Finding Reduced Density Matrix for 2 Spin-Half System

[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?

 

[Jhero]

I can provide some guidance on how to find the reduced density matrix for a system with a thermal equilibrium density matrix. First, it is important to understand what the reduced density matrix represents. It is the density matrix of one subsystem of a larger system, obtained by tracing out the other subsystem. In other words, it is the density matrix of one spin-half particle in this case, obtained by tracing out the other spin-half particle.

To find the reduced density matrix, we can use the trace operation. The trace operation is defined as the sum of the diagonal elements of a matrix. In this case, we want to trace out the second spin-half particle, so we need to sum over the second and fourth elements of the density matrix.

We can write the reduced density matrix as:

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

Simplifying this, we get:

\begin{displaymath}
\mathbf{\rho_{1}} =
\left(\begin{array}{cc}
2e^{-(1+c)/T} & 0\\
0 & 2cosh[(1-c)/T]
\end{array}\right)
\end{displaymath}

This is the reduced density matrix for one spin-half particle in a thermal equilibrium state. To obtain the reduced density matrix for the other spin-half particle, we can simply interchange the diagonal elements.

I hope this helps in understanding how to find the reduced density matrix for a system with a thermal equilibrium density matrix. It is important to note that the procedure may vary for different types of thermal equilibrium states, but the basic concept remains the same.