Quantum Information for Quantum Cats (7)

Up-Down Basis in State Space

It probably has become a little too abstract to most of you. So we bring up some specific examples. Let us consider the case of Alice preparing the quantum arrow in the angle \theta. Now let's calculate the probability of Bob finding it in the same angle \theta, and also calculate theprobability of Bob findin it in the opposite direction, \theta+\pi. They are given by

which totally conside with our intuition. We now perform the following manipulation of the vector | \theta >

Here, | up > = (1, 0), | down > = (0, 1) are the abbreviated expresion for the upward state (\theta = 0) and downward state (\theta = \pi). The above equation gets a pictorial illustration in the right.

This precisely expresses the fact that an arbitrary state | \theta > is represented as the super position (meaning the sum with possible scale change) of | up > and | down >, and thus the set { | up >, | down > } is a basis. As could be confirmed easily by direct calculation, we have

which again means Bob would find the arrow upward 100% of the time and downward 0% of the time if Alice prepare it to be upward, and vice versa for if downward. From these, we arrive at

namely, when Alice put the arrow in angle \theta, Bob, when he measures the arrow in up-down direction, will find it upward with probability |cos[\theta/2]|^2, and downward with probability |sin[\theta/2]|^2.

The sum of two probabilities is one, which attests to the fact that, in this setting, Bob can have no other chance than finding it either in up or down. This is the mathematical expression of the observer finds the arrow only in limited choice of states, in this case just two states, which is the direct result of the state of spin 1/2 being a two dimensional vector.