Table Of Contents

Operator ⬆️

• A mathematical rule that can be applied to a function to transform it into another function. One such example is the derivative operator ($\frac{d}{dx}$), which transforms functions.

• This idea can further be extended to vector space, where an operator $\hat{A}$ is a mathematical rule that transforms one ket $\ket{\psi}$ into another $\ket{\phi}$ (or one bra into another):

  $\hat{A}\ket{\psi} = \ket{\phi}$

• Linear : An operator is said to be linear, if the following relationship holds:

  $\hat{A}(\alpha\ket{\psi_1} + \beta\ket{\psi_2}) = \alpha(\hat{A}\ket{\psi_1}) + \beta(\hat{A}\ket{\psi_2})$

• Zero Operator : Transforms a vector into the zero vector:

  $\hat{0}\ket{\psi} = 0$

• Identity Operator : Leaves the vector unchanged:

  $\hat{I}\ket{\psi} = \ket{\psi}$

Observables ⬆️

• Things we measure in order to characterize the quantum state of a particle. These include : position, momentum, angular momentum & energy.
• An important postulate of quantum theory states that there is an operator that corresponds to each ovservable physical state.

The Pauli Operators ⬆️

• Operators can act on qubits (which are simply kets/state vectors in a 2D vector space).

• There are in total 4 Pauli Operators, which are denoted by:

  • $\sigma_0, \sigma_1, \sigma_2, \sigma_3$

  The following two notation sets are used time to time:

  • $\sigma_0, \sigma_x, \sigma_y, \sigma_z$
  • $I,X,Y,Z$

1. The Identity Operator $(\sigma_0 = I)$

   $\sigma_0\ket{0} = \ket{0}$, $\sigma_0\ket{1} = \ket{1}$

2. The Not Operator $(\sigma_1 = \sigma_X = X)$

   $\sigma_X\ket{0} = \ket{1}$, $\sigma_X\ket{1} = \ket{0}$

3. $(\sigma_2 = \sigma_Y = Y)$

   $\sigma_Y\ket{0} = i\ket{1}$, $\sigma_Y\ket{1} = -i\ket{0}$

4. $(\sigma_3 = \sigma_Z = Z)$

   $\sigma_Z\ket{0} = \ket{0}$, $\sigma_Z\ket{1} = -\ket{1}$

Outer Products ⬆️

• The outer product of a ket $\ket{\psi}$ and a bra $\bra{\phi}$ is written as:

  $\ket{\psi}\bra{\phi}$

• This quantity is generally used as an operator. For example, let’s apply the above outer product as an operator on $\ket{\chi}$.

  $(\ket{\psi}\bra{\phi})\ket{\chi}$ = $(\braket{\phi|\chi})\ket{\psi}$

  here, the number $\braket{\phi|\chi}$ is called the constant of proportionality.