Fisher information in quantum statistics

O.E. Barndorff-Nielsen, R.D. Gill

    Research output: Contribution to journalArticleAcademicpeer-review

    126 Citations (Scopus)


    Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.
    Original languageEnglish
    Pages (from-to)4481-4490
    JournalJournal of Physics A: Mathematical and General
    Issue number24
    Publication statusPublished - 2000

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