Viktor comments on The Born Probabilities - Less Wrong

14 01 May 2008 05:50AM

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Comment author: 05 April 2011 10:22:16AM *  1 point [-]

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|<x|x>|x>=1, <x|<0|x>|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|<1|a+<2|<2|b)(a|1>|1>+b|2>|2>)=|a|²<1|<1|1>|1>+|b|²<2|<2|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> (!),