Let’s denote the (unknown but pure) input state of the electrons in the
beam by |ii and the output state by |oi. The probability of obtaining this
particular output is given by
P(|ii → |oi) =
no. of electrons in state |oi
Total number of electrons in input beam
. (2.2)
We are going to use this information to build a mathematical picture of the
spin states of the electron. Let’s introduce a notation for this probability (in
anticipation of the next chapter). In quantum mechanics, a process such as
just described, is associated with a probability amplitude, which is in general
complex, denoted by ho|ii. Following the lead of optics, where the intensity of
a beam of light is the square of the amplitude of the resultant electric field,
the probability of getting output |oi from input |ii is represented by
P(|ii → |oi) = |ho|ii|
2
. (2.3)
The reason for amplitudes taking possibly complex values will be seen when we
look at the phenomenon of interference between amplitudes in a later section.
The importance of basis states is that when an experiment is performed
to measure the state of an electron, the result is invariably one of the basis
states. For quantum computation, these basis states represent the bits 0 and
1. They are quantum states and we write them as
|0i ≡ |↑i; |1i ≡ |↓i. (2.4)
This representation is called the computational basis. We can think of these
basis states as analogous to the classical bits 0 and 1.
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