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arXiv:2606.23185 · Journal of Physics A Mathematical and Theoretical

Entropic Uncertainty Relations for Mutually Unbiased Operator Frames

Jesni Shamsul Shaari, Stefano Mancini

quant-phmath-phSubmitted Jun 22, 2026• Announced Jun 23, 2026• 0 citations
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Abstract

We develop an operator-frame formulation of entropic uncertainty relations in the Hilbert-Schmidt space of operators. For general continuous indexed operator frames, we derive an entropic uncertainty relation for the associated coefficient distributions by combining endpoint norm estimates with Riesz-Thorin interpolation. We then identify a distinguished class of mutually unbiased operator frames, defined through constant-modulus trace overlaps. Under suitable structural conditions, the corresponding coefficient amplitudes are related by a bilinear Fourier transform, leading to a stronger Hirschman-Beckner-type entropic uncertainty relation. As canonical realizations, we consider Weyl displacement operators and Wigner kernels, as well as Cartesian dyadic frames generated by position and momentum eigenstates. These examples recover familiar continuous-variable Fourier dualities while extending entropic uncertainty relations beyond measurement outcomes to operator representations themselves.

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We develop an operator-frame formulation of entropic uncertainty relations in the Hilbert-Schmidt space of operators. For general continuous indexed operator frames, we derive an entropic uncertainty relation for the associated coefficient distributions by combining endpoint norm estimates with Riesz-Thorin interpolation. We then identify a distinguished class of mutually unbiased operator frames, defined through constant-modulus trace overlaps. Under suitable structural conditions, the corresponding coefficient amplitudes are related by a bilinear Fourier transform, leading to a stronger Hirschman-Beckner-type entropic uncertainty relation. As canonical realizations, we consider Weyl displacement operators and Wigner kernels, as well as Cartesian dyadic frames generated by position and momentum eigenstates. These examples recover familiar continuous-variable Fourier dualities while extending entropic uncertainty relations beyond measurement outcomes to operator representations themselves.
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Building on recent significant interest in entropic uncertainty relations for quantum measurements, this work explores the entropic uncertainty relation for the mutually unbiased equiangular tight frame assigned to quantum ttt-designs. Combining the mathematical structures of quantum measurements with those of quantum designs, uncertainty relations are derived in terms of Tsallis and R\'enyi $\ensuremath{\alpha}$ entropies, and rigorous theoretical analysis demonstrates the tightness of the derived entropic lower bounds. Furthermore, a specific relation between quantum design and the index of coincidence is inferred from the information diagrams, which is crucial for deriving the novel entropic uncertainty relation. The results reveal that the entropic lower bounds can be optimized by adjusting the parameters of the quantum ttt-design, broadening the theoretical framework of the entropic uncertainty relation and providing a practical tool for quantum information tasks.
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In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Rényi entropies with conjugated indexes. We consider here the more general case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes α and β in the plane (α, β). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).
Abstract Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p . This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x - p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p , but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.
A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For a given input state, different effects of each unraveling result in some probability distribution at the output. As is shown, all Tsallis' entropies of positive order as well as some of Rényi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Rényi and Tsallis entropies. The inequality with Rényi's entropies is an improvement of the previous one, whereas the inequality with Tsallis' entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a d -level system and for the angle and angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.