Worked Problems in Physics

Nielsen and Chuang Exercise 2.75

Posted in Nielsen/Chuang by rpmuller on June 1, 2010

Exercise 2.75 For each of the four Bell states, find the reduced density operator for each qubit.

$\displaystyle \rho^{00}=\frac{1}{2}\left(\left|00\right>\left<00\right|+\left|00\right>\left<11\right|+\left|11\right>\left<00\right|+\left|11\right>\left<11\right|\right)$

$\displaystyle \rho^{00}_1 = \rho^{00}_2=\frac{1}{2}\left( \left|0\right>\left<0\right| + \left|1\right>\left<1\right| \right) = I/2.$

$\displaystyle \rho^{01} = \frac{1}{2}\left(\left|00\right>\left<00\right|-\left|00\right>\left<11\right|-\left|11\right>\left<00\right|+\left|11\right>\left<11\right|\right)$

$\displaystyle \rho^{01}_1 = \rho^{01}_2 = I/2.$

$\displaystyle \rho^{10} = \frac{1}{2}\left(\left|10\right>\left<10\right|+\left|10\right>\left<01\right|+\left|01\right>\left<10\right|+\left|01\right>\left<01\right|\right)$

$\displaystyle \rho^{10}_1 = \rho^{10}_2 = I/2.$

$\displaystyle \rho^{11} = \frac{1}{2}\left(\left|10\right>\left<10\right|-\left|10\right>\left<01\right|-\left|01\right>\left<10\right|+\left|01\right>\left<01\right|\right)$

$\displaystyle \rho^{11}_1 = \rho^{11}_2 = I/2.$