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arXiv:1703.06800

Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories

Matthew A. Graydon

Published Mar 20, 2017• 2 citations
arXivHTMLPDF

Abstract

Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamiołkowski isomorphism, one and the same --- namely the homogeneous self-dual cones of positive semi-definite linear endomorphisms on finite dimensional complex Hilbert spaces. From the perspective of category theory, these cones are the sets of morphisms in finite dimensional quantum theory as a dagger compact closed category. Understanding the intricate geometry of these cones and charting the wider landscape for their host category is imperative for foundational physics. In Part I of this thesis, we study the shape of finite dimensional quantum theory in terms of quantum information. In Part II of this thesis, we move beyond quantum theory within the vein of Euclidean Jordan algebras. In posting this thesis on the arXiv, we hope that it might serve as a useful resource for those interested in its subjects.

DOI
10.48550/arxiv.1703.06800

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Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamiołkowski isomorphism, one and the same --- namely the homogeneous self-dual cones of positive semi-definite linear endomorphisms on finite dimensional complex Hilbert spaces. From the perspective of category theory, these cones are the sets of morphisms in finite dimensional quantum theory as a dagger compact closed category. Understanding the intricate geometry of these cones and charting the wider landscape for their host category is imperative for foundational physics. In Part I of this thesis, we study the shape of finite dimensional quantum theory in terms of quantum information. In Part II of this thesis, we move beyond quantum theory within the vein of Euclidean Jordan algebras. In posting this thesis on the arXiv, we hope that it might serve as a useful resource for those interested in its subjects.
Abstract The Choi–Jamiołkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C ∗ -algebras they arise from, which are distinguished by their order of composition. A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism (Choi 1975 Lin. Alg. Appl. 10 285) further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially depends on the dynamical structure in C ∗ -algebras. We explain in detail how this dependence gives rise to the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiołkowski (1972 Rep. Math. Phys. 3 275). We then overcome this subtle but pervasive issue in a number of ways: first, we prove a version of Choi’s theorem for Jamiołkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C ∗ -algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.
The interplay between the algebraic structure (operator algebras) for the quantum observables and the convex structure of the state space has been explored for a long time, and most advanced results are due to Alfsen and Shultz. Here we present a more elementary approach with a more general structure for the observables, which focuses on the transition probability of the quantum logical atoms. The binary case gives rise to the generalized qubit models and was fully developed in a preceding paper. Here we consider any case with finite information capacity (binary means that the information capacity is 2). A novel geometric property that makes any compact convex set a matching state space is presented. Generally, the transition probability is not symmetric; if it is symmetric, we get an inner product and a self-dual cone. The emerging mathematical structure comes close to the Euclidean Jordan algebras and becomes a new mathematical model for a potential extension of quantum theory.
Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.