Sponsor:
We acknowledge financial support from: the Spanish MICINN, through the RamÓn y Cajal program (JC), contract FIS2008-01236, and project QOIT
(CONSOLIDER2006-00019); from the Generalitat de
Catalunya CIRIT, contract 2005SGR-00994; and from
Alianza 4 Universidades program(JIdV).

Keywords:
[PACS] Quantum information
,
[PACS] Quantum communication
,
[PACS] Foundations of quantum mechanics; measurement theory
,
[PACS] State reconstruction, quantum tomography

We consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according to an unknown probability law, we give strategies to estimate the individual probabilities, weiWe consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according to an unknown probability law, we give strategies to estimate the individual probabilities, weights, or mixing proportions. Such strategies can be used to estimate the frequencies at which different independent signals are emitted by a source. They can also be used to estimate the weights of particular terms in a canonical decomposition of a quantum channel. The quality of these strategies is quantified by a covariance-type error matrix. According with this cost function, we give optimal strategies in both the single-shot and multiple-copy scenarios. The latter is also analyzed in the asymptotic limit of large number of copies. We give closed expressions of the error matrix for two-component quantum mixtures of qubit systems. The Fisher information plays an unusual role in the problem at hand, providing exact expressions of the minimum covariance matrix for any number of copies.[+][-]