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QI: Fachverband Quanteninformation

QI 10: Quantum Information: Concepts and Methods I

QI 10.11: Talk

Wednesday, March 11, 2026, 12:30–12:45, BEY/0137

Characterizing covariance matrix and entanglement with finite Fourier transformed Observables — •Dimpi Thakuria¹, Konrad Szymański², Shuheng Liu¹, and Giuseppe Vitagliano¹ — ¹Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria — ²Research Center for Quantum Information, Slovenská Akadémia Vied, Dúbravská cesta 9, 84511 Bratislava, Slovakia

In Quantum physics, a covariance matrix provides us a means to certify a physically valid quantum system (through its positive semi-definiteness). It constrains the allowed quantum states and fully characterizes the Gaussian states. In continuous variable systems the covariance matrix is known to capture key properties like entanglement, squeezing, the purity of the states etc.. In this work we explore these concepts in the context of the discrete phase-space observables in finite-dimension, focusing on canonical position/momentum observables linked by finite Fourier transforms. Our approach is complementary to the typical way of using finite-dimensional Heisenberg-Weyl framework (especially discrete-displacement operators) for studying such systems. We characterize the allowed states via characterization of the invariants : the trace and the determinant of the covariance matrix. We also study the structure of the allowed covariance matrix transformations in the discrete phase-space, as well as the underlying Hilbert space. Our insights help us to discuss applications like entanglement detection in finite-dimensional systems, akin to covariance matrix criteria in continuous-variable systems.

Keywords: covariance matrix; finite-dimension; phase space; entanglement