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arXiv:2411.19672

Continuous symmetry entails the Jordan algebra structure of finite-dimensional quantum theory

Gerd Niestegge

Published Nov 29, 2024• 0 citations
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Abstract

Symmetry postulates play a crucial role in various approaches to reconstruct quantum theory from a few basic principles. Discrete and continuous symmetries are under consideration. The continuous case better matches the physical needs for mathematical models of dynamical processes and is studied here. Applying the representation theory of the orthomodular lattices and a generalized version of Gleason's theorem for Jordan matrix algebras, we show that the continuous symmetry, together with three further requirements, entails that the underlying mathematical structure of a finite-dimensional generalized probabilistic theory becomes a simple Euclidean Jordan algebra. The further requirements are: spectrality, a strong state space and a condition called gbit property.

DOI
10.48550/arxiv.2411.19672

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Symmetry postulates play a crucial role in various approaches to reconstruct quantum theory from a few basic principles. Discrete and continuous symmetries are under consideration. The continuous case better matches the physical needs for mathematical models of dynamical processes and is studied here. Applying the representation theory of the orthomodular lattices and a generalized version of Gleason's theorem for Jordan matrix algebras, we show that the continuous symmetry, together with three further requirements, entails that the underlying mathematical structure of a finite-dimensional generalized probabilistic theory becomes a simple Euclidean Jordan algebra. The further requirements are: spectrality, a strong state space and a condition called gbit property.
Abstract It is quite common to use the generalized probabilistic theories (GPTs) as generic models to reconstruct quantum theory from a few basic principles and to gain a better understanding of the probabilistic or information theoretic foundations of quantum physics and quantum computing. A variety of symmetry postulates was introduced and studied in this framework, including the transitivity of the automorphism group (1) on the pure states, (2) on the pairs of orthogonal pure states [these pairs are called 2-frames] and (3) on any frames of the same size. The second postulate is Müller and Ududec’s bit symmetry , which they motivate by quantum computational needs. Here we explore these three postulates in the transition probability framework, which is more specific than the GPTs since the existence of the transition probabilities for the quantum logical atoms is presupposed either directly or indirectly via a certain geometric property of the state space. This property for compact convex sets was introduced by the author in a recent paper. We show that bit symmetry implicates the symmetry of the transition probabilities between the atoms. Using a result by Barnum and Hilgert, we can then conclude that the third rather strong symmetry postulate rules out all models but the classical cases and the simple Euclidean Jordan algebras.
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I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.