Lüders Theorem for Coherent-State POVMs

P Busch; Stefan Weigert

doi:10.1063/1.1623001

Lüders Theorem for Coherent-State POVMs

P Busch, Stefan Weigert

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Lueders theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on a Lie group. In general, operators turn out to be invariant under the Lueders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lueders map generated by the coherent states of the Heisenberg-Weyl group, H_3. The Lueders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.

Original language	English
Pages (from-to)	5474-5486
Number of pages	13
Journal	J. Phys. A: Math. Theor.
Volume	44
Issue number	12
DOIs	https://doi.org/10.1063/1.1623001
Publication status	Published - Dec 2003

Keywords

Mathematical Physics

Access to Document

10.1063/1.1623001

http://arxiv.org/abs/quant-ph/0308035

Cite this

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title = "L{\"u}ders Theorem for Coherent-State POVMs",

abstract = "Lueders theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on a Lie group. In general, operators turn out to be invariant under the Lueders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lueders map generated by the coherent states of the Heisenberg-Weyl group, H_3. The Lueders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.",

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AB - Lueders theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on a Lie group. In general, operators turn out to be invariant under the Lueders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lueders map generated by the coherent states of the Heisenberg-Weyl group, H_3. The Lueders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.

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