Conjugates, Filters and Quantum Mechanics

Alexander Wilce

Department of Mathematics and Computer Science, Susquehanna University

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.)

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Cited by

[1] Gerd Niestegge, "Local tomography and the role of the complex numbers in quantum mechanics", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 2238, 20200063 (2020).

[2] Laurie Letertre, "The operational framework for quantum theories is both epistemologically and ontologically neutral", Studies in History and Philosophy of Science Part A 89, 129 (2021).

[3] Sergey N. Filippov, Stan Gudder, Teiko Heinosaari, and Leevi Leppäjärvi, "Operational Restrictions in General Probabilistic Theories", Foundations of Physics 50 8, 850 (2020).

[4] Kenji Nakahira, "Derivation of quantum theory with superselection rules", Physical Review A 101 2, 022104 (2020).

[5] Giulio Chiribella, "Process Tomography in General Physical Theories", Symmetry 13 11, 1985 (2021).

[6] Damián Pitalúa-García, "Spacetime symmetries and the qubit Bloch ball: A physical derivation of finite-dimensional quantum theory and the number of spatial dimensions", Physical Review A 104 3, 032220 (2021).

[7] 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).

[8] Giulio Chiribella and Carlo Maria Scandolo, "Entanglement as an axiomatic foundation for statistical mechanics", arXiv:1608.04459, (2016).

[9] Howard Barnum, Matthew A. Graydon, and Alexander Wilce, "Composites and Categories of Euclidean Jordan Algebras", arXiv:1606.09331, (2016).

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

[11] Alexander Wilce, "A Royal Road to Quantum Theory (or Thereabouts)", arXiv:1606.09306, (2016).

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

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

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