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.