dorsal/arxiv
View SchemaFisher information in quantum statistics
| Authors | O. E. Barndorff-Nielsen, R. D. Gill |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9808009 |
| URL | https://arxiv.org/abs/quant-ph/9808009 |
| DOI | 10.1088/0305-4470/33/24/306 |
| Journal | J.Phys. A30 (2000) 4481-4490 |
Abstract
Braunstein and Caves (1994) proposed to use Helstrom's {\em 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 else 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.
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"abstract": "Braunstein and Caves (1994) proposed to use Helstrom\u0027s {\\em quantum\ninformation} number to define, meaningfully, a metric on the set of all\npossible states of a given quantum system. They showed that the quantum\ninformation is nothing else than the maximal Fisher information in a\nmeasurement of the quantum system, maximized over all possible measurements.\nCombining this fact with classical statistical results, they argued that the\nquantum information determines the asymptotically optimal rate at which\nneighbouring states on some smooth curve can be distinguished, based on\narbitrary measurements on $n$ identical copies of the given quantum system.\n We show that the measurement which maximizes the Fisher information typically\ndepends on the true, unknown, state of the quantum system. We close the\nresulting loophole in the argument by showing that one can still achieve the\nsame, optimal, rate of distinguishability, by a two stage adaptive measurement\nprocedure.\n When we consider states lying not on a smooth curve, but on a manifold of\nhigher dimension, the situation becomes much more complex. We show that the\nnotion of ``distinguishability of close-by states\u0027\u0027 depends strongly on the\nmeasurement resources one allows oneself, and on a further specification of the\ntask at hand. The quantum information matrix no longer seems to play a central\nrole.",
"arxiv_id": "quant-ph/9808009",
"authors": [
"O. E. Barndorff-Nielsen",
"R. D. Gill"
],
"categories": [
"quant-ph",
"math.ST",
"stat.TH"
],
"doi": "10.1088/0305-4470/33/24/306",
"journal_ref": "J.Phys. A30 (2000) 4481-4490",
"title": "Fisher information in quantum statistics",
"url": "https://arxiv.org/abs/quant-ph/9808009"
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