Entanglement collapse and relative order of events

Question

According to QM (and many confirming experiments) outcomes (e.g. spin) of entangled particles measurement are non-locally correlated (i.e. can’t be pre-configured for such correlations). It seems that physically the first measured particle dictates the outcome (probability) of the peer particle. Otherwise, the second retroactively impact the first (backward in time) which seem false (?).

OTOH, according to special relativity different (measuring) observers disagree on order of events hence breaking cause and effect. Isn’t that a good indication that SR is not a physical description of nature though mathematically accurate (e.g. without absolute frame)?

When thinking on how to simulate entanglement without breaking "precived" SR I image that the entangled (wave-like) particle having “shared hidden box” used upon measurement. The first writes its outcome and the second takes it into account. I guess that such “shared box” can comply with measurements predicated by QM including Bell inequalities, right?

Answer 1

That is an picture of the best we can argue about what happens in spacetime, avoiding non-physical notions as "instantaneous changes" when a couple of measurments are performed on an entangled state of two-particles (and nothing further happens in spacetime for that system).

In the picture, $M_1$ and $M_2$ are the causally separated regions in spacetime where measurements take place.

The state of the two particles is:

(a) the entangled one $$\psi_1\otimes \psi_2 + \psi_2 \otimes \psi_1.$$ everywhere in the region causally separated with $M_1\cup M_2$ and in the causal past of both them.

(b) the "collapsed" one $$\psi_1\otimes \psi_2$$ everywhere in the causal region which can receive physical information from the outcomes. (The other possibility is $\psi_2\otimes \psi_1$ everywhere therein.)

If we assume that observables localised in spatially separated regions commute, this physical setup cannot give rise to causal problems also adding (thus changing the figure) further measurments on the system with experiments localised in bounded regions of the spacetime.

All that was discussed, into a very general way, by K.Hellwig and K. Kraus in "Formal Description of Measurements in Local Quantum Field Theory" Phys Rev D vol 1 n.2 1970.

The puzzling issue, concerning entangled states and causally separated regions, is how the state can appear to be $$\psi_1\otimes \psi_2$$ close to the both the regions $M_1$ and $M_2$ (in their causal future). That is because no information can be transmitted between these two regions and the outcomes cannot "synchronize" with each other.

This may seem rather strange but, in principle, it does not give rise to a physical or logical contradiction. Special Relativity and Quantum Theory coexist peacefully up to now. First of all because in quantum mechanics the outcomes of measurments are intrinsically stochastic and there is no way to force a special outcome and construct a Morse telegraph between $M_1$ and $M_2$. But there are also more sophisticated arguments.

If we want to define some kind of (classical-like) mechanism that produces this sort of correlation, we know, after Bell's analysis, that the mechanism cannot be both realistic (the outcomes are already fixed before the measurements) and local (superluminal transmission of physical information is not admitted).

I emphasize that, in my view, there is no need for such a mechanism (especially of classical type). Nothing really requires it. The search for a deeper mechanism is mainly due to our philosophical prejudices. Pursuing some of them, in the past, has led us to new fundamental physics, but nothing ensures that this is the case with these strange properties of entanglement in QM.

ADDENDUM. The weirdness above does not depend on the physical solidity/inconsistency of processes such as "collapse" of the state.

We can consider the collapse simply as an effective and convenient description as I now try to explain. In this view, the quantum system interacts with the measuring instrument and the environment, but the evolution of the overall system is always unitary and no "collapse" occurs at any time.

In the measurement process the state of the system entangles with the state of the instrument and the environment. Before the interaction the states were factorized (t least the one of the measured system with the one of the rest of the overall system. The final state is instead given by a certain coherent superposition of products od instrument states and states of the system. The instrument states point out the corresponding outcomes of the measurement process.

If we intend to formally deal only with the system, we can ignore the instrument and the environment after the interaction with the instrument. This is equivalent to dealing with an effective state of the system produced by the partial trace operation.

(In the description above, a "decoherence mechanism" plays a central role, but I will not insist on this point here).

The partial trace operation produces an effective mixed state of the system as a superposition of the various possible outcomes of the instrument.

In this whole picture, I am assuming that quantum mechanics deals with ensembles and not with the individual systems that belong to the ensembles.

The specific system we have at hand (here a single pair of entangled particles) simultaneously shares three quantum states:

(1) the overall pure state of the overall system,

(2) the mixture obtained by partially tracing the previous state,

(3) a specific pure state in this mixture.

By adopting one of these three states, QM produces different statistical predictions, but these are compatible because they refer to three different ensembles to which the system belongs.

By adopting the mixture description, we can move on to focus on the subensemble (of the initial ensemble) associated with the specific branch of the effective mixture to which our specific system belongs.

This gives rise to the apparent phenomenon of "collapse".

However, even in this approach, where collapse does not play a magical role and is not the reason for the strangeness, something remains weird.

It is the fact that, even though the measurement process takes place in two causally separated parts, the outcomes in these regions are strongly correlated. The responsibility for this lies with the entangled state.

Answer 2

According to QM (and many confirming experiments) outcomes (e.g. spin) of entangled particles measurement are non-locally correlated (i.e. can’t be pre-configured for such correlations).

Bell's theorem says that if (1) the evolution of measurable quantities is described by a stochastic variable, i.e. - a number picked from the set of possible outcomes with some probability, (2) it evolves locally and (3) the measurement devices and the measurements chosen the particle don't all somehow conspire with one another to produce experimental results, then the correlations are bounded in a particular way that disagrees with the expectation values for those quantities predicted by quantum theory that have survived experimental testing. To discuss why Bell correlations happen requires a bit of a detour to discuss collapse.

The equations of motion of quantum theory describe the evolution of physical quantities in terms of Hermitian operators called observables whose eigenvalues are the possible results of measurements. Quantum theory predicts the expectation values of measurements of observables according to an equation called the Born rule involving the state, which is a unit trace Hermitian operator describing previous records of information about the relevant system.

In quantum experiments the results depend on what happened to all of the possible values of the measured observable: quantum interference. See Section 2 of this paper for an example

https://arxiv.org/abs/math/9911150

It is common in badly written textbooks to add on top of these equations of motion an unexplained assertion that what happens during a measurement is that all of the possible outcomes of the measurements vanish. Some physicists have tried to explain a process that reproduces this assertion or some approximation to it

https://arxiv.org/abs/2310.14969

Such theories don't currently reproduce any of the predictions of relativistic quantum theories

https://arxiv.org/abs/2205.00568

These collapse theories would agree with the restrictions imposed by Bell's theorem and so would have to be non-local but since they don't currently explain a load of experimental results it's unclear why anybody would adopt such a theory in the first place.

When information is copied out of a quantum system information is suppressed: a process called decoherence

https://arxiv.org/abs/1911.06282

And since the objects you see in everyday life have information copied out of them by their environment much faster than the timescales you can see and over which they change significantly you shouldn't see much interference from those objects in everyday life. As a result, quantum theory without collapse implies that on the scales of space and time in everyday life to a good approximation the world acts like a collection of autonomous classical universes:

https://arxiv.org/abs/1111.2189

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

This is commonly called the many worlds interpretation (MWI) but it is just a consequence of applying the equations of motion of quantum theory consistently without modification.

In quantum theory without collapse systems aren't described by stochastic variables, they are described by observables. So Bell's theorem doesn't imply non-locality of non-collapse quantum theory. Indeed, there is an explanation of how Bell correlations arise in quantum theory without collapse: decoherent systems such as measurement devices and electronic signals sent to computers and so on carry locally inaccessible quantum information that can't be accessed by measurements on those systems alone but is only revealed when the measurement results are compared:

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

https://arxiv.org/abs/1109.6223

As a result as long as the equations of motion of the relevant quantum theory is local, so too is the process by which the Bell correlations arise. The equations of motion used to make predictions about photons and other particles used in Bell correlation experiments are those of relativistic quantum field theory and those equations are all local and entirely consistent with special relativity, see books on quantum field theory such as "Quantum field theory for the gifted amateur" by Lancaster and Blundell or "The conceptual framework of quantum field theory" by Anthony Duncan.

Quantum theory without collapse and the Bell correlations are consistent with relativity and don't require backward in time causation or mysterious non-local influences or any other magical pixy dust or ad hockery. If you want to understand quantum theory you consistently work out the consequences of the relevant equations of motion as you would for any other theory in physics.

Answer 3

I'd say there is only one cause and one effect. Our imagination is not always the same as reality. Entangled particles react immediately, there is 0 delay, which implies the two particles "share" one property so to speak but express it differently. One cause and effect. Only one event and therefore no order.

Answer 4

Without trying to disagree with any of the other answers specifically, there are some important elements related to the question that are either incorrect or missing key information. Everything I mention below is confirmed by modern experiments by top teams, unfortunately some of their conclusions are not well known (which is why I am answering this).

Most important: Quantum Entanglement (QE) has nothing to do with Special Relativity (SE), and vice versa. You can object to this any way you want to; however, there is absolutely no limits/constraints imposed on entanglement by SR. Period.

If you assume local causality, a typical deduction from SR, you immediately run into problems with experiment. Please keep the following in mind, and you will see why QE and SR do not have any particular bearing on each other:

a) Bell's Theorem implies nonlocality, although there are some exceptions possible.

b) GHZ also tells us that reality (quantum expectation value) is dependent on choice of remote measurement basis, and is not statistical in nature. The expected outcome is certain in all cases, unlike in Bell (which depends on statistical results such as CHSH).

c) And experimentally: The order of measurement is always immaterial with 2 entangled particles A and B (as to the observed outcomes). In fact:

i) You can measure them in a certain order to all observers and see this point - it is not true that order is always relative to the observer, even in SR. For example, measure A and B at the same location, all observers must agree as to order.

ii) You can entangle A and B that have never existed in a common SR light cone. They are created from sources distant to each other, and they never interact.

iii) You can measure A and B that have never existed at the same time. A is destroyed before B is created.

iv) You can entangle A and B remotely. The choice to entangle A and B is made by a distant observer to both A and B.

v) You can entangle A and B AFTER A and B have been already measured and no longer exist. This is the delayed choice version.

Experimental delayed-choice entanglement swapping

So basically: None of the above experiments demonstrate any of the expected limitations of Special Relativity. One of the authors of the cited experiment won the 2022 Nobel for experiments such as I have described.

Answer 5

Indeed, Bell's theorem proves that any single-world theory must be nonlocal, meaning there are influences that happen faster-than-light. The two most common single-world theories of quantum mechanics, GRW theory and Bohmian mechanics, incorporate this fact in two different ways. In Bohmian mechanics, relativity is broken at the fundamental level, so there is an actual matter of fact about which particle is measured first, and that first measurement directly influences the measurement of the second particle. In GRW theory (with flash ontology), it is possible to keep relativity while breaking locality, essentially because the ontology does not have anything that evolves continuously in spacetime; it merely has flashes whose occurrence at a particular spacetime point has a probability that is independent of one's chosen reference frame. In this theory, you will get a different answer about whose measurement was first, and therefore about which measurement influenced the other, depending on which frame you choose.