 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2012

Paper 65

In the <computational basis>, write
$$
\rho=\frac12\begin{pmatrix}1+s_z&s_x-is_y\\s_x+is_y&1-s_z\end{pmatrix}.
$$
The <Kraus representation> of the <amplitude damping channel> gives
$$
\Lambda(\rho)=\begin{pmatrix}
\rho_{00}+p\rho_{11}&\sqrt{1-p}\,\rho_{01}\\
\sqrt{1-p}\,\rho_{10}&(1-p)\rho_{11}
\end{pmatrix}.
$$
Reading off its <Bloch vector> gives the <Bloch-vector map of amplitude damping>:
$$
\boxed{(s_x,s_y,s_z)\longmapsto
\left(\sqrt{1-p}\,s_x,\sqrt{1-p}\,s_y,p+(1-p)s_z\right).}
$$
Here $0\leq p\leq1$. The shift in $s_z$ shows that this <quantum channel> is not unital unless $p=0$; its fixed ground state lies at the north pole, rather than the center of the <Bloch sphere>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 65 / 2 / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 65 / 2 / ii / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 65 / 2 / ii / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)