A source prepares a qubit as the ensemble

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

Quantum Information and Technology, Ch Mixed states, generalised measurement and telling states apart

Problem

A source prepares a qubit as the ensemble ρ=0.5 ∣0⟩⟨0∣+0.5 ∣−⟩⟨−∣\rho = 0.5\,|0\rangle\langle 0| + 0.5\,|-\rangle\langle -|, where ∣−⟩=(∣0⟩−∣1⟩)/2|-\rangle = (|0\rangle - |1\rangle)/\sqrt2. ((a)) Write ρ\rho as a 2×22\times2 matrix in the computational basis, and give its Bloch vector length ∣r⃗∣|\vec r|, its eigenvalues and its purity tr ρ2\mathrm{tr}\,\rho^2. ((b)) Separately, a two-qubit pair is prepared in the pure state ∣ψ⟩=0.6 ∣00⟩+0.4 ∣11⟩|\psi\rangle = \sqrt{0.6}\,|00\rangle + \sqrt{0.4}\,|11\rangle. Compute the reduced density operator ρA\rho_A of qubit 1 by partial trace. ((c)) Taking ρ\rho from part (a) and ρA\rho_A from part (b) as the two states, compute the trace distance D(ρ,ρA)D(\rho, \rho_A) and the optimal single-shot probability of correctly telling which of the two you have been handed, assuming equal prior probabilities for the two.

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