Sum of the probabilities equals 3 in bipartite covariance ?

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Discussion Overview

The discussion revolves around the probabilities associated with a bipartite system, particularly in the context of the EPRB experiment. Participants explore how the sum of probabilities can equal 3 and examine the implications of measuring different operators in this quantum framework.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents probabilities for a bipartite system, suggesting that the sum of probabilities equals 3, and questions the reasoning behind this result.
  • Another participant asks for clarification on the specific meaning of p(+A), indicating a need for more detail on the probabilities involved.
  • A participant explains that p(+A) refers to the probability of measuring + for the A operator, describing the operators involved and the context of measuring a singlet state.
  • One participant introduces the concept of three certain events, suggesting that if p(A), p(B), and p(C) are all 1, their sum would equal 3, prompting further inquiry into the implications.
  • Another participant reflects on the theory of probability, stating that the sum of probabilities for interconnected events should equal 1, which raises questions about the nature of the events discussed.
  • There is a discussion about the interpretation of probabilities for certain events, with participants debating the conditions under which probabilities sum to 1 or 3.
  • One participant acknowledges the axioms of probability and attempts to categorize events A, B, and C, noting that some events are derived from others, which adds complexity to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of probability sums and the nature of events in the context of the bipartite system. There is no consensus on how to reconcile the sum of probabilities equaling 3 with standard probability axioms.

Contextual Notes

Participants highlight the interdependence of events and the derivation of certain probabilities from others, indicating that assumptions about event independence and definitions may affect the discussion.

jk22
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If we consider a bipartite system as in EPRB experiment we get the probabilities :

p(++)=p(--)=1/4*(1-cos(theta))
p(+-)=p(-+)=1/4*(1+cos(theta))

p(+A)=p(+B)=p(-A)=p(-B)=1/2

Thus the sum of all the probabilities equals 3...

How does that come ? Is it because in fact there are only double events out of which we consider the averages of A and B sides too, thus making 3 sample set out of one experiment ?
 
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Can you be a bit more specific? What is p(+A) for example?
 
p(+A) is the probability of measuring + for the A operator. A and B are just spin 1/2 operators and we measure a singlet state Psi=1/Sqrt(2)(0,1,-1,0)

measuring A in this bipartite system means of course measuring A1=A\otimes\mathbb{1}.

If I take A=diag(1,-1) B=\left(\begin{array}{cc}cos(\theta)&sin(\theta)\\sin(\theta)&-cos(\theta)\end{array}\right) then A1 just has two eigenvalues 1 and -1. p(+A) is the probability of measurement for the eigenvalue 1 of A1.

All this was to compute the covariance \langle A\otimes B\rangle - \langle A\rangle\langle B\rangle

Of course the average of A is p(+A)-p(-A)=0= average of B.
 
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If 3 events A B and C are certain to occur we have
p(A) = p(B) = p(C) = 1 and we have p(A) + p(B) + p(C) = 3.
No problem.
What do you want to do with that?
 
I am not sure about specific probability of these operators but if I remember the theory of probability correctly,
P(A)+P(B)+P(C) = 1 means the probability of occurring event A or event B or event C is 1. If probability of these events are interconnected it is always 1. If A, B and C are independent, dividing by total events will give 1 anyway.
 
if 3 events are sure this means they happen as a whole and nothing else so we normally write p(abc) is 1 ?
 
I would say P(A)*P(B)*P(C) = 1. It means the probability of occurring event A and event B and event C is 1 which means all three events will surely happen and as you can see each of these events need to have probability 1 individually in this case.

The P(A) + P(B) + P(C) = 1 means either of these events will surely happen.
 
but the axioms of probability say p(omega) is 1 where omega is the set of all events.
I got it :
In our case event A is +-,++,--,-+
B is +a,-a
C is +b,-b

but B and C are not real events we deduce them from A, for example +a is the reunion of ++ and +-
aso.
Thanks, i have often dumb analyses.
 
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