Uncertainty relations for quantum measurements from generalized equiangular tight frames

Abstract

The current study aims to examine uncertainty relations for measurements from generalized equiangular tight frames. Informationally overcomplete measurements are a valuable tool in quantum information processing, including tomography and state estimation. The maximal sets of mutually unbiased bases are the most common case of such measurements. The existence of $d+1$ mutually unbiased bases is proved for $d$ being a prime power. More general classes of informationally overcomplete measurements have been proposed for various purposes. Measurements of interest are typically characterized by some inner structure maintaining the required properties. It leads to restrictions imposed on generated probabilities. To apply the considered measurements, these restrictions should be converted into information-theoretic terms. It is interesting that certain restrictions hold irrespectively to overcompleteness. To describe the amount of uncertainty quantitatively, we use the Tsallis and Rényi entropies as well as probabilities of separate outcomes. The obtained results are based on estimation of the index of coincidence. The derived relations are briefly exemplified.

15 pages, two figures. Version 3 incorporates final improvements (and updates to ref. [28])

References in corpus (33)

• Symmetric Informationally Complete Quantum Measurements
• The Uncertainty Principle in the Presence of Quantum Memory
• On mutually unbiased bases
• Entropic Uncertainty Relations and their Applications
• Heisenberg's Uncertainty Principle
• Entropic uncertainty relations - A survey
• The uncertainty principle determines the non-locality of quantum mechanics
• Tight informationally complete quantum measurements
• The SIC Question: History and State of Play
• Entropic uncertainty relations and entanglement
• Mutually unbiased measurements in finite dimensions
• Unitary $n$-designs via random quenches in atomic Hubbard and Spin models: Application to the measurement of Rényi entropies
• Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies
• Experimental characterization of qutrits using SIC-POVMs
• Steering criteria from general entropic uncertainty relations
• Continuous-variable entropic uncertainty relations
• Enabling computation of correlation bounds for finite-dimensional quantum systems via symmetrisation
• A min-entropy uncertainty relation for finite size cryptography
• Tight steering inequalities from generalized entropic uncertainty relations
• Notes on general SIC-POVMs
• Minimum tomography of two entangled qutrits using local measurements of one-qutrit SIC-POVM
• On uncertainty relations and entanglement detection with mutually unbiased measurements
• Quantum Conical Designs
• Experimental certification of an informationally complete quantum measurement in a device-independent protocol
• Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems
• Entropic uncertainty relations from quantum designs
• Entropic uncertainty relations from equiangular tight frames and their applications
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• Entanglement and Designs
• Rényi formulation of uncertainty relations for POVMs assigned to a quantum design
• Separability Criteria from Uncertainty Relations
• Estimating the Shannon entropy and (un)certainty relations for design-structured POVMs