A qubit source emits

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 source emits, with equal prior probability, either ∣0⟩|0\rangle or ∣ψ⟩=12∣0⟩+32∣1⟩|\psi\rangle=\tfrac12|0\rangle+\tfrac{\sqrt3}{2}|1\rangle (real coefficients, so ⟨0∣ψ⟩=1/2\langle0|\psi\rangle=1/2). The task is unambiguous discrimination: design a three-outcome POVM {Eψ,E0,E?}\{E_\psi,E_0,E_?\} such that outcome 'ψ\psi' never occurs when the state was ∣0⟩|0\rangle, outcome '00' never occurs when the state was ∣ψ⟩|\psi\rangle, and outcome '??' (inconclusive) is allowed on either state. ((a)) Let ∣ψ⊥⟩|\psi^\perp\rangle be the state orthogonal to ∣ψ⟩|\psi\rangle. Explain why EψE_\psi must be proportional to ∣1⟩⟨1∣|1\rangle\langle1| and E0E_0 proportional to ∣ψ⊥⟩⟨ψ⊥∣|\psi^\perp\rangle\langle\psi^\perp|, and write Eψ=c∣1⟩⟨1∣E_\psi=c|1\rangle\langle1|, E0=c∣ψ⊥⟩⟨ψ⊥∣E_0=c|\psi^\perp\rangle\langle\psi^\perp| with a common constant c>0c>0. ((b)) Find the largest cc for which E?=I−Eψ−E0E_?=I-E_\psi-E_0 remains a positive operator, and state the resulting POVM. ((c)) Compute the probability that the measurement is conclusive, averaged over the two equally likely inputs, at this optimal cc. ((d)) Realise this POVM by a Naimark dilation: introduce a three-level ancilla in a fixed reference state, write the isometry VV sending the qubit's state ∣ϕ⟩|\phi\rangle to ∑kMk∣ϕ⟩⊗∣k⟩\sum_k M_k|\phi\rangle\otimes|k\rangle with Mk=EkM_k=\sqrt{E_k}, verify V†V=IV^\dagger V=I, and describe the projective measurement on the ancilla that reproduces the POVM statistics.

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