# Asymptotic cloning is state estimation

**Cite as:** http://qig.itp.uni-hannover.de/qiproblems/22

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# Problem

Fix an arbitrary probability measure on the pure states of a [math]d[/math]-dimensional quantum system. Let [math]F(N,M)[/math] be the optimal single copy fidelity for [math]N[/math]-to-[math]M[/math] cloning transformations, averaged with respect to the given probability measure and over all [math]M[/math] clones.

On the other hand, let [math]F(N,\infty)[/math] be the best mean fidelity achievable by measuring on [math]N[/math] input copies of the state, and repreparing a state according to the measured data. The problem is to decide whether one always gets [math] \lim_{M\to\infty} F(N,M) = F(N,\infty).[/math]

It is clear that the limit exists, because [math]F(N,M)[/math] is non-increasing in [math]M[/math]. Moreover, the limit will be larger or equal than the right hand side, because estimation with repreaparation is a particular cloning method. A weaker, but still interesting version of the problem is whether the above equation becomes true in the limit [math]N\to\infty[/math].

# Background

In the works of Keyl et. al. ^{[1]} and Bruss et.al. ^{[2]}, where optimal cloner and estimator have been
computed, the formula is true. The limit formula is a piece of
folklore, partly based on the idea that if one has many clones, one
could make a statistical measurement on them and thereby obtain a
good estimation. This reasoning is faulty, however, because it neglects the
correlations, and possibly the entanglement between the clones.

# Partial Results

Bae and Acín showed ^{[3]} that a channel producing an inﬁnite number of indistinguishable clones must be of the measure-and-prepare form. However, this does not clarify how the limit is approached from the finite [math]M[/math] case.

Chiribella and D'Ariano ^{[4]} solved the problem by showing that, for every finite value of [math]M[/math], the difference between the optimal cloning fidelity and the optimal estimation fidelity is bounded as [math]|F (N,M) - F(N,\infty) | \le c / M[/math], where [math]c[/math] is a positive constant depending on the dimension of the single-particle Hilbert spaces. Clearly, this implies the limit formula.

More recently, Chiribella^{[5]} proved the equivalence of asymptotic cloning and state estimation in a stronger sense , showing that the one-particle restriction of a cloning channel producing permutationally invariant output states is close to a measure-and-reprepare channel. Precisely, he proved that the diamond norm distance between the one-particle restriction of a cloning channel with [math]M[/math] output copies and the closest measure-and-reprepare channel is bounded by [math]c/M[/math], where [math]c[/math] is again a positive constant depending on the dimension of the single-particle Hilbert space.

# Literature

- ↑ M. Keyl and R.F. Werner, Optimal Cloning of Pure States, Judging Single Clones, J. Math. Phys. 40, 3283 (1999)
- ↑ D. Bruss, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, Phase covariant quantum cloning, Phys. Rev. A 62, 12302 (2000)
- ↑ J. Bae and A. Acín Phys. Rev. Lett. 97, 030402 (2006),
- ↑ G. Chiribella, G. M. D'Ariano, Quantum information becomes classical when distributed to many users, Phys. Rev. Lett. 97 250503 (2006)
- ↑ G. Chiribella, On quantum estimation, quantum cloning and finite quantum de Finetti theorems, Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science, 2011, Volume 6519/2011, 9-25, and http://arxiv.org/abs/1010.1875