Quantum Maths for Single Qubit

Inner Product and Measurement - Born’s Rule

For given matrices $u$ and $v$, The inner product is given as (uTu^T is transpose of matrix uu):

⟨u,v⟩=uTv⟨u,v⟩ = u^Tv

Given state vectors of qubits as uu and vv, the probability of measuring uu in $v$ is the square of inner product of uu and vv is given as:

prob(measuring state u in state v)=(u.v)2prob(\text{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)prob(\text{measuring state u in state v}) = prob(\text{measuring state v in state u})

Inner product can be done easily with bra-ket notations and can be show with following table:

 Notation  Value ⟨0∣∣0⟩1⟨1∣∣1⟩1⟨1∣∣0⟩0⟨0∣∣1⟩0⟨+∣∣0⟩12⟨+∣∣1⟩12⟨−∣∣0⟩−12⟨−∣∣1⟩−12\begin{aligned} &\begin{array}{cccc} \hline \hline \text { Notation } & \text { Value }\\ \hline \bra{0}\ket{0} & 1 \\ \bra{1}\ket{1} & 1 \\ \bra{1}\ket{0} & 0 \\ \bra{0}\ket{1} & 0 \\ \bra{+}\ket{0} & \frac{1}{\sqrt{2}} \\ \bra{+}\ket{1} & \frac{1}{\sqrt{2}}\\ \bra{-}\ket{0} & \frac{-1}{\sqrt{2}}\\ \bra{-}\ket{1} & \frac{-1}{\sqrt{2}}\\ \hline \end{array} \end{aligned}

Hence, Born’s rule for bra-ket notations is give as follows:

prob(measuring state a in b)=∣⟨a∣∣b⟩∣2prob(\text{measuring state a in b}) = |\bra{a}\ket{b}|^2

Gates as Matrices

gate-as-matrices

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