- #1

thatboi

- 133

- 18

I am having trouble relating probabilities with the density matrix of multiple qubits. Consider we have a system of 3 qubits: the first qubit is in the state ##\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{0}+\ket{1})## and the remaining 2 qubits are prepared in the state described by the density matrix ##\rho = a_{1}\ket{\alpha_{1}}\bra{\alpha_{1}}\otimes\ket{\beta_{1}}\bra{\beta_{1}} + a_{2}\ket{\alpha_{2}}\bra{\alpha_{2}}\otimes\ket{\beta_{2}}\bra{\beta_{2}}##

where we only know that ##\braket{\alpha_{i}|\alpha_{i}} = \braket{\beta_{i}|\beta_{i}} = 1## for ##i = 1,2## and ##a_{1} + a_{2} = 1##.

Now suppose we form the density matrix ##\rho_{tot}## of all 3 qubits.

My question is: From ##\rho_{tot}##, how do I obtain the probability of the first qubit to be in state ##\ket{0}##, which we know is ##\frac{1}{2}##. Normally I would think of performing a partial trace but I'm not sure of what to take the partial trace over in this case since we do not have further information on the other 2 qubits.