- #1
whatisreality
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Homework Statement
In a computational basis, a qubit has density matrix
##
\rho = \left( \begin{array}{ccc}
\frac{1}{2} & \frac{1}{3} \\
\frac{1}{3} & \frac{1}{2} \\
\end{array} \right)##
At t=0. Find the time dependence of ##\rho## when the Hamiltonian is given by ##AI+BY##, ##A## and ##B## are constants, ##Y## is the Pauli matrix
##
\left( \begin{array}{ccc}
0 & -i \\
i & 0 \\
\end{array} \right)##
Then the hint says to write the matrix in the form ##\rho = \frac{1}{2} (I+a(t)X+b(t)Y+c(t)Z)## where
## X=
\left( \begin{array}{ccc}
0 & 1 \\
1 & 0 \\
\end{array} \right)##,
##Z=
\left( \begin{array}{ccc}
1 & 0 \\
0 & -1 \\
\end{array} \right)##
and derive differential equations for the coefficients.
Homework Equations
The Attempt at a Solution
I thought I should be using the time shift operator ##U(t,t_0) = e^{-\frac{i}{\hbar}H(t-t_0)}##, and calculating ##\rho(t) = U\rho(t_0) U^{\dagger}##, except I'm not exactly sure what the exponential of my particular Hamiltonian would mean. But if I write the density matrix in the form suggested by the hint then I don't know what to do with it from there to get a differential equation. Is it something to do with the Schrodinger equation? Should I sub it into that and treat it like a state ##\psi##?
Thank you for any help, I really appreciate it, I'm a bit lost!