# Reduced density matrix

1. Sep 15, 2008

### yukawa

At the thermal equilibrium, the density matrix of a 2 spin-half system is given by:

$$\begin{displaymath} \mathbf{\rho} = \left(\begin{array}{cccc} e^{-(1+c)/T} & 0 & 0 & 0\\ 0 & cosh[(1-c)/T] & -sinh[(1-c)/T] & 0\\ 0 & -sinh[(1-c)/T] & cosh[(1-c)/T] & 0\\ 0 & 0 & 0 & e^{-(1+c)/T} \end{array}\right) \end{displaymath}$$

where c is a parameter.

How to find the reduced density matrix by tracing out the other spin?
i.e. $$\rho_{1} = tr_{2}\rho$$

I only know how to find the reduced density matrix for a pure state, say like $$\frac{1}{\sqrt{2}}(\left|\downarrow\uparrow> - \left|\uparrow\downarrow> )$$
But for this given density matrix, i have no idea. Are there any equations that i can use? What's the procedure?