Conjugates, Filters and Quantum Mechanics

Alexander Wilce

Department of Mathematics and Computer Science, Susquehanna University

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.)

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Cited by

[1] Carlo Maria Scandolo, "Information-theoretic foundations of thermodynamics in general probabilistic theories", arXiv:1901.08054.

[2] Peter Janotta and Haye Hinrichsen, "Generalized probability theories: what determines the structure of quantum theory?", Journal of Physics A Mathematical General 47 32, 323001 (2014).

[3] Howard Barnum, Matthew A. Graydon, and Alexander Wilce, "Some Nearly Quantum Theories", arXiv:1507.06278.

[4] Giulio Chiribella and Carlo Maria Scandolo, "Entanglement as an axiomatic foundation for statistical mechanics", arXiv:1608.04459.

[5] Howard Barnum, Matthew Graydon, and Alexander Wilce, "Composites and Categories of Euclidean Jordan Algebras", arXiv:1606.09331.

[6] Alexander Wilce, "A Royal Road to Quantum Theory (or Thereabouts)", Entropy 20 4, 227 (2018).

[7] Giulio Chiribella, "Agents, Subsystems, and the Conservation of Information", Entropy 20 5, 358 (2018).

[8] Howard Barnum and Alexander Wilce, "Local Tomography and the Jordan Structure of Quantum Theory", Foundations of Physics 44 2, 192 (2014).

[9] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti, "Quantum from principles", arXiv:1506.00398.

[10] Ding Jia, "Quantum theories from principles without assuming a definite causal structure", Physical Review A 98 3, 032112 (2018).

[11] Peter Janotta and Raymond Lal, "Non-locality in theories without the no-restriction hypothesis", arXiv:1412.8524.

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