dorsal/arxiv
View SchemaState estimation for large ensembles
| Authors | R. D. Gill, S. Massar |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9902063 |
| URL | https://arxiv.org/abs/quant-ph/9902063 |
| DOI | 10.1103/PhysRevA.61.042312 |
| Journal | Phys.Rev.A61:042312,2002 |
Abstract
We consider the problem of estimating the state of a large but finite number $N$ of identical quantum systems. In the limit of large $N$ the problem simplifies. In particular the only relevant measure of the quality of the estimation is the mean quadratic error matrix. Here we present a bound on the mean quadratic error which is a new quantum version of the Cram\'er-Rao inequality. This new bound expresses in a succinct way how in the quantum case one can trade information about one parameter for information about another parameter. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states--a striking example of non-locality without entanglement for mixed but not for pure states. Cram\'er-Rao bounds are generally derived under the assumption that the estimator is unbiased. We also prove that under additional regularity conditions our bound also holds for biased estimators. Finally we prove that when the unknown states belong to a 2 dimensional Hilbert space our quantum Cram\'er-Rao bound can always be attained and we provide an explicit measurement strategy that attains our bound. This therefore provides a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. For qubits in the same mixed state, this also provides an optimal estimation strategy if one only considers separable measurements.
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"abstract": "We consider the problem of estimating the state of a large but finite number\n$N$ of identical quantum systems. In the limit of large $N$ the problem\nsimplifies. In particular the only relevant measure of the quality of the\nestimation is the mean quadratic error matrix. Here we present a bound on the\nmean quadratic error which is a new quantum version of the Cram\\\u0027er-Rao\ninequality. This new bound expresses in a succinct way how in the quantum case\none can trade information about one parameter for information about another\nparameter. The bound holds for arbitrary measurements on pure states, but only\nfor separable measurements on mixed states--a striking example of non-locality\nwithout entanglement for mixed but not for pure states. Cram\\\u0027er-Rao bounds are\ngenerally derived under the assumption that the estimator is unbiased. We also\nprove that under additional regularity conditions our bound also holds for\nbiased estimators. Finally we prove that when the unknown states belong to a 2\ndimensional Hilbert space our quantum Cram\\\u0027er-Rao bound can always be attained\nand we provide an explicit measurement strategy that attains our bound. This\ntherefore provides a complete solution to the problem of estimating as\nefficiently as possible the unknown state of a large ensemble of qubits in the\nsame pure state. For qubits in the same mixed state, this also provides an\noptimal estimation strategy if one only considers separable measurements.",
"arxiv_id": "quant-ph/9902063",
"authors": [
"R. D. Gill",
"S. Massar"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.61.042312",
"journal_ref": "Phys.Rev.A61:042312,2002",
"title": "State estimation for large ensembles",
"url": "https://arxiv.org/abs/quant-ph/9902063"
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