Confusion around Bell's Theorem and Locality

Question

I recently got interested in foundational aspects of quantum mechanics and I have some questions:

Bell's theorem proves that any local, deterministic theory with statistical independence can't account for certain observed correlations. The standard QM formulation is usually considered local, with no determinism, but I just don't see how determinism can be false without locality also being false?

If an entangled particle pair has no pre-determined relative position, in which sense is the theory local? Furthemore, if both particle states are undetermined before the measurement, why would the local act of measuring the first one give any certain information about the second one, which is completely undisturbed by our local operation? I don't see any reasonable notion of locality that can accomplish this.

I also don't see how the notion of determinism can be rigorously defined, while locality can easily be defined in a relativistic space-time.

EDIT: I reviewed some of the discussion here, the issue i wanted to address can be stated more easily by “can a non-deterministic, relativistically local QM exist”? By deterministic, I mean a theory in which systems have defined properties before measurements.

My confusion is from the fact that these theories are not ruled out by Bell’s theorem, yet I can’t understand how they can exist at all. If a notion of locality is not even definable in non-deterministic models, then how come that QM is developed this way?

Answer 1

The standard QM formulation is usually considered local, with no determinism, but I just don't see how determinism can be false without locality also being false?

There are several ways to answer your questions, so I may as well start here. With the advent of Bell's Theorem, as well as the many experimental variations which support the predictions of QM, I would not say QM is generally considered "local". A term has arisen, "quantum nonlocality" (sometimes just "nonlocality"), to describe the kind of nonlocal effects associated with quantum mechanics. Please note that signals cannot be transmitted nonlocally in any variation or interpretation of QM. The only effect that can be transmitted non-locally are ones that yield random outcomes. Also note that there is no particular mechanism for nonlocality embedded in mainstream QM.

As to determinism: there is no known source or "cause" for the random outcomes of the probabilistic predictions of QM. It is possible that some underlying cause will be discovered in the future, but don't bet on it. No experiment has so much as pointed to the existence of such a cause (or set of hidden variables). Of course, there are interpretations of QM that posit determinism - there just isn't any hard evidence to support that hypothesis.

So there may be neither locality NOR determinism - that would also be consistent with what we know. But Bell's Theorem merely tells us that one or the other must be false.

Recent experiments have led us much further down the road. The so-called "loopholes" in Bell have been closed to the satisfaction of most scientists. For example:

1. https://arxiv.org/abs/quant-ph/9810080

Violation of Bell's inequality under strict Einstein locality conditions (1998) Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, Anton Zeilinger

"We for the first time fully enforce the condition of locality, a central assumption in the derivation of Bell's theorem. The necessary space-like separation of the observations is achieved by sufficient physical distance between the measurement stations, by ultra-fast and random setting of the analyzers, and by completely independent data registration."

And there are the incredible experiments with entanglement swapping. In these, distant photons (from independent sources) are entangled without ever existing in a common light cone. The variations on these experiments violate strict locality, and in fact don't follow the classical rules of cause and effect (where cause must precede effect).

2. https://arxiv.org/abs/quant-ph/0201134

Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping (2002-2008) Thomas Jennewein, Gregor Weihs, Jian-Wei Pan, Anton Zeilinger

"Quantum teleportation strikingly underlines the peculiar features of the quantum world. We present an experimental proof of its quantum nature, teleporting an entangled photon with such high quality that the nonlocal quantum correlations with its original partner photon are preserved. This procedure is also known as entanglement swapping. The nonlocality is confirmed by observing a violation of Bell's inequality by 4.5 standard deviations. Thus, by demonstrating quantum nonlocality for photons that never interacted our results directly confirm the quantum nature of teleportation."

And in fact the work of Zeilinger and his co-workers was recognized in the Physics Award by the 2022 Nobel Committee:

"One of the most remarkable traits of quantum mechanics is that it allows two or more particles to exist in what is called an entangled state. What happens to one of the particles in an entangled pair determines what happens to the other particle, even if they are far apart. In 1997–1998, Anton Zeilinger conducted groundbreaking experiments using entangled light particles, photons. These and other experiments confirm that quantum mechanics is correct and pave the way for quantum computers, quantum networks and quantum encrypted communication."

"Among other things, [Zeilinger's] research group has demonstrated a phenomenon called quantum teleportation, which makes it possible to move a quantum state from one particle to one at a distance [distance here meaning outside a common light cone]."

And further, the study of 3 particle entangled GHZ States casts further doubt on both locality and determinism. Papers on this get very complex very quickly. Here is one where Zeilinger is the co-author (and he is the Z in GHZ). I only provide these references to give you the idea that Bell's Theorem has led to a proliferation of ground-breaking work on entanglement and the foundations of Quantum Mechanics.

3. https://www.drchinese.com/David/Bell-MultiPhotonGHZ.pdf

Multi-Photon Entanglement and Quantum Non-Locality (2002) Jian-Wei Pan and Anton Zeilinger

"We review recent experiments concerning multi-photon Greenberger–Horne– Zeilinger (GHZ) entanglement. We have experimentally demonstrated GHZ entanglement of up to four photons by making use of pulsed parametric down conversion. On the basis of measurements on three-photon entanglement, we have realized the first experimental test of quantum non-locality following from the GHZ argument. Not only does multi-particle entanglement enable various fundamental tests of quantum mechanics versus local realism, but it also plays a crucial role in many quantum-communication and quantum computation schemes."

Answer 2

The OP made some edits to the question, and I'd like to address those specifically.  

EDIT: I reviewed some of the discussion here, the issue i wanted to address can be stated more easily by “can a non-deterministic, relativistically local QM exist”? By deterministic, I mean a theory in which systems have defined properties before measurements.

My confusion is from the fact that these theories are not ruled out by Bell’s theorem, yet I can’t understand how they can exist at all. If a notion of locality is not even definable in non-deterministic models, then how come that QM is developed this way?

QM is mostly silent as to whether it is a) local non-realistic(or local non-deterministic or local but no hidden variables); b) nonlocal realistic; or c) nonlocal non-realistic. But you are asking specifically about a), so let's discuss that.  Note that some will argue that non-realistic, non-deterministic, and no hidden variables are different things.  But for this discussion, let's use your definition and refer to it as non-realistic (which is more often used).

The 1935 EPR paper (assuming you are familiar with it) defines "an element of reality" as being a probabilistic quantum outcome that can be predicted with certainty in advance. Realism is the assumption that if any quantum outcome can individually be predicted in advance, then all such must also be predictable in advance - even if such outcomes can be predicted simultaneously.  So spin (on any basis), momentum, etc. of entangled pairs would fall into this category. If all these outcomes could be predicted in advance, then they must be predetermined. The key point here: "pre" implying before.  The reality of a quantum property therefore should not depend on how the observer might choose to measure a system.

So what is an indeterministic/non-realistic interpretation?  It is any that does not have all of its properties predetermined.  The implication is: a) something far away affects the outcome (but you want to exclude this); or b) something in the future contributes to the outcome in some manner. It could also be something that is part of the measurement process itself.  In terms of the EPR paper, we might also say that indeterministic QM is observer dependent, i.e. subjective.  (You can see how quickly and easily we get lost in words...)  In fact, simply accepting the Heisenberg Uncertainty Principle as complete/true/factual could be seen as accepting indeterminism.  

@KenWharton, a respected contributor here, has studied and written on this subject extensively.  I would call attention to a series of his papers on Bell's Theorem and retrocausal class interpretations.  These have the features you are asking about. 

1. https://arxiv.org/abs/1906.04313

"Bell's Theorem rules out many potential reformulations of quantum mechanics, but within a generalized framework, it does not exclude all "locally-mediated" models. Such models describe the correlations between entangled particles as mediated by intermediate parameters which track the particle world-lines and respect Lorentz covariance. These locally-mediated models require the relaxation of an arrow-of-time assumption which is typically taken for granted. Specifically, some of the mediating parameters in these models must functionally depend on measurement settings in their future, i.e., on input parameters associated with later times. This option (often called "retrocausal") has been repeatedly pointed out in the literature, but the exploration of explicit locally-mediated toy-models capable of describing specific entanglement phenomena has begun only in the past decade. A brief survey of such models is included here. These models provide a continuous and consistent description of events associated with spacetime locations, with aspects that are solved "all-at-once" rather than unfolding from the past to the future. The tension between quantum mechanics and relativity which is usually associated with Bell's Theorem does not occur here." 

2. https://arxiv.org/pdf/2101.05370.pdf

This addresses issues around entanglement swapping (which can occur without any specific speed restraint) while attempting to maintain locality (which you want).  

In addition, there are other interpretations that are "local non-realistic" or perhaps better termed "local acausal".  The best known of these (but not all that well known) is Relational Blockworld.  Unfortunately, none of my references on it were in a format to really address the question at hand without excessive study.  Instead: 

3. https://www.physicsforums.com/insights/retrocausality/

This touches on Relational Blockworld and its relationship to some of the ideas of Wharton et all in references 1. and 2.

So to answer your question: there are interpretations of QM that are explicitly local and non-realistic.  I would personally call MWI non-realistic as well, but its proponents would scream and shout that it is the most realistic.  At any rate, the interpretations debate rages.  My friends that are ardent professionals in the area of Quantum Field Theory point out that QFT is completely relativistic, and features local causality.  On the other hand, they acknowledge the obvious issues with FTL quantum teleportation and limits due to the uncertainty principle.

Answer 3

If an entangled particle pair has no pre-determined relative position, in which sense is the theory local? Furthemore, if both particle states are undetermined before the measurement, why would the local act of measuring the first one give any certain information about the second one, which is completely undisturbed by our local operation? I don't see any reasonable notion of locality that can accomplish this

I think there's some confusion in this sentence.

You can think of entangled particles as sharing a stronger form of correlation. Classically you can imagine them as being correlated in the sense that, for example, if one is black then the other is white, and vice versa. In the quantum case, the correlations are stronger than that, and whether you observe such correlations depends on the way you measure each one.

So the "local act of measuring the first one" can give "certain information about the second one" — provided we're talking about a maximally entangled pair and suitable choices of measurements — because they were correlated to begin with. Nothing particularly weird about this. It only gets weird when you realise that such correlations remain for (infinitely) many choices of measurement bases, in a way that can't be replicated classically.

This is also why the act of measuring gives information about the second particle even though the second particle is not disturbed by the act of measurement. You get information because there was a prior correlation (and you assume you know the structure of this correlation beforehand in this context). At the same time, even though after the measurement your description of the state of the second particle is changed (think of it as you updating your description of the second particle), the second particle is unperturbed in the sense that its reduced state is unaffected by the measurement outcome.

Nevertheless, you cannot explain the observation as simply due to shared correlations in the classical sense, hence why it's tempting to say that measuring one "actually affects" the other one, whatever that is taken to mean. See also Does Bell's theorem imply a causal connection between the measurement outcomes? for more info on this.

In conclusion, the theory remains local because the local state of the other particle is not affected by whatever you are doing to yours. At the same time, the correlations are (can be) "quantum", meaning that they cannot be simply explained by our lack of knowledge of some hidden variable, and therefore we might conclude that the link between the particles is "stronger than just correlations". But they still are "just correlations" because you cannot exploit them for superluminal information transfer in any way (which is just another way to phrase the fact that the local states are not affected by the measurements on the other side). So at the end of the day, they are both "just correlations" and "not just correlations", depending on how you choose to understand those words.

Answer 4

Bell's theorem proves that any local, deterministic theory with statistical independence can't account for certain observed correlations. The standard QM formulation is usually considered local, with no determinism, but I just don't see how determinism can be false without locality also being false?

The account of QM given by many sources is a kludge. It takes quantum mechanical equations of motion such as the Schrodinger equation and then adds an idea called collapse on top. Collapse is a process that supposedly picks one of the possible values of an observable and sez that is the value. In most accounts no explanation of collapse is given so we can't say much about such theories and don't need to discuss them further.

Some physicists try to come up with a variant of quantum theory that explains how collapse takes place. Such theories make predictions that differ from those of quantum theory

https://arxiv.org/abs/2310.14969

They are also non-local since Bell's theorem only actually relies on describing measurement outcomes in terms of stochastic variables. A stochastic variable is a physical quantity that has values that are picked with some probability out of a set of possible values. Such a variable could be local if the probability of measurement outcomes depended locally on the values of other physical quantities. Collapse theories effectively change quantum observables into stochastic variables so they are non-local.

If an entangled particle pair has no pre-determined relative position, in which sense is the theory local? Furthemore, if both particle states are undetermined before the measurement, why would the local act of measuring the first one give any certain information about the second one, which is completely undisturbed by our local operation? I don't see any reasonable notion of locality that can accomplish this.

The answer to this question is a matter of controversy because the standard way of discussing quantum theory is a kludge that can be used to discuss predictions but not to give any account of what is happening in reality to produce those predictions.

There is a version of quantum theory that isn't a kludge. It involves taking the equations of quantum theory and trying to work out their implications without collapse. (What follows is a significant simplification for brevity.) What happens in this version of quantum theory is that measurable physical quantities are described by quantum observables represented by Hermitian operators. The possible results of measurements of these observables are the eigenvalues of those operators. These different possible values can interfere with one another under some circumstances such as single particle interference experiments. There is no collapse but when information about a measurement result is copied that prevents interference between the different values of an observable:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/0707.2832

As a result this theory predicts the existence of many variants of the world we see around us and is commonly called the many worlds interpretation. This is a somewhat sensational term for just taking QM equations of motion seriously as a description of how the world works.

If the observables of a system depend only on the observables of systems with which it has interacted the the relevant quantum mechanical theory is local. There is an explanation of how entanglement correlations arise from such a local theory as a result of quantum information being carried in decoherent quantum systems in a locally inaccessible form:

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

https://arxiv.org/abs/1109.6223

https://arxiv.org/abs/2008.02328

The correlations don't arise until information about the measurement results from each system are brought together, e.g. - by light from one side of the lab travelling to the other side of the lab.

There is a local theory about how entanglement works: quantum theory. I haven't found any other account of the mechanism by which entanglement correlations are created.