
Vectors to describe quantum states can get tedious really quick, so we introduce a new notation.
We can rewrite a quantum state in the following way:
![[Pasted image 20260730101012.png]]

To represent the vector $e_i$ as its classical possibility we use $\ket{}$ .
For a classical possibility x, which is the i-th possibility represented by $e_i$ we use:
![[Pasted image 20260730101435.png]]

Example:

![[Pasted image 20260730101723.png]]

Also we have 2 special cases:

![[Pasted image 20260730102024.png]]

$\ket{+} = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}$ 
$\ket{-} = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix}$


The Teleportation example from [[Quantum Circuits]] with ket notation:

![[Pasted image 20260730102305.png]]
![[Pasted image 20260730102312.png]]
![[Pasted image 20260730102318.png]]


## Bra and Braket notation
Additionally to ket notation, bra notation, $\bra{}$ denotes the conjugate transpose of $\ket{}$:

$\bra{\psi} := \ket{\psi}^\dagger$ 

Example:
![[Pasted image 20260730103526.png]]

Combined, we get the Braket notation $\braket{\psi||\phi}$ 

Example:

![[Pasted image 20260730103836.png]]

