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What is the arbitrary density matrix of a mixed state qubit?
Of course. Thanks for clearing that up.The density matrix is always Hermitian, and it's trace is always 1. This is regardless of whether it represents a pure state or a mixed state. Once you diagonalize it, the condition of whether it is a pure state or a mixed state depends on whether one of the eigenvalues is 1 or not. If one is 1, and the others are zero, then it is a pure state. If more than one eigenvalue is non-zero, then it is a mixed state.
Remember what the density matrix represents.