Quantum Fourier analysis

Abstract

Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined Lp spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

The uncertainty principle: Variations on a theme

“…Thus, Fourier transform matrices are n-Hadamard n × n matrices, where n = |G|. More generally, quantum analogues of the Fourier transform can also be seen as k-Hadamard matrices; for more information, see [22]. Other explicit square matrices: We do not insist that k = n in our n × n Hadamard matrices.…”

On the spectral nature of entanglement

“…In the last two decades, QFT has participated in the most diverse studies, such as the analysis of spectral effects in quantum teleportation [4], its experimental implementation using the local classical entanglement state between the polarization and orbital angular momentum [5], the relation between spectral theory and quantum mechanics [6], the teleportation of multiple qubits based on QFT [7,8] where this tool is used to find orthogonal bases on which the sender projects the states to be teleported, the explanation about the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the QFT [9], its implementation in an entangled system of multilevel atoms [10], the realization of the two-qubit QFT and entangled protected using a Hamiltonian, which constitutes a matrix with circulant symmetry [11], its use in multiparty quantum telecommunication [12], its use as a tool to investigate phenomena such as quantum symmetry [13], and in an interesting study that showed herein that under the constraints imposed by quantum decoherence [14], only a parallel approach of QFT can guarantee a reliable solution or, alternatively, improve scalability [15] of this important tool.…”

Predicting plant growth response under fluctuating temperature by carbon balance modelling

“…These attributes provide valuable information about system properties like stability or predictability which need to be essentially addressed for predictive modeling 32 , 33 . Conclusively, Fourier polynomial-based balance modeling provides a mathematical approach that can essentially support nonlinear modeling of metabolism, and which might, in future studies, even serve as a mathematical framework to connect oscillations in metabolism with quantum theory 34 , 35 .…”