Density matrix spin half, Pauli vector

[Lapetude]

A nice discussion of the density operator for a qubit can be found here: http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf

 

[Asim Riaz]

The density operator for a qubit is a 2x2 matrix that describes the state of the qubit. It is given by ρ = ρ0|0⟩⟨0| + ρ1|1⟩⟨1|. Here, ρ0 and ρ1 are the probabilities for the qubit to be in the 0 and 1 states, respectively. The matrix elements of this matrix give the probability amplitudes for each state. For example, the element ⟨0|ρ|1⟩ gives the amplitude for the qubit to transition from the 0 state to the 1 state. The density operator can also be used to calculate the expectation values of observables. For example, the expectation value of the Pauli Z operator can be calculated using the density operator as follows:⟨Z⟩ = Tr(ρZ) = ρ00 − ρ11. This can be seen as a measure of the qubit's polarization along the Z direction.The density operator can also be used to calculate other quantities, such as the entropy of the qubit. This is done by calculating the von Neumann entropy of the density operator:S = −Tr(ρlog(ρ)). The entropy gives an indication of the amount of disorder in the qubit's state.