2019 International Workshop: Beyond Bell’s theorem

[Richard Healey]

https://www.ijqf.org/wps/wp-content/uploads/2019/08/Beyond-Bell-post.pdf

In a recent archive post ([1]) Shan Gao has argued that quantum theory is incompatible with relativity. He calls this a new proof beyond Bell’s theorem, arguing elsewhere ([2]) that it closes the superdeterminism loophole in Bell’s theorem. Such strong claims must be backed up by irrefutable arguments. My aim in this post to the workshop “Beyond
Bell’s theorem” is to refute Gao’s “proof” and to show how quantum theory is compatible with relativity theory and so why Gao’s “proof” does not take us beyond Bell’s theorem.

[editor]

Hi Richard,

Thanks for your comments! In your paper, you said:

“In the second scenario (i.e. Bob precedes Alice) Gao argued that the Born rules yields probability 1 for the sequence of outcomes of Alice’s repeated z-spin measurements in which every outcome is the opposite of Bob’s single outcome of a measurement at point 3, and probability 0 to every other possible sequence of outcomes of Alice’s z-spin measurements. This is incorrect.”

I cannot agree. I think amost all people will agree that this is a correction application of the Born rule (at least in one frame).

If this is not correct, then QM will be unable to explain the perfect anti-correlation between the results of Alice’s and Bob’s spacelike measurements.

Best,

Shan

[Richard Healey]

The Born Rule should always be applied not at a time (that would already introduce a preferred frame in relativity) but from what I have elsewhere called an agent situation—a physically specified situation that may or may not actually be occupied by an agent. In this Gedankenexperiment we may take an agent situation to be adequately specified by a spacetime point, which may be thought to mark the momentary location of an (actual or merely hypothetical) idealized agent applying the Rule.

In the second scenario there are several relevant agent situations–points on Alice’s world line immediately prior to her measurements a- e, and points just after point 3 on Bob’s worldline.

In the passage from my previous post quoted in Shan’s reply I was assuming that the Born Rule is being applied from points on Alice’s world line immediately prior to her measurements a – e. From those agent situations, the application correctly issues in the probability assignments I gave in that post: the chance of a +1 outcome of Alice’s measurement and the chance of a -1 outcome of Alice’s
measurement both equal 1/2, as I stated in the original post.

I now add that the Born Rule may also be applied from points just after point 3 on Bob’s worldline (in the lab frame before any of Alice’s measurements). From those agent situations, the application correctly issues in a chance of either 1 or 0, depending on the outcome of Bob’s measurement. It is these chances that enable Bob (or anyone else who might have been located at such a point on Bob’s world line) to predict Alice’s outcomes with certainty, and afterwards to explain the perfect anti-correlation between the results of Alice’s and Bob’s spacelike separated measurements.

It is only if one illicitly smuggles in the nonrelativistic assumption that the chances of Alice’s outcomes change at an instant everywhere at once consequent upon Bob’s measurement having one definite outcome rather than the other that one can one take an application of the Born Rule from points just after point 3 on Bob’s worldline to give “the real” chances of Alice’s outcomes at that instant.

It is interesting that even Bell makes that illicit assumption when giving his version of the EPR argument in “La Nouvelle Cuisine” to conclude that after Bob’s measurement the outcomes of Alice’s (first) measurement “is predetermined”. It is certain relative to the contents of the back light cone of Bob’s outcome, but it is not certain relative to the contents of Alice’s back light cone at a point on her world line just before she makes her (first) measurement, even though her world line reaches that point after (in the lab frame) the outcome of Bob’s measurement. The whole notion of “predetermination” must be reinterpreted in relativity by identifying “the past” of an event with its back light cone—so events that occur at different places at the same time in the lab frame have different pasts.

If Shan is right that almost everyone agrees with his application of the Born Rule in his second scenario, then almost everyone is wrong!

[editor]

Thanks for your clarification, Richard!

But I cannot agree with you. I think the Born Rule must be applied at a time, while this would not necessarily introduce a preferred frame in relativity as some people think, although my new argument says the opposite.

The reason is that if the Born Rule is not applied at a time for a joint measurement of Alice and Bob on the EPR pair, then the theory will be unable to explain the perfect anti-correlation between the results of Alice’s and Bob’s spacelike measurements. At least in one frame, immediately after Bob’s measurement, the Born rules must yield probability 1 for the sequence of outcomes of Alice’s repeated z-spin measurements in which every outcome is the opposite of Bob’s outcome, whether for Alice or other observers in the frame. Otherwise the predictions will contradict experiments. This is my point in the last post.

[Richard Healey]

In my previous reply I said that when the Born Rule is applied from an agent situation indexed to a spacetime point p on Bob’s worldline that is timelike later than Bob’s outcome (such as point 3 in Figure 1) it yields chance 0 or 1 for each of Alice’s measurement outcomes in regions a – e. So from that agent situation, each of Alice’s outcomes is certain to be opposite to Bob’s. But from a point on Alice’s worldline at the same lab time t as point p, application of the Born Rule yields chance 1/2 for each of Alice’s possible measurement outcomes in regions a – e. If the Born Rule were to be (incorrectly) applied at lab time t it would yield incompatible values for “the” chance at that lab time of each of Alice’s possible measurement outcomes in regions a – e.

But there is no such thing as the chance at t of each of Alice’s outcomes. There are just the chances at each spacetime point of these events and these chances differ from one point to another on the spacelike hyperplane indexed by t.

Knowing his outcome, Bob can predict Alice’s outcomes with certainty. Since Alice cannot know Bob’s outcome before that event enters her back light cone, all she can predict with certainty at points on her worldline before making those measurements is that, whatever they are, they will be opposite to Bob’s. That is how correct applications of the Born Rule from different agent situations correctly predict the perfect anticorrelations between Bob’s and Alice’s measurement outcomes in the second scenario, in which Bob’s measurement precedes (in the lab frame) all of Alice’s measurements. (Relativistically, there is no relevant difference between the first and second scenarios.)

It is easy to fall into the trap of asking what “we” can predict using quantum theory, as if prediction is a concept that makes sense without regard to the spacetime location of the predictor. Taking the “God’s eye” perspective of one not situated in a spacetime such as that depicted in Figure 1 it may seem natural to think of prediction as just a logical or probabilistic relation between events at different lab times, each represented by a horizontal line in the diagram. But prediction is a concept that makes sense only from the perspective of a situated agent with access to certain information but prevented by that situation from accessing other information (s)he would like to have. In this situation prediction is just what is needed to form reliable beliefs about this inaccessible information.

PS In my previous post I meant to cite Bell’s 1964 paper “On the Einstein Podolsky Rosen Paradox”, not his later paper “La Nouvelle Cuisine”. He does not talk about ‘predetermination’ in the more recent paper, but he continues to assume that it is meaningful at a time to assign a unique chance to an event at a later time. But assignments of probability and chance cannot be made that way without assuming an absolute time or a preferred frame. In a relativistic spacetime with no preferred frame, each meaningful chance assignment must be made from a spacetime point, not at a time.

[editor]

So, you also agree that “in the second scenario, in which Bob’s measurement precedes (in the lab frame) all of Alice’s measurements”, “Since Alice cannot know Bob’s outcome before that event enters her back light cone, all she can predict with certainty at points on her worldline before making those measurements is that, whatever they are, they will be opposite to Bob’s.”

But this is just my analysis.

“In the second scenario Gao argued that the Born rules yields probability 1 for the sequence of outcomes of Alice’s repeated z-spin measurements in which every outcome is the opposite of Bob’s single outcome of a measurement at point 3, and probability 0 to every other possible sequence of outcomes of Alice’s z-spin measurements.”

[Jerry Finkelstein]

In the case in which Bob’s measurement occurs after Alice’s, Richard
Healey and Shan Gao agree that Alice would expect to obtain the result
“+1” about 50% of the time. Let’s think about what Bob would expect
Alice’s results to be. I will suppose, to make the story more
definite, that Bob’s own measurement has the result “-1”.
Consider first a simplified scenario: Alice measures only once,
and the superobserver does nothing. As soon as Bob records his own
result, he knows that Alice’s result (was/is/will be) “+1”, but (as
Healey points out) Alice must apply the Born rule to the original
entangled wavefunction, so she assigns probability 50% to the result
“+1”. There is no contradiction here; Bob knows something that Alice
does not know, so it is not surprising that she might assign
probability 50% to an event which Bob thinks is certain.
In the full scenario, Alice measures several times, and each time
everything in her laboratory is “reset” by a superobserver. In the full scenario
Bob will expect that Alice will obtain the result “+1” each time.
(Alice’s first measurement, before the superobserver does anything, is
just like the simplified scenario, so Bob knows that Alice’s first
result (was/is/will be) “+1”; after the superobserver has reset the
first measurement, the situation is again the same as it was before
the first measurement, so Bob knows that Alice’s second result
(was/is/will be) “+1”; etc.) So Bob expects that Alice will see the
same result each time, while (as Healey and Gao agree) Alice expects
the result “+1” only 50% of the time.
On the other hand, neither expectation could be verified. The
superobserver has used its super-powers to erase anything which could
serve as a record of Alice’s measurement results, including contents
of Alice’s memory, and so it is not clear that it is meaningful to
even talk about results of Alice’s measurements. But to the extent
that this is meaningful, it would seem that unitary quantum theory has
induced Alice and Bob to have contradictory expectations. If that is
not an acceptable conclusion, then perhaps the moral of the story should be
(as Asher Peres might have said): Measurements reset by a
superobserver have no results.

[editor]

Thanks for your useful comments, Jerry!

[Richard Healey]

Thanks, Jerry.

A little clarification:
You say

Healey and Gao agree Alice expects the result “+1” only 50% of the time

.

In her situation prior to each of her individual measurements Alice expects each of its possible results to be equally likely. But in her situation prior to the whole sequence of her future measurements Alice expects either a sequence of all +1s or a sequence of all -1s, each with probability 1/2, and expects no other sequence (in particular, she assigns probability 0 to a sequence of an equal number of +1s and -1s).
All of these expectations are “consistent” with each other and with Bob’s expectations since a rational agent’s expectation varies as that agent’s situation changes so as to give the gent access to additional information.

[Jerry Finkelstein]

Hi Richard –
Thank you for your response to my earlier posting. I had understood that
you were agreeing with Shan that Alice would expect 50% of her results to
be “+1”; this clarification eliminates my worry about a contradiction
between the expectations of Alice and Bob. Alice would say that each
individual result has a probability of 50% to be “+1”, but these
results are not statistically independent; since each is constrained
to be opposite to Bob’s single result, they would all agree with each
other.
Now I am wondering what you would say about the case in which there
was also a superobserver who could reset Bob’s measurements. Suppose
for example that Alice and Bob each measure twice, that their labs are reset
after each measurement, and that each of Bob’s measurements are
at space-like separation from each of Alice’s. Question: would the following
set of results be possible: Alice’s first result is “+1”, her second
is “-1”, Bob’s first result is “-1”, and his second result is “+1”?
If the answer to that question is “yes”, then the superobserver on
Bob’s side would be able to superluminally influence Alice’s results
(because if it chose not to reset Bob’s measurement, Bob would only
measure once, in which case it would not be possible for Alice’s first
result to be “+1” and her second result to be “-1”). That might seem strange,
but would not actually be a superluminal signal since after the resets the results
are unobservable. However, I am guessing that your answer is “no” and
that the only allowed possibilities are that Alice sees “+1” both times and that
Bob sees “-1” both times, or visa versa.
Of course if (as I am imagining that Asher Peres would say) it is
meaningless to talk about results of measurements which have been
reset, then the questions above would not apply.

[Richard Healey]

Thanks, Jerry: that’s interesting.

You say:

I am guessing that your answer is “no” and that the only allowed possibilities are that Alice sees “+1” both times and that Bob sees “-1” both times, or visa versa.

For the spacelike separated case you describe, “no” is indeed my answer.
You are right to distinguish superluminal influence from superluminal signalling, and to see that (for anyone who answered “yes”) there would be the possibility of superluminal influence without superluminal signalling in that case.

The application of Asher Peres’s dictum “Unperformed measurements have no results” is a bit unclear here. If a measurement is performed but its result is then completely erased, does that count as a performed or an unperformed measurement? Asher was happy to be called a positivist, so maybe he would have said that counts as an unperformed measurement. I am not a positivist, and I don’t think it is meaningless to talk about results of measurements which have been reset. So I think his dictum doesn’t apply here. That’s why I think it matters to say “no” instead of “yes” in answer to your question.

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