Bypassing the Normalization
Get familiar with the concept of bypassing the normalization process.
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Normalization
The angle controls the probabilities of measuring the qubit in either state 0
or 1
. Therefore, also determines and .
Let’s take a look at the figure 2-dimensional qubit system.
Any valid qubit state vector must be normalized:
It states that vectors have the same magnitude (length). Since they all originate in the center, they form a circle with a radius of their magnitude, that is,half of the circle diameter.
In such a situation, Thales’ theorem states that if two conditions, where the first condition states that A, B, and C are distinct points on a circle, and the second condition states that the line AC is a diameter, are met, then the angle (the angle at point B) is right.
In our case, the heads of , , and represent the points A, B, and C, respectively. This satisfies the first condition. The line between and is the diameter, which satisfies the second condition. Therefore, the angle at the head of is a right angle.
Now, the Pythagorean theorem states that the area of the square whose side is opposite the right angle (hypotenuse, ) is equal to the sum of the areas of the squares on the other two sides (legs , ).
When looking at the figure 2-dimensional qubit system, again, we can see that and are the two legs of the rectangular triangle and the diameter of the circle is the hypotenuse.
Therefore, we can insert the normalization as follows:
The diameter is two times the radius and is therefore two times the magnitude of any vector . The length of is thus .
Since all qubit state vectors have the same length, including and , there are two isosceles triangles ( and ).
Let’s look at the following figure.
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