On quantum estimation, quantum cloning and finite quantum de Finetti theorems

TL;DR

This paper explores the relationship between quantum estimation, cloning, and finite de Finetti theorems, revealing how measure-and-prepare channels approximate optimal cloning or partial trace depending on the ratio of input to output copies, with implications for quantum information distribution.

Contribution

It establishes a unified framework connecting quantum estimation, cloning, and de Finetti theorems, providing new bounds and convergence results for symmetric broadcast channels.

Findings

Measure-and-prepare channels approximate universal cloning for large k.

Estimation becomes nearly perfect when M is large compared to k.

Finite de Finetti theorems yield bounds on quantum information distribution.

Abstract

This paper presents a series of results on the interplay between quantum estimation, cloning and finite de Finetti theorems. First, we consider the measure-and-prepare channel that uses optimal estimation to convert M copies into k approximate copies of an unknown pure state and we show that this channel is equal to a random loss of all but s particles followed by cloning from s to k copies. When the number k of output copies is large with respect to the number M of input copies the measure-and-prepare channel converges in diamond norm to the optimal universal cloning. In the opposite case, when M is large compared to k, the estimation becomes almost perfect and the measure-and-prepare channel converges in diamond norm to the partial trace over all but k systems. This result is then used to derive de Finetti-type results for quantum states and for symmetric broadcast channels, that is, channels that distribute quantum information to many receivers in a permutationally invariant fashion. Applications of the finite de Finetti theorem for symmetric broadcast channels include the derivation of diamond-norm bounds on the asymptotic convergence of quantum cloning to state estimation and the derivation of bounds on the amount of quantum information that can be jointly decoded by a group of k receivers at the output of a symmetric broadcast channel.