Classical Information

 

In this post we will use a method to present classical information. This basic method will be used in later posts to discuss quantum information. This is based on the information in the Qikit course.

Classical State and Probability Vectors

If X represents a bit, whose state can be 0 with probability 1/3 and 1 with probability 2/3, then we mark it as follows.

X bit

Σ = {0,1}

Pr (X=0) = 1/3   and    Pr (X=1) = 2/3

This state can also be presented as probabilistic column vector:    

Notice that:

1. all numbers in the vector are non-negative real numbers
2. The sum of the vector is 1

Bra and Ket

Bra is a row vector with 1 set in a single position, and all others are zeros.

(1,0) is bra zero, and the shorthand is marked as <0|

(0,1) is bra one, and the shorthand is marked as <1|

Ket is a probability column vector representing the X bit in only one state.

X is bit 0, and the shorthand for that is ket zero, marked as |0>

X is bit 1, and the shorthand for that is ket one, marked as |1>

We can use bra and ket to create vectors and matrices, for example:

 = 1/3 |0>   + 2/3 |1>

Deterministic Operations

We can define a function to make a change to the bit X.

For example: 

f1(X) = 1, will always convert the value of the bit to 1

f2(X) = !X, will always change the value of the bit to the opposite value.

The functions can be represented as matrices, so that M |X> = |f(X)>

For example, the corresponding matrices for the functions above are:

M1 = 

M2 = 

Probabilistic Operations

We can also configure probabilistic functions that have a probability of changing a bit.

Notice that the sum of each column in the matrix must be one.

For example:

M = 

and then

M |0> = always |0>

M |1> = 50% |0> and 50% |1>