Quantum-teleportation benchmarks for independent and identically distributed spin states and displaced thermal states
Mădălin Guţǎ, Peter Bowles, Gerardo Adesso · Physical Review A · 2010
A successful state-transfer (or teleportation) experiment must perform better than the benchmark set by the ``best'' measure and prepare procedure. We consider the benchmark problem for the following families of states: (i) displaced thermal equilibrium states of a given temperature; (ii) independent identically prepared qubits with a completely unknown state. For the first family we show that the optimal procedure is heterodyne measurement followed by the preparation of a coherent state. This procedure was known to be optimal for coherent states and for squeezed states with the ``overlap fidelity'' as the figure of merit. Here, we prove its optimality with respect to the trace norm distance and supremum risk. For the second problem we consider $n$ independent and identically distributed (i.i.d.) spin-$\frac{1}{2}$ systems in an arbitrary unknown state $\ensuremath{\rho}$ and look for the measurement-preparation pair $({M}_{n},{P}_{n})$ for which the reconstructed state ${\ensuremath{\omega}}_{n}:={P}_{n}○{M}_{n}({\ensuremath{\rho}}^{\ensuremath{\bigotimes}n})$ is as close as possible to the input state (i.e., $\ensuremath{\Vert}{\ensuremath{\omega}}_{n}\ensuremath{-}{\ensuremath{\rho}}^{\ensuremath{\bigotimes}n}\ensuremath{\Vert}{}_{1}$ is small). The figure of merit is based on the trace norm distance between the input and output states. We show that asymptotically with $n$ this problem is equivalent to the first one. The proof and construction of $({M}_{n},{P}_{n})$ uses the theory of local asymptotic normality developed for state estimation which shows that i.i.d. quantum models can be approximated in a strong sense by quantum Gaussian models. The measurement part is identical to ``optimal estimation,'' showing that ``benchmarking'' and estimation are closely related problems in the asymptotic set up.