Notes - Quantum Information HT24, Bra-ket notation


Flashcards

How are $ \vert 0\rangle$ and $ \vert 1\rangle \in \mathbb C^2$ defined in bra-ket notation, and more generally, how is $ \vert k\rangle \in \mathbb C^2$ defined?


\[\begin{aligned} |0\rangle &:= {1 \choose 0} \in \mathbb C^2 \\\\ |1\rangle &:= {0 \choose 1} \in \mathbb C^2 \\\\ |k\rangle &:= e_k \in \mathbb C^d \\\\ \end{aligned}\]

where $e _ k$ is the $k$-th unit vector.

Suppose $\psi = {\alpha \choose \beta} \in \mathbb C^2$. What would this equal in bra-ket notation?


\[\psi = \alpha |0\rangle + \beta|1\rangle\]

Suppose $\psi = {\alpha \choose \beta} \in \mathbb C^2$. What would $\langle \psi \vert $ equal in ordinary vector notation?


\[\psi = \overline \alpha \langle 0 | + \overline \beta\langle 1| = \overline \alpha \begin{pmatrix} 1 & 0 \end{pmatrix} + \overline \beta \begin{pmatrix} 1 & 0 \end{pmatrix}\]

Suppose $\psi = {\alpha \choose \beta}, \psi’ = {\alpha’ \choose \beta’} \in \mathbb C^2$. What is $\langle\psi \vert \psi’\rangle$


\[\begin{pmatrix}\overline \alpha & \overline \beta\end{pmatrix} \begin{pmatrix}\alpha' \\\\ \beta'\end{pmatrix} = \overline \alpha \alpha' + \overline \beta \beta'\]

Suppose $\psi = {\alpha \choose \beta}$. How can you write $\langle \psi \vert $ more familiarly in terms of $\psi$?


\[\psi^\dagger\]

where $\dagger$ represents the conjugate transpose.

What are column vectors called in bra-ket notation?


Kets.

What are row vectors called in bra-ket notation?


Bras.

Suppose you have

\[|\alpha \rangle = \begin{pmatrix}\alpha_0 \\\\ \alpha_1 \\\\\vdots \\\\ \alpha_{d-1}\end{pmatrix}\]

and

\[|\beta \rangle = \begin{pmatrix}\beta_0 \\\\ \beta_1 \\\\\vdots \\\\ \beta_{d-1}\end{pmatrix}\]

What does $ \vert \alpha \rangle \langle \beta \vert $ equal?


\[\begin{pmatrix} \alpha_0 \overline \beta_0 & \alpha_0\overline{\beta_1} & \cdots \\\\ \alpha_1 \overline \beta_0& \alpha_1 \overline \beta_1 & \cdots \\\\ \vdots & \vdots & \ddots \end{pmatrix}\]

Suppose $A = \vert \alpha \rangle \langle \beta \vert $. What is $A^\dagger$?


\[|\beta\rangle\langle \alpha |\]

Suppose you have an orthonormal basis $\{ \vert \psi _ k\rangle : k = 0, \cdots, d-1 \}$. What is

\[\sum^{d-1}_{m = 0} |\psi_m\rangle\langle\psi_m|\]

\[I\]

Can you define $ \vert 0, \theta \rangle$ and $ \vert 1, \theta\rangle$?


\[\begin{aligned} |0, \theta\rangle &:= \cos \frac \theta 2 |0\rangle + \sin \frac \theta 2 |1\rangle \\\\ |1, \theta\rangle &:= \sin \frac \theta 2 |0\rangle - \cos \frac \theta 2 |1\rangle \end{aligned}\]

Proofs




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