Partial trace of density matrix

[climbon]

I am unsure how to (mathematically) do the partial trace of a density matrix so that I can find the expectation value of an observable.

I am working on a model similar to the Jaynes cummings model. My density matrix is of the form;

$$\rho = [\rho_{11}, \rho_{12}, \rho_{21}, \rho_{22}]$$

As a 2x2 matrix. My system is a composite system as;

$$H_{A} \otimes H_{B}$$

I want to find the partial trace over the field so I can use an observable M to find the population inversion of the atom;

$$\rho^{A}(t) = Tr_{F}\rho(t)$$

That way I can;

$$Tr(M \bullet \rho^{A})$$

To find the inversion of the atom.

How do i do the trace over the field...I understand the principle but struggling how to do this mathematically??

Thanks

 

[azntoon]

!The partial trace of a density matrix is a mathematical operation that allows us to trace out one subsystem from a composite system. To do this mathematically, you need to use the definition of the trace operation: Tr(A) = Σi Ai,i Where A is an NxN matrix and Ai,i is the ith diagonal element of A. In the case of a 2x2 density matrix, you can calculate the partial trace over one subsystem (let's say B for simplicity) by summing all the elements in the diagonal of the matrix: Tr_B(ρ) = Σi ρii where ρii is the ith diagonal element of ρ.Once you have calculated the partial trace, you can then use it to calculate the expectation value of an observable M: <M> = Tr(M ρ) Where M is your observable and ρ is the density matrix.