Asymptotic cloning is state estimation
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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 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
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 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 .
In the works of Keyl et. al.  and Bruss et.al. , 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.
Bae and Acín showed  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 M case.
Chiribella and D'Ariano  solved the problem by showing that, for every finite value of M, the difference between the optimal cloning fidelity and the optimal estimation fidelity is bounded as , where c is a positive constant depending on the dimension of the single-particle Hilbert spaces. Clearly, this implies the limit formula.
More recently, Chiribella 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 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.
- ↑ 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