Quantum teleportation benchmarks for independent and identically-distributed spin states and displaced thermal states

TL;DR

This paper establishes optimal benchmarks for quantum teleportation of specific state families, demonstrating that heterodyne measurement and estimation techniques are asymptotically optimal for independent, identically prepared quantum states.

Contribution

It proves the optimality of heterodyne measurement for displaced thermal states and links benchmarking to quantum state estimation via local asymptotic normality.

Findings

Heterodyne measurement is optimal for displaced thermal states.

Asymptotic equivalence between benchmarking and state estimation.

Optimal measurement-preparation pairs minimize trace norm distance.

Abstract

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 given temperature; (ii) independent identically prepared qubits with 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 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 i.i.d. spin-1/2 systems in an arbitrary unknown state $\rho$ and look for the measurement-preparation pair $(M_{n},P_{n})$ for which the reconstructed state $\omega_{n}:=P_{n}\circ M_{n} (\rho^{\otimes n})$ is as close as possible to the input state, i.e. $\|\omega_{n}- \rho^{\otimes n}\|_{1}$ is small. The figure of merit is based on the trace norm distance between input and output states. We show that asymptotically with $n$ the 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 with `optimal estimation', showing that `benchmarking' and estimation are closely related problems in the asymptotic set-up.