Conjugates, Filters and Quantum Mechanics

Alexander Wilce

doi:10.22331/q-2019-07-08-158

Conjugates, Filters and Quantum Mechanics

Alexander Wilce

Department of Mathematics and Computer Science, Susquehanna University

Published:	2019-07-08, volume 3, page 158
Eprint:	arXiv:1206.2897v8
Doi:	https://doi.org/10.22331/q-2019-07-08-158
Citation:	Quantum 3, 158 (2019).

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Abstract

The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system $A$ can be paired with an isomorphic $\textit{conjugate}$ system, $\overline{A}$, by means of a non-signaling bipartite state $\eta_A$ perfectly and uniformly correlating each basic measurement on $A$ with its counterpart on $\overline{A}$. In the case of a quantum-mechanical system associated with a complex Hilbert space $H$, the conjugate system is that associated with the conjugate Hilbert space $H$, and $\eta_A$ corresponds to the standard maximally entangled EPR state on $H \otimes \overline{H}$. A second ingredient is the notion of a $\textit{reversible filter}$, that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the ``no restriction" hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. (An appendix shows how a version of Hardy's ``subpace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.)

► BibTeX data

@article{Wilce2019conjugatesfilters,
  doi = {10.22331/q-2019-07-08-158},
  url = {https://doi.org/10.22331/q-2019-07-08-158},
  title = {Conjugates, {F}ilters and {Q}uantum {M}echanics},
  author = {Wilce, Alexander},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {158},
  month = jul,
  year = {2019}
}

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Cited by

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The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 01:59:40) and SAO/NASA ADS (last updated successfully 2024-05-21 01:59:41). The list may be incomplete as not all publishers provide suitable and complete citation data.

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