Statistical Distance vs. Actual Distance, I am grappling with this concept and need a succinct explanation of the difference between statistical distance vs actual distance with some examples of how to calculate the numerical values for each distance.
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Here it is.
I think the main point of confusion here is that Wootters uses teh same notation for outcome probability and configuration probability.
What do I mean by that?
say you have a dice that you throw 50 times. You have a probability of 1/6 to get a certain value. The number 1/6 is an outcome probability.
the configuration probability is the answer to the question after these 50 trials - what is the probability to have 10 results as "1", 20 results of "2", 5 results of "3", 15 results of "4" and no "6" or "5"?
Obviously, these are not the same probabilities, but Wootters uses the same notations, namely p_i interchangeably. Or in other, more elegant words: #$%%^^#@$#$!!
Anyway, if you have any questions - you know where I am.
Statistical distance and Hilbert Space.
Wootters starts his discussion with a simple question - how do we know that two states are distinguishable if we have finite number of measurements.
If we inspect the results of a finite number of probabilistic events we can only estimate the appropriate quantities (such as average) because due to the finite number of samples we have fluctuations. If we measure say in 10 samples an average outcome of 0.25, is it different from an outcome of 0.255? What is the resolution of our measurement? If we take 100 samples instead - ill then can we say that the two measurements are distinguishable?
If we look at the different outcomes as tick marks on a ruler - what is the minimal distance between the ticks that which we can identify two separate ticks? What does this distance mean?
He then goes on to explore a very simple system that has only two outcomes. A photon can pass or be reflected through a filter that is rotated by an angle . The photon probability to pass the filter is given by
We throw a finite number of photons on the filter and we count how many passed through. From this information we assume we can find p (the ratio of photons that passed to the photons that we threw at the filter), and from that we can calculate the angle of the filter.
However, because we have finite number of photons, we get a certain value for p which has uncertainty value associated with it (in the extreme, think about throwing only one photon that passes. We can say that p=1 but obviously this measurement has a huge error associated with it).
This uncertainty value (root mean square) is given by:
Where n is the number of photons that pass.
As we can see, we will have uncertainty as long as the number n is finite - which is always the case. We can only try to make it as large as possible.
If there is uncertainty in p and p is a function of , then we will have uncertainty in the calculated value of based on the value of ...
The expert examines statistical distance versus the actual distance.