A qubit is prepared

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, with equal prior probability, in ∣0⟩|0\rangle or in ∣ϕ⟩=cos⁡30∘ ∣0⟩+sin⁡30∘ ∣1⟩|\phi\rangle = \cos30^\circ\,|0\rangle + \sin30^\circ\,|1\rangle (Bloch polar angle 60∘60^\circ from the north pole, in the xx-zz plane). ((a)) Compute the overlap ∣⟨0∣ϕ⟩∣|\langle0|\phi\rangle| and the trace distance D(∣0⟩⟨0∣,∣ϕ⟩⟨ϕ∣)D(|0\rangle\langle0|, |\phi\rangle\langle\phi|). ((b)) Find the optimal single-shot success probability psuccp_{\rm succ} for guessing which state was prepared. ((c)) Give the polar angle, measured from ∣0⟩|0\rangle's Bloch axis, of the measurement direction r⃗−s⃗\vec r - \vec s that achieves this, where r⃗,s⃗\vec r, \vec s are the two Bloch vectors.

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