A while ago I've enjoyed Mermin's accessible introduction to quantum
physics for those who are already familiar with probability theory and
computer science: http://arxiv.org/abs/quant-ph/0207118
Andrei wrote:
However, one may say: well we have a rather strange description of
quantum waves-particles, namely, by using the tensor product of Hilbert
spaces to describe composite systems, but it may be, nevertheless, a
purely classical and local model behind this? Of course, Bell would say
you: not at all.
So we have the notion of the wave function \Phi. \Phi is a superposition
of all possible classical states, and each classical state has an
associated weight.
As for our disagreement regarding the bathtub, and classical versus
quantum waves. I agree that quantum physics would model the wave as a
weighted superposition of all possible measurements (AB, notAnotB,
notAB, AnotB). On the other hand, the classical wave, under the
assumption of determinism, only carries a single outcome. But we do not
know what that outcome is! In quantum physics we thus model the
uncertainty in a single experiment: the wave function is a model of the
underlying process.
In a classical context we can also model the uncertainty about what
exactly will be the outcome of the full experiment. While each
individual experiment might be deterministic (here we do not distinguish
between inherent randomness and observer's ignorance), across a number
of experiments, we cannot predict the outcome of the full experiment:
sometimes it will be AB, sometimes notAnotB, sometimes AnotB and yet
sometimes notAB. In that respect, switching off the light in the room
with the bathtub would correspond to the quantum wave: as we don't know
what pattern of the splashes into detectors' eyes will happen, we have
to allow for all four possibilities (each with its probability) prior to
learning the actual results of the measurement.
In that respect, once we learn the value of A, we obtain some
information about the value of B. I would express entanglement by the
inequality I(A;B) > 0 (mutual information between A and B is greater
than zero) which means that A and B have something in common. If so,
H(B|A) < H(B) (entropy of B when knowing A is less than the entropy of B).
Thus, the joint probability mass P(A,B) is constant. However, P(B|A=a)
is not the same as P(B) if A and B are not independent (ie. are entangled).
My suggestion would be to reconsider the wave function and to interpret
the "weight" corresponding to each Hilbert space base vector as a
classical probability. But, I might be missing something. If I am, what
is the difference between quantum probabilities and joint probability
distributions? Can quantum probabilities do something that joint
probability distributions can't? Is the process of quantum measurement
the same as the operation of taking the conditional distribution?
There is one thing that couple both detectors. This is nothing else than
time. In all experiments there is such a thing as TIME WINDOW and
experimenter identify two clicks as belonging to an entangled pair
if these clicks are inside the time window.
Agreed, this relates to the problem of defining what is an event in
classical probability. In that respect, the definition of an event
depends on the time window within which we interpret two detections as
relating to the same event. This is how we have chosen to conceptualize
the world.
the time window we identify a special series of clicks in labs, so we
extract a special ensemble S_AB of particles. If we choose another
orientations, say C and D, then through the time window we shall get
another ensemble S_CD. If the situation is really such and ensembles are
really statistically different, then this gives us purely classical
explanation of the violation of Bell\'s inequality. However, it is common
to say that S_AB has the same statistical properties as S_CD.
I would be interested in how to express A,B,C and D in terms of
properties. Thus, if we have detectors A and B, and o(A) and o(B) are
their orientations, I would be interested in the probabilistic model of
P(A,B|o(A),o(B)) or the joint probability distribution of clicks in A
and B given the orientations of A and B.
called FAIR SAMPLING ASSUMPTION. I thing it is not justified, see
http://www.arxiv.org/abs/quant-ph/0309010
[Experimental Scheme to Test the Fair Sampling Assumption in EPR-Bell
Experiments]
This is intriguing. I would be curious about the results of such
experiment. Indeed, the above model P(A,B|o(A),o(B)) would require the
probabilities to sum up to 1 for normalization purposes, perhaps simply
by having a high probability of the 0/0 event (no detection in either
detector).
Finally, I would recommend Cerf&Adami's paper "Information theory of
quantum entanglement and measurement"
http://quic.ulb.ac.be/publications/1998-PhysicaD-120-62.pdf - It is
closely related to my own work in a statistical/information theoretic
context (http://kt.ijs.si/aleks/Int/)
Best regards,
Aleks Jakulin
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