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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.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.
