Rényi formulation of uncertainty relations for POVMs assigned to a quantum design

Abstract

Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and Rényi entropies for POVMs assigned to a quantum design. Relations of the Landau--Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.

arXiv comment14 pages, four figures. [v3] Final corrections, will be published in J. Phys. A: Mathematical and Theoretical

References in corpus (57)

• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Symmetric Informationally Complete Quantum Measurements
• The Uncertainty Principle in the Presence of Quantum Memory
• On mutually unbiased bases
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Entropic Uncertainty Relations and their Applications
• Quantum Steering
• Heisenberg's Uncertainty Principle
• Entropic uncertainty relations - A survey
• Evenly distributed unitaries: on the structure of unitary designs
• Stronger uncertainty relations for the sum of variances
• SIC-POVMs and the Extended Clifford Group
• SIC-POVMs: A new computer study
• Tight informationally complete quantum measurements
• The SIC Question: History and State of Play
• Improved entropic uncertainty relations and information exclusion relations
• Uncertainty relations from simple entropic properties
• Universal Uncertainty Relations
• Majorization entropic uncertainty relations
• Strong Majorization Entropic Uncertainty Relations
• Entropic Uncertainty Relations in Quantum Physics
• Optimizing quantum process tomography with unitary 2-designs
• Unitary designs and codes
• 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
• Entropic uncertainty and measurement reversibility
• Variance-based uncertainty relations
• Continuous-variable entropic uncertainty relations
• Characterizing multipartite entanglement with moments of random correlations
• General entropy-like uncertainty relations in finite dimensions
• A min-entropy uncertainty relation for finite size cryptography
• Tight steering inequalities from generalized entropic uncertainty relations
• Rényi entropy uncertainty relation for successive projective measurements
• On uncertainty relations and entanglement detection with mutually unbiased measurements
• SICs and Algebraic Number Theory
• Separability conditions from the Landau-Pollak uncertainty relation
• Uncertainties in Successive Measurements
• Optimized entropic uncertainty relation for successive measurement
• Entropic uncertainty relations from quantum designs
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Uncertainty, joint uncertainty, and the quantum uncertainty principle
• Entropic uncertainty relations for successive measurements of canonically conjugate observables
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• Majorization entropic uncertainty relations for quantum operations
• On a generalized entropic uncertainty relation in the case of the qubit
• Generalized Landau-Pollak Uncertainty Relation
• Majorization uncertainty relations for mixed quantum states
• Optimal Entropic Uncertainty Relation for Successive Measurements in Quantum Information Theory
• A transform of complementary aspects with applications to entropic uncertainty relations
• Simplified exact SICs
• Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures
• The world according to Renyi: Thermodynamics of multifractal systems
• Quantum preparation uncertainty and lack of information
• Secret Sharing Schemes Based on Min-Entropies
• Entropic uncertainty relations for successive measurements in the presence of a minimal length
• Notes on use of generalized entropies in counting
• Individual attacks with generalized discrimination and inadequacy of some information measures

Cited by in corpus (8)

• Entropic uncertainty relations from equiangular tight frames and their applications
• Estimating the Shannon entropy and (un)certainty relations for design-structured POVMs
• Entropic uncertainty relations for SIC-POVMs and MUMs
• Uncertainty relations for quantum measurements from generalized equiangular tight frames
• Entropic uncertainty relations for multiple measurements assigned with biased weights
• Uncertainty and certainty relations for quantum coherence with respect to design-structured POVMs
• General framework of quantum complementarity from a measurement-based perspective
• Complementarity relations for design-structured POVMs in terms of generalized entropies of order $\alpha \in (0,2)$