dorsal/arxiv
View SchemaModal dynamics for positive operator measures
| Authors | Jay Gambetta, H. M. Wiseman |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0306145 |
| URL | https://arxiv.org/abs/quant-ph/0306145 |
| DOI | 10.1023/B:FOOP.0000019622.81881.f8 |
| Journal | Foundations of Physics 34, 419 (2004) |
Abstract
The modal interpretation of quantum mechanics allows one to keep the standard classical definition of realism intact. That is, variables have a definite status for all time and a measurement only tells us which value it had. However, at present modal dynamics are only applicable to situations that are described in the orthodox theory by projective measures. In this paper we extend modal dynamics to include positive operator measures (POMs). That is, for example, rather than using a complete set of orthogonal projectors, we can use an overcomplete set of nonorthogonal projectors. We derive the conditions under which Bell's stochastic modal dynamics for projective measures reduce to deterministic dynamics, showing (incidentally) that Brown and Hiley's generalization of Bohmian mechanics [quant-ph/0005026, (2000)] cannot be thus derived. We then show how {\em deterministic} dynamics for positive operators can also be derived. As a simple case, we consider a Harmonic oscillator, and the overcomplete set of coherent state projectors (i.e. the Husimi POM). We show that the modal dynamics for this POM in the classical limit correspond to the classical dynamics, even for the nonclassical number state $\ket{n}$. This is in contrast to the Bohmian dynamics, which for energy eigenstates, the dynamics are always non-classical.
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"abstract": "The modal interpretation of quantum mechanics allows one to keep the standard\nclassical definition of realism intact. That is, variables have a definite\nstatus for all time and a measurement only tells us which value it had.\nHowever, at present modal dynamics are only applicable to situations that are\ndescribed in the orthodox theory by projective measures. In this paper we\nextend modal dynamics to include positive operator measures (POMs). That is,\nfor example, rather than using a complete set of orthogonal projectors, we can\nuse an overcomplete set of nonorthogonal projectors. We derive the conditions\nunder which Bell\u0027s stochastic modal dynamics for projective measures reduce to\ndeterministic dynamics, showing (incidentally) that Brown and Hiley\u0027s\ngeneralization of Bohmian mechanics [quant-ph/0005026, (2000)] cannot be thus\nderived. We then show how {\\em deterministic} dynamics for positive operators\ncan also be derived. As a simple case, we consider a Harmonic oscillator, and\nthe overcomplete set of coherent state projectors (i.e. the Husimi POM). We\nshow that the modal dynamics for this POM in the classical limit correspond to\nthe classical dynamics, even for the nonclassical number state $\\ket{n}$. This\nis in contrast to the Bohmian dynamics, which for energy eigenstates, the\ndynamics are always non-classical.",
"arxiv_id": "quant-ph/0306145",
"authors": [
"Jay Gambetta",
"H. M. Wiseman"
],
"categories": [
"quant-ph"
],
"doi": "10.1023/B:FOOP.0000019622.81881.f8",
"journal_ref": "Foundations of Physics 34, 419 (2004)",
"title": "Modal dynamics for positive operator measures",
"url": "https://arxiv.org/abs/quant-ph/0306145"
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