpret the correlation between strong and weak measurements, for instance, as causal from strong measurements to weak ones independently of their time ordering. Hence, the correlation between weak and strong measurements is considered by these authors as causal when weak measurements are performed after strong measurements and retrocausal when weak measurements are performed before strong measurements.

There is no inconsistency there. Indeed, if the statistical thermodynamic grid of reading of the macroscopic observer is considered as irrelevant contrary to a positivist interpretation (cf. Incomplete descriptions and relevant entropies, Balian, 1999) then the irreversible flow of time is interpreted as an illusion as proposed by Einstein (2) or by Thibault Damour (3) for instance (due to their realist interpretation of physics aim).

Consequently, for Aharonov, Cohen, Elitzur, Vaidman and physicists committed to their school of thought, there is no reason to ascribe a causal character to time-symmetric correlations only in a direction consistent with the causality principle. Indeed, due to their observer independent interpretation of physics, this time-asymmetric principle and the irreversible flow of time disappear. They are considered as pure illusions as any other time-asymmetric effects (noteworthy tracks of the past and classification of events as past or future events, cf. The arrow of time issue, an overview, Chaverondier, April 2024)…

…as well, nevertheless, in my positivist opinion, as any physics properties. Indeed, physics properties are encoded as information recorded in tracks of the past, and these tracks exist thanks to the macroscopic observer's grid of reading. That’s why the retrocausal interpretation of the correlation between strong measurements and previous weak measurements for instance is, in my opinion, an unnecessary additional assumption to their outstanding studies and experimental verifications.

(1) Note, however, that E.T. Jaynes supports a realist interpretation of physics and its role despite, paradoxically, his insistence on the importance of Bayesian inference and the broad development he gave to this approach (cf. Maxent)

For people like us who believe in physics, the separation between past, present and future has only the importance of an admittedly tenacious illusion.

Thibault Damour, Permanent Professor at the Institut des Hautes Études Scientifiques, delves on the experience of Time through the work of Marcel Proust “À la recherche du temps perdu” and Albert Einstein… Both the novelist and the physicist held the view that the flow of time is an illusion.

• I don't even know what this is supposed to mean. Well written quantum mechanics textbooks don't use the term "collapse" at all. I have not found a workable definition in the physics literature. Can you tell us what you mean by "collapses the state instantaneously"? A quantum mechanical state is a mathematical abstract. It doesn't exist in physical reality. How can an abstract "collapse"? And what is "instantaneous" in a relativistic universe? Commented Jul 11 at 21:03
• @FlatterMann Can you please give a name and author of such well-written quantum mechanics textbook? Commented Jul 11 at 23:40
• @FlatterMann "instantaneous collapse" means that when Alice makes spin/polarization measurement on her part of the system, the appropriate state vector, in mind of Alice, for the whole system (including Bob's part) is no longer what it was before the measurement (let's assume a linear combination of product states), but changes to a product state. Commented Jul 11 at 23:51
• @JánLalinský Sakurai is a standard textbook, is it not? It seems reasonably well written. I tried to find "wave function collapse" in it. It's not in there for all I can tell. Commented Jul 12 at 1:49
• @JánLalinský What happens in the mind of Alice stays in the mind of Alice. The universe does not care about it. Neither should Bob or other physicists. Bob may or may not have detected a quantum of energy himself. If he did, then his detector got a little warmer. If he didn't, then the detector temperature didn't change. Now that is a physical process. "collapse of the wave function" is not. That is just a horrible meme that one shall not use, ever, because it serves no identifiable scientific function but causes a lot of confusion among lay people and specialists alike. Commented Jul 12 at 1:52

The equations of motion of quantum theory, such as the Schrödinger equation or equations of motion for relativistic quantum theory don't predict collapse and aren't compatible with it. For this reason alone it's worth considering whether there is an explanation of Bell correlations in quantum theory without collapse.

In quantum theory without collapse a system's measurable quantities can be described in terms of its Heisenberg picture observables represented by Hermitian matrices whose eigenvalues are the possible results of measurements. In general the motion of a quantum system can't be explained in terms of just one of the possible values, e.g. - single particle interference, see "The Fabric of Reality" by David Deutsch Chapter 2. So there are multiple versions of every system in quantum theory without collapse. A measurement is just an interaction that produces a record that can be copied. When such a record is produced that suppresses quantum interference: an effect called decoherence:

https://arxiv.org/abs/1911.06282

Decoherence tends to make the different versions of systems sort themselves into layers that evolve autonomously of one another to a good (but, importantly, not perfect) approximation:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

This is commonly called the many worlds interpretation but it is just an implication of quantum theory without modifications. There are explanations of how probabilities arise in this theory and I'll leave links to a couple:

https://arxiv.org/abs/0906.2718

https://arxiv.org/abs/quant-ph/9906015

https://arxiv.org/abs/quant-ph/0405161

It is common for people to say that Bell's theorem shows quantum theory is non-local, but this is false. Bell's theorem sez that if you have a theory in which (1) the evolution of each physical quantity is described by stochastic variables (random numbers), (2) the measurement system and measured system are not somehow endowed with knowledge about what measurements you will choose and (3) the expectation values of measurements agree with those of quantum theory, then the resulting theory must be non-local. Quantum theory describes the evolution of physical quantities in terms of Heisenberg picture observables not stochastic variables so Bell's theorem doesn't prohibit quantum theory from being local. The relevant equations of motion for observables in quantum theory without collapse are local so that rules out non-local explanations of Bell correlations in such a theory.

The local explanation goes like this. When Alice and Bob have quantum systems that are entangled with one another and they do a measurement both of the possible outcomes happen and there are two versions of the measurement result. The observables of the measurement device and any other system that contains the measurement result has quantum information that can't be accessed by measurements on that system alone: locally inaccessible information. The correlations only arise when the measurement results are compared and the information about what the correlations should be is carried to the comparison as locally inaccessible information:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

The questioner writes:

But how can we say that there is not actually a collapse if the model of the collapse gives us the right probability distribution for the angles?

This is wrong. There are variants of quantum theory that model the collapse process:

https://arxiv.org/abs/2310.14969

However, none of those variants can reproduce the predictions of relativistic quantum theories:

https://arxiv.org/abs/2205.00568

This means that they can't reproduce the predictions of any quantum experiments involving photons for which there is no non-relativistic theory and so they can't reproduce the vast bulk of quantum experiments, including almost all Bell correlation experiments.

One a more general note, many theories make some correct predictions while being wrong, e.g. Newton's theory of gravity. So making some correct predictions isn't any kind of guarantee of correctness.

But in this particular case, collapse theories don't make correct predictions so the issue doesn't arise.