A University of the Philippines Diliman mathematician has developed a groundbreaking framework to describe complex quantum operators. Dr. Arvin Lamando, from the Institute of Mathematics, collaborated with Dr. Henry McNulty of the Norwegian University of Science and Technology. Their work shows how intricate quantum “machines” can be broken down into simpler parts and rebuilt. This offers fresh insights for quantum mechanics and advanced signal processing technologies.
Dr. Lamando specializes in harmonic analysis. This field explores whether any signal can be split into pure frequencies, like sines and cosines. The famous Fourier transform solves this problem. Think of a musical chord. The Fourier transform separates it into individual pure notes. Just as you replay the chord by pressing those notes together, abstract signals can be rebuilt from their pure frequencies. “While harmonic analysis started with real-world problems,” Dr. Lamando explained, “the Fourier transform ideas became very abstract. Surprisingly, they connect to many branches of pure mathematics.”
Classical harmonic analysis deals with signals. Quantum harmonic analysis applies similar ideas to operators. These operators follow rules used when moving from classical physics to quantum physics. Dr. Lamando and Dr. McNulty introduced a key concept: the ‘modulation’ of an operator in phase space. “This notion fits quantum harmonic analysis perfectly,” Dr. Lamando said. “The operator Fourier transform of a modulated operator gives a translated version of the original transform.” They focused on operators that stay the same, even when shifted or modulated over grid patterns in phase space.
“We have shown these invariant operators have properties similar to classical cases,” Dr. Lamando shared. The team used a structure called the Heisenberg module to understand them better. They discovered these complex quantum operators can be closely approximated by much simpler ones. These simpler operators, called finite-rank operators, have outputs describable in only a few dimensions. Their work bridges deep abstract algebra with concrete structures needed for quantum math. It provides new tools to model quantum systems and signals.
This research, titled “On Modulation and Translation Invariant Operators and the Heisenberg Module,” appears in the Journal of Fourier Analysis and Applications. This respected journal covers topics from abstract harmonic analysis to real-world tech applications. The study demonstrates how pure mathematical exploration, like Dr. Lamando’s work at UP Diliman, fuels progress in cutting-edge fields like quantum computing. It highlights the vital role of foundational mathematics in future technologies.
The paper was published in Volume 31, Issue 4 (2025). It is available online: Lamando, A., & McNulty, H. (2025). On modulation and translation invariant operators and the Heisenberg module. Journal of Fourier Analysis and Applications, 31(4): https://doi.org/10.1007/s00041-025-10176-5. For further information, contact the UP Diliman College of Science Institute of Mathematics at (+632) 8-920-0000 loc. 3001 or email: imath.dcs@upd.edu.ph.
UP Mathematician Unlocks Quantum Complexity with New Mathematical Framework
