Is global asymptotic cloning state estimation?

TL;DR

This paper investigates whether quantum state estimation can asymptotically match the fidelity of quantum cloning when considering all copies jointly, revealing that entangled states outperform simple measure-and-prepare strategies.

Contribution

It demonstrates that measure-and-prepare protocols with entangled states can asymptotically achieve optimal fidelity, challenging the assumption that product states are sufficient.

Findings

Asymptotic measure-and-prepare fidelity matches optimal cloning fidelity for equatorial qubits.

Entangled states outperform product states in asymptotic fidelity for maximally entangled two-qubit states.

Simple estimation and re-preparation of identical copies is suboptimal even at large M.

Abstract

We pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at single-copy level, still holds when all the copies are examined jointly. For an N-to-M cloner, we consider the overall fidelity between the state of the M output systems and the state of M ideal copies, and we ask whether the optimal fidelity is attained by a measure-and- prepare protocol in the limit of large M. In order to gain intuition into the general problem, we analyze two concrete examples: i) cloning qubit states on the equator of the Bloch sphere and ii) cloning two-qubit maximally entangled states. In the first case, we show that the optimal measure-and- prepare fidelity converges to the fidelity of the optimal cloner in the limit of large M. In the second case, we restrict our attention to economical covariant cloners, and again, we exhibit a measure- and-prepare protocol that achieves asymptotically the optimal fidelity. Quite counterintuitively, in both cases the optimal states that have to be prepared in order to maximize the overall fidelity are not product states corresponding to M identical copies, but instead suitable M-partite entangled states: the simple protocol where one estimates the input state and re-prepares M identical copies of the estimated state is strictly suboptimal, even in the asymptotic limit.