Assignment 20.1.1

(Due 28/01/2020)

20.1.1 Show that the trace of an operator does not depend on the basis in which it is expressed.

20.1.2 Consider states $\ket{\psi} = 3i\ket{\phi_1} - 7i\ket{\phi_2}$ and $\ket{\chi} = -\ket{\phi_1} + 2i \ket{\phi_2}$ where $\ket{\phi_1}$ and $\ket{\phi_2}$ are orthonormal

    a. Calculate $\ket{\psi+\chi}$ and $\bra{\psi+\chi}$
    
    b. Are both of them equal ?
    
    c. Show that $\ket{\psi}$ and $\ket{\chi} satisfy Schwarz and triangle inequality 

20.1.3. Shoe that the length of a vector is not changed by a unitary operator.