# Quantum nonlocality without hidden variables: An algorithmic approach

1. Mar 9, 2007

### Demystifier

Is quantum mechanics (QM) local or nonlocal? Different formulations/interpretations (FI) of QM, with or without hidden variables, suggest different answers. Different FI's can be viewed as different algorithms, which leads me to propose an algorithmic definition of locality according to which a theory is local if and only if there exists at least one FI in which all irreducible elements of that FI are local:
http://arxiv.org/abs/quant-ph/0703071
The fact that no such FI of QM is known strongly supports quantum nonlocality.

2. Mar 9, 2007

### vanesch

Staff Emeritus
The problem I see is that the definition of "algorithmic" locality is what it is: a definition. The *real* issue is not a new definition of "locality", but rather: to see whether there exists a formulation of quantum theory which is entirely compatible with a spacetime manifold.
Now, as far as I understand, in the Heisenberg formulation, all operators that have any use in quantum theory can be formulated over spacetime (that is, they live on a fibre bundle of operators over hilbert space with base space spacetime). The part that is "non-local" according to your definition is just the non-evolving "state vector" in Hilbert space, but which doesn't need to be a dynamical quantity, and can be, in fact, just any vector in Hilbert space (on the condition that we apply a unitary transformation).

As such, no *dynamics* in quantum theory is non-local. But there is indeed an element to quantum theory which is not defined over spacetime, and that is the initial state vector. One could argue that that initial state vector doesn't even matter much, because whatever experiment of entanglement comes about because of entanglements due to interactions that occur much later. That is: we could even start with a specific initial state vector, which is entirely factorized over spacetime (that's a peculiar state), and have our dynamics (build up by operators which are entirely indexed over spacetime) such that we obtain entanglement later, and do "measurements" (also using these operators) still later.

As such, at no point, we needed an explicitly non-local element (not even the starting state vector). What do you think ?

3. Mar 9, 2007

### Demystifier

Vanesch, I agree with you that my definition is just - a definition.
The real issue is: Is there a better definition?

Let me put some critical remarks on your attempt to "define" locality.
On the one hand, you require a compatibility with the spacetime manifold. This suggests that space and time should be treated on an equal footing.
On the other hand, you note that (in the Heisenberg picture) nonlocality is not *dynamical*, where "dynamics" clearly refers to quantities that depend on time. However, such a dynamical notion of locality clearly does not respect the requirement that space and time should be treated on an equal footing.

Further, even if an initial product state may lead to entanglement later, this is certainly not the most general situation. QM allows different cases as well. In addition, the words "initial" and "later" again give a special role to time, which may ruin your first requirement above again.

To conclude, no matter how you reinterpret QM to make it local as much as possible, it seems there will always remain *something* nonlocal about it. This, indeed, is the point of my paper.

Last edited: Mar 9, 2007
4. Mar 9, 2007

### vanesch

Staff Emeritus
The point is: what it suggests is not important. After all, the entire importance to "locality" is its compatibility with relativity, and the central idea to relativity is that things should be defined over the spacetime manifold, and not over a specific preferred foliation of it. The most general requirement is then that all that is important in the theory should be defined as elements, sections, whatever over a fibre bundle of which the base space is the spacetime manifold. This is the same in classical (relativistic) physics: fields are sections over the (co)tangent bundle of spacetime, or of "tensorisations" of such.

Now, and that's where I have to give in a small part to "non-locality", in quantum theory (in the unitary part of quantum theory at least), all "interesting" operators are "indexed" over spacetime (that's another way of saying that they are members of a fibre bundle with base space spacetime, and in many cases, are a section), but they ACT upon a totally different space, which is Hilbert space. However, this hilbert space itself doesn't matter that much, in that we can always "rotate" the "initial" vector to any vector we like with a unitary transformation. So the "initial" vector doesn't even matter, we can make it what we want, by a right choice of unitary transformation. For instance, in QFT, we have the habit of just sandwiching any operator expression between a bra and a ket of the "vacuum". This is just a generic state, which doesn't depend on spacetime, in the same way as the number pi doesn't depend on spacetime.

No, dynamics refers to a timelike regularity, but doesn't need to introduce a specific foliation. GIVEN a foliation, it becomes an "evolution in time", but dynamics can be a spacetime concept. It actually means: "interaction" or "coupling" (between subsystems, such as the electron field, the EM field, ...).

In relativity, there is no *equal* footing between space and time, just a similar treatment. But there is a fundamental difference between "spacelike" and "timelike" all together (given by the causal structure, or the signature of the metric).

True, but one might think that the quantum case where one starts with "non-entangled" systems is probably largely sufficient to reproduce all observable phenomena. After all, that's how things are done in the lab! We use a specific interaction to have "entangled systems", starting from non-entangled product states.

It is an interesting question whether it is necessary to consider initial states which are not product states, or whether this is essentially impossible to find out. I tend to think the latter.

No, there is a precise sense in relativity to causal relationships. Not all pairs of events can be classified as "earlier" and "later", but those who can, are unambiguous.

True. The point is: in how much is that important ? The whole discussion about "locality" (as I understand it) is essentially: relativity or not ? Out of relativity (and the requirement of things to be defined over a spacetime manifold, independent of a specific foliation) came the idea of "locality", but maybe the concept of locality got a life of itself, and is somewhat more restrictive than what is strictly needed for relativity. QM seems to hide in the little margin between both.

5. Mar 9, 2007

### Demystifier

If the laws of nature are a big "algorithm", and if the algorithmic nonlocality conjecture is correct, then the laws of nature are nonlocal. I think it is important from a fundamental point of view, even if it is irrelevant from a practical point of view. This can be compared with the question whether the fundamental laws of nature are deterministic or indeterministic, despite the fact that it does not influence our practical abilities to control nature.

First, the requirement of locality is much more general than the principle of relativity. For example, the laws that govern the behavior of nonrelativistic fluids are local.

Second, relativistic nonlocal laws are also conceivable. For example, consider an action of the form
$$\int d^4x \int d^4y L(x,y)$$
where L is a Lagrangian density transforming as a biscalar. By varying such an action you will obtain nonlocal but relativistic-covariant equations of motion.

6. Mar 9, 2007

### Demystifier

If gravity is also quantized, then you have a problem. How to define causality without a background metric? BTW, what do you think about quantum gravity in general, what is your favored approach to it?

7. Mar 9, 2007

### vanesch

Staff Emeritus
I don't know much about it. I don't like the background-dependent approach of string theory (however, it could contain a background-dependent formulation of a background-independent theory, and also work ok for low-order corrections as a kind of effective theory). I like more the quantum-gravity approach, but unfortunately I do not understand enough of it to be able to say anything sensible about it.
In any case, I think we're in for a paradigm change (as such, I consider all interpretation problems of quantum mechanics as just a warm-up for the bigger fireworks). That said, I'm quite pessimistic about a paradigm change which is not driven by experimental input. The only guy who succeeded doing such a feat was Einstein, but - with hindsight - he had quite some cards on the table already, and the problem (again, with hindsight) was easier than it is now. Moreover, experimental results flowed in quite quickly.
So let us not get too excited too early. I think the first big hurdle is:
super symmetry or not ? LHC will try to give an answer in a few years.
Next, there are some precision tests of general relativity under way.
Physics is, after all, an experimental science.

8. Mar 9, 2007