Entropic uncertainty relations for extremal unravelings of super-operators

Abstract

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 given input state, different effects of each unraveling result in some probability distribution at the output. As it is shown, all Tsallis' entropies of positive order as well as some of Renyi'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 Renyi and Tsallis entropies. The inequality with Renyi'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 the angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.

8 pages, one figure. More explanations are given for Eq. (2.19) and Example III.5. One reference is added

References in corpus (4)

• Heisenberg's Uncertainty Principle
• Formulation of the uncertainty relations in terms of the Renyi entropies
• Almost-periodic time observables for bound quantum systems
• Continuity and Stability of Partial Entropic Sums

Cited by in corpus (25)

• Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies
• Some general properties of unified entropies
• General entropy-like uncertainty relations in finite dimensions
• Real-Time Source Independent Quantum Random Number Generator with Squeezed States
• Notes on general SIC-POVMs
• Contradictory uncertainty relations
• Entropic uncertainty relations for successive measurements of canonically conjugate observables
• Majorization entropic uncertainty relations for quantum operations
• Entropic formulation of the uncertainty principle for the number and annihilation operators
• Unsharpness of generalized measurement and its effects in entropic uncertainty relations
• Entropic trade-off relations for quantum operations
• Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis' entropies
• Alternative measures of uncertainty in quantum metrology: Contradictions and limits
• Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy
• On entropic uncertainty relations in the presence of a minimal length
• Upper continuity bounds on the relative $q$-entropy for $q>1$
• Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures
• On unified-entropy characterization of quantum channels
• Unified-entropy trade-off relations for a single quantum channel
• Rényi formulation of entanglement criteria for continuous variables
• On entropic uncertainty relations for measurements of energy and its "complement"
• On Kirkwood--Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
• Notes on entropic characteristics of quantum channels
• Rényi and Tsallis formulations of separability conditions in finite dimensions