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All of Viktor's Comments + Replies

The Born Probabilities
Viktor20

That gave me, if I am not mistaken, the last piece of the puzzle. Let's just take the naive definition of probability - the relative frequency of outcomes as N goes to infinity. Now prepare N systems independently in the state a|0>+b|1>. Now measure one after another - couple the measurement device to the system. At first we have (a|0>+b|1>)^N |0>. Now the first one is measured: (a|0>+b|1>)^(N-1) (a|0,0>+b|1,1>) where the number after the comma denotes the state of the measuring device, which just counts the number of measured ... (read more)

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5drnickbone
Yes, this is called the Finkelstein-Hartle theorem (D. Finkelstein, Transactions of the New York Academy of Sciences 25, 621 (1963); J. B. Hartle, Am. J. Phys. 36, 704 (1968)). This theorem is the basis for constructing a limit operator for the relative frequency when there are infinitely many independent repetitions of a measurement, and showing that the product wave-function is an exact eigenstate of the relative frequency operator. Unfortunately, it seems that Hartle's construction of the frequency operator wasn't quite right, and needed to be generalized. (E. Farhi, J. Goldstone, and S. Gutmann, Ann. Phys. 192, 368 (1989)). Even so, the critics are still picky about the construction. There is a line of criticism that infinite frequency operators can be constructed arbitrarily as functions over Hilbert space, and unless you already know the Born rule, you won't know how to construct one sensibly (so that the Hartle derivation is circular). However this seems unfair, because if you want the relative frequency operator to obey the Kolmogorov axioms of probability then it has to coincide with the Born rule, something which is another long-standing result called Gleason's theorem. (The squared modulus of the amplitude is the only function of the measure which follows the axioms of probability.) Hence the full derivation is: 1) (Postulate) If the wavefunction is in an eigenstate of a measurement operator, then the measurement will with certainty have the corresponding eigenvalue. 2) (Postulate) Probability is relative frequency over infinitely many independent repetitions. 3) (Postulate) Relative frequency follows the Kolmogorov axioms of probability. 4) (Gleason's theorem) Relative frequency must converge to the Born rule (squared modulus of amplitude) over infinitely many repetitions, or it won't be able to follow the Kolmogorov axioms. 5) (Hartle's theorem, as strengthened by Farhi et al) There is a unique definition of the relative frequency operator over i
Spooky Action at a Distance: The No-Communication Theorem
Viktor-10

Not exactly. By writing down a density matrix you specify a special basis (via the eigenvectors) and cannot change this basis and still have some meaning. Eliezer, back in the beginning of the series you wrote about classical phase space and that it is an optional addition to classical mechanics, while QM inherently takes place in configuration space. Well, the density matrix is sort of the same addition to QM. Whether we have (in some basis) a simple state or a linear combination thereof is a physical fact that does not say anything about our beliefs. But... (read more)

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Bell's Theorem: No EPR "Reality"
Viktor90

You actually get neater numbers if you take 0°, 30° and 60°. Then the probabilities are 1/8,1/8 and 3/8. :)

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The Born Probabilities
Viktor00

Hm, just read the article again and saw that many of this was already explained there. But the essential point is that although the full information of a system is given by the amplitude distribution over all possible configurations, this information is not accessible to another system. When we try to couple the system to another (for example, by copying the state), this only respects the pure 'classical' states as described above. Thus it is possible to ask the question 'how much have these two states in common', where one classical state compared with it... (read more)

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The Born Probabilities
Viktor10

First of all - great sequence! I had a lot of 'I see!'-moments reading it. I study physics, but often the clear picture gets lost in the standard approach and one is left with a lot of calculating techniques without any intuitive grasp of the subject. After reading this I became very fond of tutoring the course on quantum mechanics and always tried to give some deeper insight (many of which was taken from here) in addition to just explaining the exercises. If I am correct, the world mangling theory just tries to explain some anomalies, but the rule of squa... (read more)

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0Viktor
Hm, just read the article again and saw that many of this was already explained there. But the essential point is that although the full information of a system is given by the amplitude distribution over all possible configurations, this information is not accessible to another system. When we try to couple the system to another (for example, by copying the state), this only respects the pure 'classical' states as described above. Thus it is possible to ask the question 'how much have these two states in common', where one classical state compared with itself gives one and with another one 0. If we want to also be able to compare mixed states, the notion of a scalar product comes in. The squared modulus is just the comparison of a state with itself, which is constantly 1 - obviously, the state has a hell lot in common with itself.