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Asymptotic cloning is state estimation? |
| contact: |
M. Keyl |
date: |
10 Feb 2005 |
last progress: |
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solved by: |
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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 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 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 N\to\infty.
In the examples [KW99,BCDM00], 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.
| [KW99] | M. Keyl and R.F. Werner, »Optimal Cloning of Pure
States, Judging Single Clones«, J. Math. Phys. 40, 3283 (1999) and
quant-ph/9807010 (1998). |
| [BCDM00] | D. Bruss, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello,
»Phase covariant quantum cloning«, Phys. Rev. A 62, 12302
(2000) and quant-ph/9909046
(1999). |
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