# How to connect the idea of inner product, outer product, matrix element in 3 level composite system?

I was solving steady state solution given by Lindblad equation, and I got stuck on the calculation part like what will be the value of $$\langle e, 0|b, 0\rangle$$?

And how should I rewrite $$|d, 0\rangle\langle e, 0| \rho |e, 0\rangle\langle d, 0|$$, where, $$\rho$$ is a density matrix.

The possible transition is given in the attached image