dorsal/arxiv
View SchemaClassical Vs Quantum Probability in Sequential Measurements
| Authors | Charis Anastopoulos |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0509019 |
| URL | https://arxiv.org/abs/quant-ph/0509019 |
| DOI | 10.1007/s10701-006-9077-5 |
| Journal | Found. Phys. 36, 1601 (2006) |
Abstract
We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we consider a quantum mechanical measurement device or the presence of an environment. We then examine the same issues in alternative interpretations of quantum theory. We first show that multi-time probabilities cannot be naturally defined in terms of a frequency operator. We next prove that local hidden variable theories cannot reproduce the predictions of quantum theory for sequential measurements, even when the degrees of freedom of the measuring apparatus are taken into account. Bohmian mechanics, however, does not fall in this category. We finally examine an alternative proposal that sequential measurements can be modelled by a process that does not satisfy the Kolmogorov axioms of probability. This removes contextuality without introducing non-locality, but implies that the empirical probabilities cannot be always defined (the event frequencies do not converge). We argue that the predictions of this hypothesis are not ruled out by existing experimental results (examining in particular the "which way" experiments); they are, however, distinguishable in principle.
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"abstract": "We demonstrate in this paper that the probabilities for sequential\nmeasurements have features very different from those of single-time\nmeasurements. First, they cannot be modelled by a classical stochastic process.\nSecond, they are contextual, namely they depend strongly on the specific\nmeasurement scheme through which they are determined. We construct\nPositive-Operator-Valued measures (POVM) that provide such probabilities. For\nobservables with continuous spectrum, the constructed POVMs depend strongly on\nthe resolution of the measurement device, a conclusion that persists even if we\nconsider a quantum mechanical measurement device or the presence of an\nenvironment. We then examine the same issues in alternative interpretations of\nquantum theory. We first show that multi-time probabilities cannot be naturally\ndefined in terms of a frequency operator. We next prove that local hidden\nvariable theories cannot reproduce the predictions of quantum theory for\nsequential measurements, even when the degrees of freedom of the measuring\napparatus are taken into account. Bohmian mechanics, however, does not fall in\nthis category. We finally examine an alternative proposal that sequential\nmeasurements can be modelled by a process that does not satisfy the Kolmogorov\naxioms of probability. This removes contextuality without introducing\nnon-locality, but implies that the empirical probabilities cannot be always\ndefined (the event frequencies do not converge). We argue that the predictions\nof this hypothesis are not ruled out by existing experimental results\n(examining in particular the \"which way\" experiments); they are, however,\ndistinguishable in principle.",
"arxiv_id": "quant-ph/0509019",
"authors": [
"Charis Anastopoulos"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s10701-006-9077-5",
"journal_ref": "Found. Phys. 36, 1601 (2006)",
"title": "Classical Vs Quantum Probability in Sequential Measurements",
"url": "https://arxiv.org/abs/quant-ph/0509019"
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