Frequently asked (and answered) questions for “Quantum Theory Cannot Be Extended

Note that many of these questions were made based on version 1 on the arXiv.


Dated: 10th October, 2010.


Question

What about the following model:

X=f(A,U)
Y=f(A,B,W)

i.e. Alice's outcome is completely determined by a local HV U, and Bob's outcome is completely determined by a non-local variable W. "Alice" being the designation for the party that makes the measurement first in some preferred frame (e.g. that of the cosmological 3K radiation).

Why does it not contradict your claim?


Note that this model would allow signalling between Alice and Bob if the could observe the hidden variables (which they need to be able to do in order to use them to make better predictions):

The sender times his/her measurement to choose whether to be Alice or Bob. The receiver can use U to find out who Alice and Bob are.

More precisely, suppose X depends non-trivially on U. To make the argument simpler, consider the extreme case, where X=U. The receiver is to measure at some pre-agreed time. The sender signals by either measuring his particle before this time (so that he becomes Alice) or not (so that he becomes Bob). The receiver checks if the outcome she gets is equal to U or not. If it is not, she knows the sender has measured, which is signalling. (If the sender measures first, there is always some probability that X is not equal to U, otherwise the quantum correlations are not observed.)

Since this model allows signalling it necessarily violates the freedom of choice assumption (FR), since in the first part of the proof we show that this assumption implies non-signalling (see also the answer to the related question below).



Question
Consider the Toner-Bacon model of the singlet state. This is nonrelativistic, but we can put it into relativistic form by tweaking it a little, as follows:
(i) Introduce a preferred time-ordering on spacetime, in a covariant manner, i.e., such that all observers agree on the ordering. This is easy enough, whether in GR or SR - for example, one could specify a (slightly idealised) preferred frame of reference, as that in which the cosmic background radiation is isotropic, and specify the ordering as per this frame. Note that nothing in relativity theory rules out having a preferred frame of reference; the theory only requires that the laws of of
physics are the same in all frames.
(ii) For two observers performing a joint spin measurement on a singlet state, one of measurements will always be earlier than the other according to the preferred time-ordering. For each such measurement, call this earlier observer Alice, and the other observer Bob. The label will in general change from measurement to measurement, but is covariantly defined in each case.
(iii) Now introduce the Toner-Bacon model. In particular, Alice's result is determined by her choice of direction, as per this model, and a physical bit, c, is assumed to be propagated, faster than light (eg, along Alice's backward lightcone if you like). Bob's result is then determined by his choice of direction and c, as per this model.

In the above, the two observers have free will, QM is assumed to be correct, and all measurements are made in spacelike separated regions. The model is nonlocal, but you don't claim to assume locality. The above example must violate one of your assumptions, but it is not clear to me which one?


The reason this example does not contradict our result is that it violates our assumption FR which assumes the freedom of choice in all reference frames. In the Toner-Bacon model, we cannot satisfy this if (using the notation of
quant-ph/0304076) the communication bit, c, propagates faster than light, since then there is a frame in which c (as well as lambda_1 and lambda_2) is in the past when Alice makes her choice, a. However, since c is a function of a, lambda_1 and lambda_2, clearly a can not be considered free in this frame.



Question

Barrett-Kent-Pironio had shown that the probability point that violates maximally the chained inequality for a singlet state is arbitrarily close to the boundary of the no-signalling polytope for N large. Here, you prove that it is actually arbitrarily close to an extremal point of that polytope. You don't phrase it this way, but it seems to us that it must be the case: otherwise, if P=p1P1+p2P2, then P can be extended by "purifying" this mixture.


We don't agree with this. The chained Bell correlations are not extremal in the space of NS correlations (although, as you say, in the limit they touch a facet). However, when written as a convex mixture of extremal NS correlations, all must have the same marginal. Hence the knowledge of which extremal point is present in a particular run does not help to predict the outcome. In our paper, we rule out additional systems that convey information about the outcomes, hence this does not contradict our claim.



Question

What about models which provide higher explanation of the correlations, but for which certain parameters or systems remain hidden (as a physical principle), either forever or until after the quantum measurements have been performed?



We do not rule out this possibility. However, this does not contradict our claim since such hidden systems do not provide additional predictive power at the level of experimentalists, and hence are not extensions in our terminology.



Question

Predictions in QM can be interpreted in at least 2 ways:

a. predictions that already have been checked experimentally

or b. all predictions that could be checked in principle

For your assumption QM, do you mean a. or b.?



We mean b. Note also that we do not require all aspects of quantum theory to be correct for our result. Furthermore, this assumption is falsifiable: experiments could later show contradictions with particular aspects of the theory.



Response to arXiv:1007.0536

We do not think that the Scarani-Suarez model stated in this comment contradicts our claim. In that model, Alice could signal to Bob using the following procedure.  Alice and Bob agree
on a reference frame, and Alice agrees to measure at time t in that frame.  Bob signals to Alice by measuring either before or after time t in that same frame.  According to the model,
Alice's outcome is then given either by equation (1) or (2) of arXiv:1007.0536.  Alice could calculate these functions herself and thus guess whether Bob measured before or after, which
allows her to receive information superluminally (before could represent 0 and after could represent 1 for example).  Since the non-signalling conditions are not obeyed, this model
cannot be compatible with FR according to the first part of our proof.

In addition, we are confused by the argument that our extensions are self-contradictory.  In fact, we would see the argument given on the last page of this comment as confirmation of
ours:  It posits an extension with D=0.25, then argue that such an extension is contradictory. However, it is then suggested this makes the notion of such an extension contradictory.  We
would instead simply conclude that an extension with D=0.25 cannot exist. We could argue similarly for any D > 0, and thus conclude that all extensions must have D=0.