dorsal/arxiv
View SchemaBhattacharyya inequality for quantum state estimation
| Authors | Yoshiyuki Tsuda |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611182 |
| URL | https://arxiv.org/abs/quant-ph/0611182 |
| DOI | 10.1088/1751-8113/40/4/015 |
| Journal | J. Phys. A: Math. Theor. 40 793-810 (2007) |
Abstract
Using higher-order derivative with respect to the parameter, we will give lower bounds for variance of unbiased estimators in quantum estimation problems. This is a quantum version of the Bhattacharyya inequality in the classical statistical estimation. Because of non-commutativity of operator multiplication, we obtain three different types of lower bounds; Type S, Type R and Type L. If the parameter is a real number, the Type S bound is useful. If the parameter is complex, the Type R and L bounds are useful. As an application, we will consider estimation of polynomials of the complex amplitude of the quantum Gaussian state. For the case where the amplitude lies in the real axis, a uniformly optimum estimator for the square of the amplitude will be derived using the Type S bound. It will be shown that there is no unbiased estimator uniformly optimum as a polynomial of annihilation and/or creation operators for the cube of the amplitude. For the case where the amplitude does not necessarily lie in the real axis, uniformly optimum estimators for holomorphic, antiholomorphic and real-valued polynomials of the amplitude will be derived. Those estimators for the holomorphic and real-valued cases attains the Type R bound, and those for the antiholomorphic and real-valued cases attains the Type L bound. This article clarifies what is the best method to measure energy of laser.
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"abstract": "Using higher-order derivative with respect to the parameter, we will give\nlower bounds for variance of unbiased estimators in quantum estimation\nproblems. This is a quantum version of the Bhattacharyya inequality in the\nclassical statistical estimation. Because of non-commutativity of operator\nmultiplication, we obtain three different types of lower bounds; Type S, Type R\nand Type L. If the parameter is a real number, the Type S bound is useful. If\nthe parameter is complex, the Type R and L bounds are useful. As an\napplication, we will consider estimation of polynomials of the complex\namplitude of the quantum Gaussian state. For the case where the amplitude lies\nin the real axis, a uniformly optimum estimator for the square of the amplitude\nwill be derived using the Type S bound. It will be shown that there is no\nunbiased estimator uniformly optimum as a polynomial of annihilation and/or\ncreation operators for the cube of the amplitude. For the case where the\namplitude does not necessarily lie in the real axis, uniformly optimum\nestimators for holomorphic, antiholomorphic and real-valued polynomials of the\namplitude will be derived. Those estimators for the holomorphic and real-valued\ncases attains the Type R bound, and those for the antiholomorphic and\nreal-valued cases attains the Type L bound. This article clarifies what is the\nbest method to measure energy of laser.",
"arxiv_id": "quant-ph/0611182",
"authors": [
"Yoshiyuki Tsuda"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/1751-8113/40/4/015",
"journal_ref": "J. Phys. A: Math. Theor. 40 793-810 (2007)",
"title": "Bhattacharyya inequality for quantum state estimation",
"url": "https://arxiv.org/abs/quant-ph/0611182"
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