[Paper Review] Encoding spatial data into quantum observables

Artur Sowa|arXiv (Cornell University)|Sep 6, 2016

Quantum chaos and dynamical systems26 references3 citations

TL;DR

This paper introduces the Q-transform, a unitary isomorphism mapping doubly periodic generalized functions to quantum observables, enabling the reformulation of open quantum system dynamics as classical nonlocal evolution. It proves well-posedness of the initial value problem in Sobolev spaces and identifies a novel phenomenon—broadband redundancy—linked to the uniform distribution of Riemann zeta zeros, with implications for quantum information stability and classical-quantum hybrid systems.

ABSTRACT

The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic Analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a classical observable. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified---in particular, by the results presented here---to pose such a question.

Motivation & Objective

• To establish a rigorous mathematical correspondence between quantum observables and doubly periodic generalized functions via the Q-transform.
• To analyze the well-posedness of open quantum system dynamics governed by asymptotically harmonic Hamiltonians and compact-type Lindblad operators.
• To investigate the role of Sobolev-type norms in preserving regularity during quantum evolution.
• To generalize the Q-transform using non-orthonormal bases and uncover a new phenomenon: broadband redundancy.
• To assess the physical relevance of broadband redundancy in quantum dynamics and its potential implications for quantum information processing.

Proposed method

• Define the Q-transform as an invertible linear map between Hilbert space operators and doubly periodic generalized functions.
• Use the Q-transform to translate the Heisenberg equation with an essentially harmonic Hamiltonian into a first-order PDE with constant-velocity drift.
• Employ Sobolev spaces on the 2D torus, leveraging their connection to Fourier series for regularity analysis.
• Establish submultiplicativity of Q-transform-induced norms on compact operators, refining Hilbert-Schmidt norm properties.
• Generalize the Q-transform using non-orthonormal bases in $L_2[0,1]$ to study broadband redundancy.
• Apply results on the uniform distribution modulo one of the ordinates of nontrivial Riemann zeta zeros to derive convergence estimates.

Experimental results

Research questions

• RQ1 Can the dynamics of a quantum observable in an open system with an asymptotically harmonic Hamiltonian be reformulated as a classical nonlocal evolution?
• RQ2 Is the initial value problem for the resulting classical evolution well-posed in Sobolev spaces?
• RQ3 What is the nature and origin of the broadband redundancy phenomenon observed in generalized Q-transforms?
• RQ4 How does the distribution of Riemann zeta function zeros relate to the convergence properties of the generalized Q-transform?
• RQ5 Can the Q-transform framework enable stable encoding and processing of classical spatial data in quantum systems?

Key findings

• The initial value problem for the classical nonlocal evolution derived via the Q-transform is well-posed in the appropriate Sobolev spaces, as established in Theorem 1.
• The Q-transform preserves Sobolev regularity of quantum observables during evolution, indicating potential for compressibility of quantum information.
• Convergence of the generalized Q-transform to the standard Q-transform occurs in $\ell_2(\mathbb{Z}^2)$-norm at a rate dependent on $f$'s $2\alpha$-norm, with convergence rate $O(1/\log T)$ under $\sigma > \alpha + 1$.
• The broadband redundancy phenomenon arises from the uniform distribution modulo one of the ordinates of nontrivial Riemann zeta zeros, as formalized in Theorems 2 and 3.
• The convergence of the generalized Q-transform is bounded by $\left(\frac{C}{\log T}\zeta''(2(\sigma - \alpha) - 1)\right)^2\|f\|_{2\alpha}^2 \sum_{k \neq 0} |k|^{-2\alpha} \sum_{l \neq 0} |l|^{-2\alpha}$, which is finite for $\alpha > 1/2$.
• The Q-transform is distinct from the Wigner transform in both symmetry treatment and Fourier transform dimensionality, though a planar extension is possible.