# reduced density matrix of state

given a multi particle state I have to calculate the reduced density matrix where I trace out the third particle

for this I first calculate the corresponding 2D density matrix with the bra vector of the state;

the partial trace is then given by the expression below where I sum over the Eigenbasis of the third particle

but I don't get how they get from the first line to the second one in the partial trace calculation; how does all the terms cancel? I think they cancel somehow due to orthogonality reasons but I don't see how the third bra/ket vector acts on the density matrix.. can someone explain in detail how this calculation works?

• Please use MathJax to format your equations rather than upload pictures – Aaron Stevens Feb 5 at 17:57

$$\sum_{|s_1, s_2, s_3 \rangle\ \in\ \mathrm{Basis(\mathcal{H})} \mathrm{}} |s_1, s_2, s_3\rangle \langle s_1, s_2, s_3 | = \hat{1}. \tag{1}$$
$$| \uparrow\rangle\langle \uparrow| + |\downarrow\rangle\langle\downarrow| = \hat{1}.$$
• thank you for your answer. i see want you mean but nontheless i dont get how the unity acts on the density operator such that from a triple state remains a state with only two particles; can you please explain this step by step for $$_3\langle\uparrow|\psi\rangle\langle\psi|\uparrow\rangle_3$$ – jeffs Feb 5 at 18:59