A trapped-ion experiment (reported 2021

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 trapped-ion experiment (reported 2021, the pair of ions measured in isolation) prepares two ions in the Bell state

|\Phi^+\rangle

and reports, from measurements that reconstruct the full two-ion density operator, a fidelity

F^2=0.998

in the squared convention used by most hardware papers. ((a)) Convert this figure to the root fidelity

F=\sqrt{F^2}

, the convention with

F=\mathrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}

that reduces to the overlap

|\langle\psi|\phi\rangle|

for pure states. ((b)) Use the Fuchs-van de Graaf inequalities

1-F\le D\le\sqrt{1-F^2}

to bound the trace distance

D

between the prepared state and the ideal Bell state. ((c)) Use the Helstrom bound

p_{\rm succ}=\tfrac12(1+D)

to convert your bounds on

D

into a range for the best possible single-shot probability of correctly telling the prepared state apart from a perfect copy of

|\Phi^+\rangle

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