Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures

Abstract

It is known that the variance and entropy of quantum observables decompose into intrinsically quantum and classical contributions. Here a general method of constructing quantum-classical decompositions of resources such as uncertainty is discussed, with the quantum contribution specified by a measure of the noncommutativity of a given set of operators relative to the quantum state, and the classical contribution generated by the mixedness of the state. Suitable measures of noncommutativity or 'quantumness' include quantum Fisher information, and the asymmetry of a given set, group or algebra of operators, and are generalised to nonprojective observables and quantum channels. Strong entropic uncertainty relations and lower bounds for Rényi entropies are obtained, valid for arbitrary discrete observables, that take the mixedness of the state into account via a classical contribution to the lower bound. These relations can also be interpreted without reference to quantum-classical decompositions, as tradeoff relations that bound the asymmetry of one observable in terms of the entropy of another.

v1: 12+3 pages, 0 figures, 1 bazinga! v2: comparison with classical lower bound for Renyi entropy added (including 1 Figure), plus minor corrections/clarifications

References in corpus (7)

• Observable measure of quantum coherence in finite dimensional systems
• Extending Noether's theorem by quantifying the asymmetry of quantum states
• Formulation of the uncertainty relations in terms of the Renyi entropies
• Measuring the quality of a quantum reference frame: the relative entropy of frameness
• Does nonlinear metrology offer improved resolution? Answers from quantum information theory
• Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices
• Better Heisenberg limits, coherence bounds, and energy-time tradeoffs via quantum Rényi information

Cited by in corpus (7)

• Operational interpretation and estimation of quantum trace-norm asymmetry based on weak value measurement and some bounds
• General quantum correlation from nonreal values of Kirkwood-Dirac quasiprobability over orthonormal product bases
• Unveiling quantum steering by quantum-classical uncertainty complementarity
• Separation of measurement uncertainty into quantum and classical parts based on Kirkwood-Dirac quasiprobability and generalized entropy
• The polygon relation and subadditivity of entropic measures for discrete and continuous multipartite entanglement
• Quantum speed limit for observables from quantum asymmetry
• Spin minimum uncertainty states for refined uncertainty relations