QuantumCore

Hands-on lab · 1 step

Bloch vector and purity of a mixed qubit

Read the Bloch vector r⃗=(rx,ry,rz)\vec r=(r_x,r_y,r_z) with rj=Tr(ρσj)r_j=\mathrm{Tr}(\rho\sigma_j) and the purity γ=Trρ2=12(1+∥r⃗∥2)\gamma=\mathrm{Tr}\rho^2=\tfrac12(1+\lVert\vec r\rVert^2) directly off the entries of a 2×22\times2 density matrix, using the Bloch-ball form ρ=12(I+rxX+ryY+rzZ)\rho=\tfrac12(I+r_xX+r_yY+r_zZ).

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Step 1: Bloch vector and purity

A qubit is in the mixed state

ρ=(0.70.2−0.1i0.2+0.1i0.3).\rho=\begin{pmatrix}0.7&0.2-0.1i\\0.2+0.1i&0.3\end{pmatrix}.

Give its Bloch vector r⃗=(rx,ry,rz)\vec r=(r_x,r_y,r_z), where rj=Tr(ρσj)r_j=\mathrm{Tr}(\rho\sigma_j) for σ1=X\sigma_1=X, σ2=Y\sigma_2=Y, σ3=Z\sigma_3=Z, and its purity γ=Trρ2\gamma=\mathrm{Tr}\rho^2. Enter the four real numbers rxr_x, ryr_y, rzr_z, γ\gamma in that order; answers are accepted within 10−310^{-3}.

Enter exact expressions or decimals, e.g. 1/sqrt(2), pi/3. Values must be real.

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