Inner Product Spaces

In quantum computing, state vectors of an n-qubit system reside in the ${\mathbb{C}}^2$ vector space. These must be unit vectors.

Dirac Notation

Column Vectors and Row Vectors

Matrix: rectangular array of numbers, represented as:

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right]m\times n$

Entries $a_{11},a_{12}\dots$ are called elements.

Column vector: $\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]$    Row vector: $\left[\begin{array}{cccc}{v}_1,& v_2,& \dots ,& v_n\end{array}\right]$

Matrix transposition swaps columns by rows

$A^T=\left[\begin{array}{cccc}{a}_{11}& a_{21}& \cdots & a_{m1}\\ a_{12}& a_{22}& \cdots & a_{m2}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \cdots & a_{mn}\end{array}\right]n\times m$

E.g.

$A=\left[\begin{array}{cc}0& i\\ -i& 0\end{array}\right]$

$A^T=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$

Kets for Column Vectors

Given a general vector $v=(v_1,v_2,\dots ,v_n)$ in ${\mathbb{C}}^n$, we use $|v⟩$ to denote its column vector form

$|v⟩\equiv \left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]\text{where }|v⟩\text{ is referred to as a ket.}$

Other common vectors:

• $|\psi ⟩$ for a general state vector
• $|{\lambda }_i⟩$ for eigenvectors corresponding to the eigenvalue ${\lambda }_i$

Reserved nomenclature:

• $|0⟩$, $|1⟩$ $\Rightarrow$ computational basis vectors in a single qubit system
• $|V⟩$, $|H⟩$ $\Rightarrow$ vertical and horizontal linear polarisation states of a photon
• $|{\Phi }^{+}⟩$, $|{\Phi }^{-}⟩$, $|{\Psi }^{+}⟩$, $|{\Psi }^{-}⟩$ for Bell states of two qubits

Computational Basis

$|0⟩\equiv \left[\begin{array}{c}1\\ 0\end{array}\right],|1⟩=\left[\begin{array}{c}0\\ 1\end{array}\right]$

Addition and Multiplication

$\frac{1}{\sqrt{2}}|0⟩+\frac{1}{\sqrt{2}}|1⟩=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 0\end{array}\right]+\frac{1}{\sqrt{2}}\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]$

A general vector in ${\mathbb{C}}^2$ can be expressed as

$\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\alpha |0⟩+\beta |1⟩,\text{for }\alpha ,\beta \in \mathbb{C}$

E.g. $|\psi ⟩=\left[\begin{array}{c}1\\ i\end{array}\right]=1|0⟩+i|1⟩=|0⟩+i|1⟩$

Hermitian Adjoint

For each vector $|v⟩$ in ${\mathbb{C}}^n$, there is an associated row vector obtained through the conjugate.

E.g. $\left[\begin{array}{c}1+i\\ 1-i\end{array}\right]$ is computed as:

${\left({\left[\begin{array}{c}1+i\\ 1-i\end{array}\right]}^{\ast }\right)}^T=\left[\begin{array}{cc}(1+i{)}^{\ast },& (1-i{)}^{\ast }\end{array}\right]=\left[\begin{array}{cc}1-i& 1+i\end{array}\right]$

Consider a column vector in ${\mathbb{C}}^n$:

$v=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]$

its Hermitian adjoint is defined as its conjugate transpose denoted as:

$v^{†}=(v^{\ast }{)}^T=(v^T{)}^{\ast }=\left[\begin{array}{cccc}{v}_1^{\ast },& v_2^{\ast },& \dots & v_n^{\ast }\end{array}\right]$

Bra, denoted as $⟨v|$, is the Hermitian adjoint of $|v⟩$:

$⟨v|\equiv |v{⟩}^{†}$

E.g.

$⟨0|=|0{⟩}^{†}={\left[\begin{array}{c}1\\ 0\end{array}\right]}^{†}=\left[\begin{array}{cc}1& 0\end{array}\right]$

$⟨1|=|1{⟩}^{†}={\left[\begin{array}{c}0\\ 1\end{array}\right]}^{†}=\left[\begin{array}{cc}0& 1\end{array}\right]$

Given vectors $|u⟩$, $|v⟩$ in ${\mathbb{C}}^n$ and scalars $\alpha ,\beta \in \mathbb{C}$, bras exhibit the following conjugate-linear properties:

$⟨u+v|=⟨u|+⟨v|$

$⟨\alpha v|={\alpha }^{\ast }⟨v|$

$⟨\alpha u+\beta v|={\alpha }^{\ast }⟨v|+{\beta }^{\ast }⟨u|$

$|\psi ⟩=\frac{1+i}{2}|0⟩+\frac{1-i}{2}|1⟩$

$⟨\psi |={\left(\frac{1+i}{2}\right)}^{\ast }⟨0|+{\left(\frac{1-i}{2}\right)}^{\ast }⟨1|$

$=\frac{1-i}{2}⟨0|+\frac{1+i}{2}⟨1|$