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arXiv:1906.04939 · Symmetry Integrability and Geometry Methods and Applications

Geometric Approach to Quantum Theory

UC Davis, USA, Albert Schwarz

Published Apr 1, 2020• 6 citations
arXivHTMLPDF

Abstract

We formulate quantum theory taking as a starting point the cone of states.
DOI
10.3842/sigma.2020.020

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The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.
Generalized Parton Distribution functions (GPDs) are off-diagonal light-cone matrix elements that encode the internal structure of hadrons in terms of quark and gluon degrees of freedom. In this work, we present the first nonperturbative study of quasi-GPDs in the massive Schwinger model, quantum electrodynamics in 1+1 dimensions (QED2), within the Hamiltonian formulation of lattice field theory. Quasi-distributions are spatial correlation functions of boosted states, which approach the relevant light-cone distributions in the luminal limit. Using tensor networks, we prepare the first excited state in the strongly coupled regime and boost it to close to the light-cone on lattices of up to 400 lattice sites. We compute both quasi-parton distribution functions and, for the first time, quasi-GPDs, and study their convergence for increasingly boosted states. In addition, we perform analytic calculations of GPDs in the two-particle Fock-space approximation and in the Reggeized limit, providing qualitative benchmarks for the tensor network results. Our analysis establishes computational benchmarks for accessing partonic observables in low-dimensional gauge theories, offering a starting point for future extensions to higher dimensions, non-Abelian theories, and quantum simulations.