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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 ones. After the second measurement we have (a|0>+b|1>)^(N-2) (a²|00,0>+ab|01,1>+ab|10,1>+b²|11,2>) Since the two states ab|01,1> and ab|10,1> are not distinguished by the measurement, the basis should be changed - and this is the crucial point: |01>+|10> has a length of sqrt(2), so if we change the basis to |+>=(|01>+|10>)/sqrt(2), we have (a|0>+b|1>)^(N-2) (a²|00,0>+absqrt(2)|+,1>+b²|11,2>).
The coefficiants are like in the binomial theorem, but note the sqare root!
Continuing, we will get something similar to a binomial distribution:
sum(k=0..N: sqrt(N!/(k!(N-k)!))a^k b^(N-k) |...,k>).
Now it remains to prove that for j/N not equal to a² the amplitudes go to zero as N goes to infinity. This is equivalent to the square of the amplitude going to zero (this is just to make the calculation easier, it does not have anything to do with the Born rule). It is, for |...,k>,
ck² = N!/(k!(N-k)!) a²^k b²^(N-k)
which becomes a Gaussian distribution for large N, with mean at k=Na² and width Na²b². So at k/N=a²+d it has a value proportional to exp(-(Nd)²/(2Na²b²))=exp(-Nd²/(2a²b²)) --> 0 as N --> inf.
So a time capsule where the records indicate that some quantum experiment has been performed a great number of times and the Born rule is broken will have an amplitude that goes to zero (yeah, I just read Barbour's book).
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 if we want to express that a system could be in one of several states with different probabilities (or, equivalently, a lot of equally prepared systems are in one state each and the percentage of systems in a particular state is given), you use a density matrix. Because a hermitian matrix encodes exactly two informations: it picks an orthogonal basis (the eigenvectors) and stores a number for each of those basis vectors (the eigenvalue). In the case of the density matrix these are the possible states with their probabilities. By the Way, that's also why we use hermitian operators/matrices to represent observables: they specify the states the measurement is designed to distinguish together with the results of the measurement in each case.
You actually get neater numbers if you take 0°, 30° and 60°. Then the probabilities are 1/8,1/8 and 3/8. :)
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.
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 squared moduli is well established and can be derived. Let me try an easy explanation:
The basic principle is that if one defines how the measurement equipment reacts to all pure states (amplitude 1 for one configuration, 0 for all else), one has no freedom left to define how it reacts to mixed states. I think the only prerequisite is that time evolution is linear. From here one can derive the No-Cloning theorem: Suppose you have two systems, one being in the 'ready to store a copy' state |0> and one having the two possibilities |1> and |2> (and of course every linear combination of those, so a combination of a|1>+b|2> will have an amplitude of a for the configuration |1> and b for |2>). Now you set up some interaction which tries to copy the state of the second system onto the first. So:
- |0>|1> evolves into |1>|1>.
- |0>|2> evolves into |2>|2>.
But if we have a combination
- |0>(a|1>+b|2>)=a|0>|1>+b|0>|2>,
this will be mapped onto
- a|1>|1>+b|2>|2>
and not just clone the state, which would give
- (a|1>+b|2>)(a|1>+b|2>)=a²|1>|1>+ab|1>|2>+ab|2>|1>+b²|2>|2>.
So it is not possible to copy the whole state of a system, but it is possible to choose a basis and then copy the state if it is one of the basis vectors. So the basic measurement process would just copy the state of the system onto another system as good as possible (hence the so-called Heisenberg Uncertainty Principle - one has to choose according to which basis the measurement is coupled to the system). From the basis states of the composite system (|0>|x>, |x>|x>, x=1,2) one can construct a scalar product such that every vector has length 1 and they are orthogonal to each other:
- |x>=1, |x>=0 etc.
So the time evolution obviously conserves the length of the basis vectors - but since we could also have chosen another basis, it has to conserve also the length of mixed states (this step may be not so rigorous but at least makes the square rule much more plausible that any other). So the state (a|1>|1>+b|2>|2>) has to have length 1 and if we compute it we get
- 1=(|1>+b|2>|2>)=|a|²|1>+|b|²|2>+0=|a|²+|b|².
So the squared moduli add to 1 (Pythagoras sends his regards). Furthermore, if the 'original' system had three possibilities, but the copy process mapped
- |0>|1> onto |1>|1>
- |0>|2> onto |2>|2>
- |0>|3> onto |1>|3> (!),
we had
- |0>(a|1>+b|2>+c|3>) --> |1>(a|1>+c|3>)+b|2>|2>.
Mathematically, one can 'trace out' the influence of the original system - graphicly one just sees that the length of the part with the copied system in |1> is the length of the vector a|1>+c|3>, namely |a|²+|c|², while the other part has the length |b|². Thus the Born probabilities are added when grouping states together in the process of copying them - which could be responsible for the connection of the Born rule to the process of creating anticipations and so forth. Of course a measurement and the coupling of our brains to a system is not just copying the states - but the same argumentation holds since every sensitive coupling of another system to the original system can only be defined on some basis - the way the measurement reacts to combinations of states is determined from there and is not open to manipulation. So the Born rule is not a great mystery - although some of the steps may lack some rigor, it is far more plausible than for example just the modulus or some other power of it.
I hope this clears up some confusion,
Viktor