# Locality vs Separability

1. Oct 9, 2009

### Demystifier

Some papers on foundations of quantum mechanics distinguish the notions of locality and separability (or non-locality and non-separability). Can someone explain to me what is the difference between locality and separability? Or can someone point to a reference where this difference is explained?

2. Oct 9, 2009

### WaveJumper

I would guess that the fine line between the two is the scenario where 'infinite speed'(or much greater than the speed of light), at least hypothetically, can exist as a phenomenon, while the universe as a system remains separable(i.e. everything is not one wholeness). I think Non-separability includes Non-locality, while Non-locality, as a term, must not necessarily include Non-separability.

Last edited: Oct 9, 2009
3. Oct 9, 2009

### ThomasT

I doubt that I'm able to explain anything to you in any way better than you probably already understand it, while I have learned from your contributions here -- but since you're starting a discussion on the difference between the notions of nonlocality and nonseparability, then here's my two cents.

Nonlocality refers to violations of the principle of locality. Bell tests structure out the possibility of local causal interactions/transmissions between spacelike separated events at A and B during any given coincidence interval.

Nonseparability refers to statistical dependence between events at A and B. The experimental violation of Bell inequalities requires that paired events (A,B) be statistically dependent (detection at one end affects the sample space at the other end).

The most direct way to model locality is via a factorable representation of the joint probability (Bell locality condition of an lhv). The most direct way to model statistical dependence or nonseparability is via a nonfactorable representation of the joint probability (standard qm).

The reason that nonlocality (in nature) cannot be inferred from experimental violations of Bell inequalities is because the Bell locality condition isn't, exclusively, a locality condition.

An article by Peres:

http://arxiv.org/abs/quant-ph/9609016v1

4. Oct 9, 2009

### DrChinese

I think you know that this area has been sliced and diced a lot. There are those who point to Bell's paper as the starting point (or the ending point) for the discussion. Unfortunately, as much as I am enamored of the Bell paper for its science: I think some of the terminology and structure of the argument has been something of a breeding ground for some confusion. So while I don't see there as being much debate on the conclusion itself, there is a lot of debate on the explicit and implied elements of the argument.

Fundamentally, I see the separability requirement as a denial of 2 particle state entanglement. So Bell's (2) is a statement of the EPR implied position that entanglement cannot exist for space-like separated particles. That position, in turn, assumes locality, i.e. that there do not exist physical connections (exceeding c) between any 2 particles.

You are asking whether separability and locality are the same. In your non-local view, there is no locality as ALL particles influenced (and are influenced by) all other particles. But that view does not mean that all particles are entangled; clearly you still have that special behavior that is a result of shining a laser into a PDC crystal - which creates entangled photon pairs. If you believe that those pairs are sharing a wave state, you believe in entanglement and deny separability. ON THE OTHER HAND: I don't yet share your non-local view and yet I share your denial of separability. So I would say that locality and separability are NOT the same thing. If you can picture a alternative universe in which there are non-local forces but there is no entanglement of particle states (hardly something that is a direct deduction from non-locality), then they must not be the same thing.

How are they related? Is one a subset of the other? In my mind, entanglement is contained within QM. On the other hand, locality is not. So I see Bell as proving: separability (denial of entanglement) is NOT possible if QM is correct. But at the same time, he ALSO proves the more important result that local realism is not possible if QM is correct. Bell had to assume realism (see his 14 to 15) to prove that separability is incompatible with QM.

Seen another way: I assume that you can at least acknowledge the possibility that we could live in a time-symmetric universe in which locality is otherwise respected. Then we could have locality without separability (or realism) - so again they must not be the same thing. I wouldn't imagine that you could find a definitive discussion of this subject, because everyone has a certain historical twist on the matter. References you might be interested in:

1. More from Peres, Quantum Theory: Concepts and Methods (2002, see page 160 of 464): http://www.fisica.net/quantica/Peres - Quantum Theory Concepts and Methods.pdf

"The title of Bell’s second paper is 'On the Einstein Podolsky Rosen paradox,' but, contrary to the EPR argument, Bell’s is not about quantum mechanics. Rather, it is a general proof, independent of any specific physical theory, that there is an upper limit to the correlation of distant events, if one just assumes the validity of the principle of local causes. This principle (also called Einstein locality, but conjectured well before Einstein) asserts that events occurring in a given spacetime region are independent of external parameters that may be controlled, at the same moment, by agents located in distant spacetime regions."

2. From Marinescu & Marinescu, Quantum Information and Error Correction From Classical to Quantum Concepts(2009, page 139 of 702): http://www.eecs.ucf.edu/~dcm/QCV2.pdf [Broken]

"Bell's inequality is derived using a very simple model of a physical system that makes only two common sense assumptions: physical properties are independent on observations, the realism principle, and the measurement of different physical properties of different objects carried out by different observers at distinct locations cannot influence each other, the locality principle."

3. Norsen has written on this, of course, and I assume you are already familiar with his work on that - such as: http://arxiv.org/abs/0707.0401 or http://arxiv.org/abs/quant-ph/0601205 (and keeping in mind that I think Norsen is wrong on much of the history and semantics of this issue):

"Bell Locality then entails the following: once we specify a complete description of the pre-measurement state of the particle pair, the probability for Alice to obtain a certain outcome A for a measurement along a certain direction ˆa is independent of the setting (ˆb) and outcome (B) of Bob’s experiment. In particular, the probability in question does not change depending on whether we do or do not specify this information about Bob’s experiment."

So Norsen sees Bell locality as equivalent to a kind of statistical independence, as expressed by the separability formula. I say that should be called Bell separability instead, so as to clarify that violation of separability does not require that there is spooky action at a distance. Norsen sees action at a distance as essentially being a deduction from a Bell Inequality violation, whereas this conclusion is nearly universally rejected elsewhere.

Last edited by a moderator: May 4, 2017
5. Oct 9, 2009

### RUTA

The ontological distinction, as I see it, is that in non-locality you have a superluminal causal mechanism or exchange of information between the entangled particles. In non-separability you don't really have "particles" but just one entity that is responsible for the detector clicks.

6. Oct 12, 2009

### Demystifier

Thank you all!
Now the difference is completely clear to me.

7. Oct 12, 2009

### DrChinese

Ha ha!! Now can you explain it to me then... :rofl:

8. Oct 13, 2009

### Demystifier

To avoid repeating of the nice explanations above, I will try with a short indirect explanation:

Non-separability of QM means that, in the system of many particles, there is no separate wave function for each particle (due to entanglement). In other words, there are statistical correlations among particles. All physicists agree that QM is non-separable.

If we assume that reality exists even without measurements, then, according to the Bell theorem, distant particles must influence each other instantaneously (or faster than light). This is non-locality. Since not all physicists agree that reality exists even without measurements, not all phsicists agree that QM is non-local.

9. Oct 13, 2009

### DrChinese

Well said! I like it...

10. Oct 14, 2009

### ThomasT

Bell's theorem says that factorable lhv entangled state formulations are incompatible with qm and experiments. They're incompatible because the nonseparability of entangled subsystems manifests experimentally via statistical dependency between the separately accumulated data sets via the data matching process.

Violation of Bell inequalities tells us nothing whatsoever about whether Nature is local or nonlocal, or about whether there is a Nature independent of our observations. It tells us only that one of the assumptions (statistical independence) embodied in the formulation of the Bell theorem-inequality has been contradicted.

The defacto standard mainstream scientific assumptions are (1) Nature exists independent of observation, and there are deeper levels of Nature than that revealed via our senses, (2) Nature obeys the principle of locality, and (3) Nature is deterministic.

Whether one wants to call qm local or nonlocal or acausal is irrelevant.

11. Oct 14, 2009

### DrChinese

I would say that there is NOT mainstream agreement on any of these. Reality, locality and determinism are hotly debated across the board. Sorta surprised to see you make these assertions given the discussions you have been a part of here.

12. Oct 15, 2009

### zonde

From Bell theorem we can conclude that for real entanglement experiments fair sampling assumption is not applicable.

Idea that distant particles must influence each other instantaneously will fail very quickly if you would examine mind experiment with three entangled photons where conflicting "influences" should take place.

13. Oct 15, 2009

### Demystifier

Experiments (that test Bell inequalities) show that either (1) or (2) is wrong.
If by "science" one means "physics", then (3) is certainly not the mainstream assumption.

14. Oct 15, 2009

### Demystifier

Fair sampling assumption has more to do with actual experiments (having low detector efficiency) than with the Bell theorem.

Who said that the influences are conflicting? They are not.

15. Oct 15, 2009

### zonde

Fair sampling assumption is required if you want to apply Bell's theorem to actual experiments.

And if you have two photons with polarizers at 45deg relative angle and third photon with polarizer in between first two (22.5 and 22.5 deg)?
Clearly third photon has conflicting "influences" form first two photons.
Or you think otherwise?

16. Oct 15, 2009

### Demystifier

With that, I agree.

Yes I do. In fact, without both influences, the photon would not even know how to behave, because the information would be incomplete. The behavior of the photon is a single-valued function of both variables (influences).

This is like a 3-body problem in classical Newtonian mechanics. The acceleration of the first body is determined if and only if the positions of both second and third body are known.

17. Oct 15, 2009

### zonde

I guess you mean that without influence photon will behave in random fashion.

But let's see about two influences in the case I mentioned ... quantitatively.
Two entangled photons from polarizers at 45deg relative angle will have 50% coincidences in idealized case as cos^2(45deg)=0.5 (I assume fair sampling obviously).
Third entangled photon has to have 85% coincidences with the first photon and 85% coincidences with second photon as cos^2(22.5deg)=0.85.
The maximum amount for what all three photons can coincidence is 50% (that's because that is the number for first two photon coincidences). So it means that the rest of 35% from both of 85% coincidences of the third photon should be separate for first photon and second photon. But now for the third photon we have:
50%+35%+35%=120%
Ups, something wrong :uhh:

Where is the problem?

18. Oct 15, 2009

### Demystifier

No, I don't. I mean without the influence the photon will not behave at all, because the influence is a part of the physical laws. See the 3-body-problem analogy that I mentioned.

The maximum amount (for what all three photons can coincidence) should not be identified with the actual amount. The actual amount is not 50%, but less.

19. Oct 15, 2009

### zonde

I do not see connection. Singlet photon stream will interact with polarizer anyways. In that case you have Malus law.

Ok, but if you look at it this way then these 35% is minimum so that those final 120% can only increase but not decrease.
replace 50% with x <= 50% and we have:
x+(85%-x)+(85%-x)=170%-x >= 120%

So this is not a solution to the problem.

20. Oct 15, 2009

### Demystifier

You are talking about statistics in ordinary QM. I am talking about hypothetical deterministic influences in a nonlocal hidden-variable formulation of QM.

I don't see your point. Are you saying that probabilities in ordinary QM do not sum up to 1?