Mutually unbiased frames

Fabián Caro Pérez, Victor Gonzalez Avella, and Dardo Goyeneche

Departamento de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile

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In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown $d$-dimensional fiducial states are parameterized, a priori, with roughly $3d/2$ real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of $d+1$ orthonormal bases are shown, in every dimension $d$. These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of $d+1$ mutually unbiased bases, when $d$ is a prime number.

In quantum mechanics, two Von Neumann observables are complementary if they have unbiased eigenvectors bases. In this work, we introduce the most general notion of unbiasedness for sets of linearly independent vectors that span the entire Hilbert space where they are defined. This generalization allows us to find a series of new results in a remarkably simple way, related to a special kind of constellations known as Symmetric Informationally Complete (SIC)-POVM.

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[1] L. Innocenti, S. Lorenzo, I. Palmisano, F. Albarelli, A. Ferraro, M. Paternostro, and G. M. Palma, "Shadow Tomography on General Measurement Frames", PRX Quantum 4 4, 040328 (2023).

[2] Xian Shi, "The entanglement criteria based on equiangular tight frames", Journal of Physics A: Mathematical and Theoretical 57 7, 075302 (2024).

[3] Alexey E. Rastegin, "Entropic uncertainty relations from equiangular tight frames and their applications", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 479 2274, 20220546 (2023).

[4] Ingemar Bengtsson and Basudha Srivastava, "Dimension towers of SICS: II. Some constructions", Journal of Physics A Mathematical General 55 21, 215302 (2022).

[5] Beata Derȩgowska, Matthew Fickus, Simon Foucart, and Barbara Lewandowska, "On the value of the fifth maximal projection constant", arXiv:2206.01596, (2022).

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