A qubit is prepared in the mixed state .

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 qubit is prepared in the mixed state ρ=0.7∣0⟩⟨0∣+0.3∣1⟩⟨1∣\rho=0.7|0\rangle\langle0|+0.3|1\rangle\langle1|. ((a)) Using the spectral decomposition of ρ\rho, construct a purification ∣Ψ⟩|\Psi\rangle of ρ\rho on the qubit together with a reference qubit RR, writing ∣Ψ⟩|\Psi\rangle explicitly as a normalised two-qubit ket. ((b)) Apply the unitary V=XV=X to the reference qubit RR alone, leaving the first qubit untouched, to obtain ∣Ψ′⟩=(I⊗X)∣Ψ⟩|\Psi'\rangle=(I\otimes X)|\Psi\rangle. Compute trR∣Ψ′⟩⟨Ψ′∣\mathrm{tr}_R|\Psi'\rangle\langle\Psi'| explicitly and confirm it still equals ρ\rho. ((c)) Prove, using the Schmidt decomposition, that any two purifications of the same ρ\rho on a reference system of the same size must be related in this way, ∣Ψ′⟩=(I⊗V)∣Ψ⟩|\Psi'\rangle=(I\otimes V)|\Psi\rangle for some unitary VV on the reference alone.

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