What is probabilistic state?
‘Probabilistic state’ is a vehicle for reasoning about the state of a system in the presence of uncertainty.
What is ‘ket’
Ket is a basis vector that describes a state. Said another way, it is 1 in the entry corresponding to a and 0 for all other entries, denoted by |a\rangle.
\Sigma = (ON, OFF) \implies |ON\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, |OFF\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
What are Classical Operations
Consider a function f defined as follows:
f: \Sigma \rightarrow \Sigma
This function f is known as a classical operation. It transforms any classical state a \in \Sigma into another state f(a), where f(a) \in \Sigma.
What is a way we represent classical operations?
Let f be a function and a \in \Sigma where \Sigma is a set of states. Then, the classical operation f(a) can be expressed as a transformation on the basis vector |a\rangle.
Such a transformation a matrix-vector multiplication:
M |a\rangle = |f(a) \rangle, \forall a \in \Sigma