The long dark teatime of the notation

Consider this a break to dive a little deeper in the notation. This is a page you can look up whenever you get lost in the sea of symbols.

Dirac notation

Dirac notation is basicly composed by two symbols: a $\textbf{bra}$ ($\bra{\cdot}$) and a $\textbf{ket}$ ($\ket{\cdot}$). Given a vector $v$ in the Hilbert space ($\mathbb{C}^n$,$ \langle \cdot,\cdot \rangle$) such as

$$v = \begin{pmatrix}v_1\\ v_2\\ \vdots \\ v_n \end{pmatrix},$$

then

$$\ket{v}=\begin{pmatrix}v_1\\ v_2\\ \vdots \\ v_n \end{pmatrix}$$

while

$$\bra{v}= \begin{pmatrix}v_1\\ v_2\\ \vdots \\ v_n \end{pmatrix}^\dagger = \begin{pmatrix}v_1^* & v_2^* & \dots & v_n^* \end{pmatrix}$$

where $v_i^*$ is the complex conjugate of $v_i$. So, if

$$w = \begin{pmatrix}w_1\\ w_2\\ \vdots \\ w_n \end{pmatrix},$$

we can write

$$\braket{v | w} = \begin{pmatrix}v_1^* & v_2^* & \dots & v_n^* \end{pmatrix} \begin{pmatrix}w_1\\ w_2\\ \vdots \\ w_n \end{pmatrix} = \langle v, w \rangle \in \mathbb{C}, $$

the inner product in the Hilbert space $\mathbb{C}^n$.

Moreover:

$$\ket{v}\bra{w} =\begin{pmatrix}v_1\\ v_2\\ \vdots \\ v_n \end{pmatrix} \begin{pmatrix}w_1^* & w_2^* & \dots & w_n^* \end{pmatrix}= \begin{bmatrix}v_1w_1^* & v_1w_2^* & \dots & v_1w_n^* \\ v_2w_1^* & v_2w_2^* & \dots & v_2w_n^* \\ \dots & \dots & \dots & \dots \\ v_nw_1^* & v_1w_2^* & \dots & v_nw_n^* \end{bmatrix}\in \mathbb{C}^{n \times n}$$

The Riesz theorem

In a more general setting, the bra notation is used for linear and continuous functions

$$f:\mathbb{C}^n\rightarrow \mathbb{C},$$

so that

$$\braket{f|v} = f(v) \in \mathbb{C}.$$

This is where Riesz representation theorem comes into play. It essentialy states that every function such as $f$ can be represented as the action of a vector $w\in \mathbb{C}^n$. More in detail, for every $f:\mathbb{C}^n\rightarrow \mathbb{C}$ linear and continuous, there exists a unique $w\in \mathbb{C}^n$ such that

$$f(v) = w^\dagger v$$

for every $v\in \mathbb{C}^n$. This justifies the use of the bra notation for vectors.

In quantum computing

We use the Dirac notation in a more flexible way. For example, we represent vectors of the canonical basis with $\ket{0}$ and $\ket{1}$ instead of $\ket{(1, 0)}$ and $\ket{(0, 1)}$. This is for the sake of simplicity. As you know, systems of $n$ qubits are represented on $\mathbb{C}^{2^n}$: the dimension scales very fast! It would be impractical to write vectors in their extended version.

Moreover, we often take poetic licenses such as $\ket{+}$ and $\ket{-}$, which are even less significant if we take the Dirac notation too (let’s say) seriously, but are very meaningful in our context because they provide compactness and clarity! Never taken a shortcut before?

More on notation

Since

$$\ket{x_1}\ket{x_2}\dots \ket{x_m} = \ket{x_1}\otimes \ket{x_2}\otimes\dots\ket{x_n} $$

we will write these two representations interchangeably, depending on the context.

Moreover, we will often use the more compact notation

$$\ket{x_1x_2\dots x_m}$$

instead of both, and when the states are known we may write something like, for example:

$$\ket{00100}_{x_1x_2x_3x_4x_5}$$

to stress the fact that those 0’s and 1’s are the states of the system relative to the corresponding $x_i$.

And another thing…

We aren’t redundant, the subject is redundant! There’s another way to express vectors of the computational basis. For example, take:

$$\ket{00100}$$

and note that $00100$ is the binary representation of $4$. Then, we write

$$\ket{4}_5=\ket{00100}$$

where the subscript “5” is there because we are working with a 5-qubit system. In fact, even $0000100$ is a binary representation of $4$ but $\ket{0000100}$ is a state of a
$7$-qubit system.

Thus, in general, we write $\ket{x}_n$ with $x=0, \dots, 2^n-1$ to denote the computational basis of a $n$-qubit system.