Linear Algebra of State Vectors; Inner Product and probability
This section has some technical details involving linear algebra. If you would like to really master the quantum information to the point of utility, you certainly have to follow it. But if your math is not up to this, just take note of the result and might skip the logic leading to it.
The concept of "state" is so abstract and far removed from such intuitive concepts as the angle and angular velocity, and far harder to comprehend. But the state has to have some connection to those comprehensible quantities. In mathematical term, a state is represented as a vector in Hilbert space, and the observable quantity is represented by the inner product of two Hilbert vectors, one representing the initial state of Alice's preparation, and he other representing the final state of Bob's observation.
It might be easier to learn it operatively than to have a formal definitions. The easiest way to conceptualize the Hilbert vector is to imagine it to be a usual vector you have learnt in high school math. Suppose Alice prepares the spin 1/2 in the state in which the direction of the arrow is off from the perpendicular direction by angle \theta. We call this state as | \theta > and write it in the two component representation.
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| \theta > = ( \cos[ \theta/2 ], \sin[ \theta/2 ] )
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In truth, we want to have it in a column vector form, but because that is harder on Web papges, let us be content with this expression. Suppose Bob prepares his detection device in such a way that he expects to see it in < \psi | which represents a state with direction \psi, whose explicit representation we write
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< \psi | = ( \cos[ \psi/2 ], \sin[ \psi/2 ] )
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This "final" vector with oposite bracket <| is a genuine raw vector in comparison to the "initial" vector |>. For now, we surpress such natural question as why stste has to be represented by such vectors, and aslo whether this vector with direction has anything to do with the physical direction in our "arrow" model. We just say that these vectors are related to the reality (aka observables) through the square of the inner product of the vectors < \psi | \theta >
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| < \psi | \theta > |^2 = | cos[\psi/2]cos[\theta/2]+sin[\psi/2]sin[\theta/2] |^2
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In quantum physics, this quantity represents the probability of Bob finding the spin 1/2 to be in the state < \psi | after observation of the state | \theta > prepared by Alice.
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