On entropic uncertainty relations for measurements of energy and its "complement"

Abstract

Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays the important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non-commuting observables. Since quantum theory was originating, many approaches to energy-time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, we obtain uncertainty relations of the "energy-time" type in terms of the Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state-dependent and state-independent formulations are of interest. Some of the derived state-independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.

10 pages, one figure. Minor corrections in v3. Except for the style, it matches the journal version

References in corpus (12)

• Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
• Heisenberg's Uncertainty Principle
• The uncertainty principle determines the non-locality of quantum mechanics
• Stronger uncertainty relations for the sum of variances
• Entropic uncertainty relation for mutually unbiased bases
• Entropic uncertainty relations for Markovian and non-Markovian processes under a structured bosonic reservoir
• Effects of Hawking radiation on the entropic uncertainty in a Schwarzschild space-time
• Rényi entropy uncertainty relation for successive projective measurements
• On uncertainty relations and entanglement detection with mutually unbiased measurements
• Generalized Landau-Pollak Uncertainty Relation
• Almost-periodic time observables for bound quantum systems
• The world according to Renyi: Thermodynamics of multifractal systems

Cited by in corpus (5)

• Quantum-memory-assisted entropic uncertainty relations
• Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields
• Entropic time-energy uncertainty relations: An algebraic approach
• Better Heisenberg limits, coherence bounds, and energy-time tradeoffs via quantum Rényi information
• Confidence uncertainty: position and momentum can be jointly determined with a guaranteed probability