A computer scientist’s reconstruction of quantum theory*

Westerbaan, Bas; Wetering, John van de

doi:10.1088/1751-8121/ac8459

Cited by 2 publications

(1 citation statement)

References 68 publications

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“…Since their introduction in [46] in the context of the foundations of quantum mechanics, Jordan algebras proved to be extremely versatile in both mathematics and physics. For instance, we mention the link between Jordan algebras, symmetric and harmonic analysis [18,19,30,52,78], the connection between Jordan algebras and quantum theories [7,27,29], and the role Jordan algebras play in the reconstruction of quantum theories [62,79]. We also refer to [43] for an extensive discussion of different fields of application of Jordan algebras in both mathematics and physics.…”

Section: Introductionmentioning

confidence: 99%

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Ciaglia,

Jost,

Schwachhöfer

2023

SIGMA

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Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$, we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

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Section: Introductionmentioning

confidence: 99%

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Ciaglia,

Jost,

Schwachhöfer

2023

SIGMA

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Complete extension: the non-signaling analog of quantum purification

Winczewski,

Das,

Selby

et al. 2023

Quantum

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Deriving quantum mechanics from information-theoretic postulates is a recent research direction taken, in part, with the view of finding a beyond-quantum theory; once the postulates are clear, we can consider modifications to them. A key postulate is the purification postulate, which we propose to replace by a more generally applicable postulate that we call the complete extension postulate (CEP), i.e., the existence of an extension of a physical system from which one can generate any other extension. This new concept leads to a plethora of open questions and research directions in the study of general theories satisfying the CEP (which may include a theory that hyper-decoheres to quantum theory). For example, we show that the CEP implies the impossibility of bit-commitment. This is exemplified by a case study of the theory of non-signalling behaviors which we show satisfies the CEP. We moreover show that in certain cases the complete extension will not be pure, highlighting the key divergence from the purification postulate.

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A computer scientist’s reconstruction of quantum theory*

Cited by 2 publications

References 68 publications

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Complete extension: the non-signaling analog of quantum purification

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