Quantum Maths for Single Qubit
Inner Product and Measurement - Born’s Rule
For given matrices $u$ and $v$, The inner product is given as (uT is transpose of matrix u):
⟨u,v⟩=uTv Given state vectors of qubits as u and v, the probability of measuring u in $v$ is the square of inner product of u and v is given as:
prob(measuring state u in state v)=(u.v)2 It can also be said that:
prob(measuring state u in state v)=prob(measuring state v in state u) Inner product can be done easily with bra-ket notations and can be show with following table:
Notation ⟨0∣∣0⟩⟨1∣∣1⟩⟨1∣∣0⟩⟨0∣∣1⟩⟨+∣∣0⟩⟨+∣∣1⟩⟨−∣∣0⟩⟨−∣∣1⟩ Value 110021212−12−1 Hence, Born’s rule for bra-ket notations is give as follows:
prob(measuring state a in b)=∣⟨a∣∣b⟩∣2 Gates as Matrices

The transformation go quantum states using gates is given by: Gate * State=New State