Department of Physics & Astronomy
University of New Mexico

CQuIC Seminars

Principle-based formulations of quantum theory

Presented by Howard N. Barnum

I give two characterizations of finite-dimensional quantum theory's framework of
density matrices (states) and POVM elements (measurement outcomes) for describing systems, from simple postulates whose physical and informational meaning and appeal is clear. Each first characterizes a class of Euclidean Jordan-algebraic (EJA) systems. The simple EJAs were classified by Jordan, von Neumann, and Wigner: real, complex, and quaternionic quantum theory, systems whose state spaces are balls, and 3-dimensional octonionic quantum theory. Complex quantum theory then follows from "local tomography", or "energy observability": the generators of continuous symmetries of the state space (potential reversible dynamics) are observables.

The first characterization uses:
(1) Spectrality: every state is a convex combination of perfectly distinguishable pure states, (2) Strong Symmetry: every set of perfectly distinguishable pure states may be taken to any other such set (of the same size) by a symmetry of the state space.

The second uses:
(1) Homogeneity: any strictly positive element of the cone of unnormalized states may be taken to any other by a symmetry of this cone,
(2) Pure Transitivity: any pure state may be taken to any other pure state by a symmetry of the normalized state space

The physical, informational, and operational significance of the postulates will be explained.

Collaborators: Joachim Hilgert, Cozmin Ududec, John van de Wetering

3:30 pm, Thursday, September 14, 2023

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