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Journal of Physics A Mathematical and General

Continuum quantum systems as limits of discrete quantum systems: II. State functions

Laurence Barker

Published May 24, 2001• 15 citations
Publisher

Abstract

In this second of four papers on the eponymous topic, pointwise convergence of a `discrete' state function to a `continuum' state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel. As examples (and as a prerequisite for the sequels), the normal approximation theorem and the convergence of the Kravchuk functions to the Hermite-Gaussians are expressed in terms of the algebraic notion of convergence.

DOI
10.1088/0305-4470/34/22/308

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In this second of four papers on the eponymous topic, pointwise convergence of a discrete' state function to a continuum' state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel. As examples (and as a prerequisite for the sequels), the normal approximation theorem and the convergence of the Kravchuk functions to the Hermite-Gaussians are expressed in terms of the algebraic notion of convergence.
We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
Gaussian states play a central role in continuous quantum mechanics, and an appropriate generalization might play a similar role in discrete quantum mechanics. A common-sense discretization of these turns out to be trivial. In this work, we describe two new discretizations based on recent developments in discrete phase space methods. We also extend existing notions of convergence between continuous and discrete systems and apply some of them to our constructions. Accompanying this work is a numerical software package "qWeyl" which we use to give examples and provide visualizations.
We establish a deterministic discrete-to-continuum theorem for free-information fields whose values are finite collections of quantum-register density operators. The local state space is the compact convex configuration space introduced in the companion finite theory, embedded in a finite-dimensional Hilbert space with the Hilbert–Schmidt metric. A mesh field carries two contributions: a finite-range quadratic register-mismatch energy and a local free-information density consisting of an energetic term, the relational-entropy hierarchy, and an optional log-determinant faithfulness barrier. Under explicit assumptions on cell quadrature, finite interaction range, directional second moments, discrete Poincaré compactness, and divergence consistency, the discrete Dirichlet forms Mosco-converge to an anisotropic continuum form. We then prove equicoercivity, the full Gamma-liminf inequality, a recovery theorem that respects the density-matrix constraints and the barrier, convergence of minima and almost minimizers, and mesh universality: any two admissible mesh sequences with the same limiting second-moment tensor have the same continuum functional. Natural Neumann and imposed Dirichlet boundary conditions are separated at the form level. In the convex regime the convergence strengthens to Mosco convergence of the complete functionals and therefore to convergence of resolvents and variational semigroups. We also give sharp failure statements: without a convergent directional tensor there need not be a unique full-sequence limit; the local continuum theorem does not solve the overlapping quantum marginal problem; and static Gamma-convergence alone does not derive a physical open-system dynamics. No random-substrate, Lorentzian, gravitational, gauge, particle-physics, cosmological, or empirical claim is made.