Asymptotic cloning is state estimation

Cite this problem as Problem 22.

Problem

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

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

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

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.

Solution

Bae and Acín solved the problem in [3], by arguing that the Choi operator of the optimal channel (for an arbitrary distribution of states) producing k indistinguishable clones must be k-extendible. By the Bolzano-Weierstrass theorem, this implies that, in finite dimensions, there exists a subsequence of optimal channels for increasing k that in the limit k\to\infty tends to an \infty-extendible (and thus separable [4]) Choi matrix. Hence the channel must be entanglement-breaking and therefore of the measure-and-prepare form. In particular, the monotone sequence of values (F(N,k))_k must converge to F(N,\infty).

Shortly after, Chiribella and D’Ariano showed that, for every finite value of M, the difference between the optimal cloning fidelity and the optimal estimation fidelity is bounded as |F (N,M) - F(N,\infty) | \le c / M, where c is a positive constant depending on the dimension of the single-particle Hilbert spaces [5]. Chiribella extended this result in [6] by proving that the diamond norm distance between the one-particle restriction of a cloning channel with M output copies and the closest measure-and-reprepare channel is bounded by c/M, where c is again a positive constant depending on the dimension of the single-particle Hilbert space.

References

[1] M. Keyl and R.F. Werner, Optimal Cloning of Pure States, Judging Single Clones, J. Math. Phys. 40, 3283 (1999)

[2] D. Bruss, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello, Phase covariant quantum cloning, Phys. Rev. A 62, 12302 (2000)

[3] J. Bae and A. Acín Phys. Rev. Lett. 97, 030402 (2006),

[4] R. F. Werner, Lett. Math. Phys. 17, 359 (1989).

[5] G. Chiribella, G. M. D’Ariano, Quantum information becomes classical when distributed to many users, Phys. Rev. Lett. 97 250503 (2006)

[6] 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