Uncertainty relations for general unitary operators

Abstract

We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators, we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum-uncertainty states, we derive the minimum-uncertainty state equation by the analytic method and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher-dimensional limit of the uncertainty relations and minimum-uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general noncommuting unitary operators is nontrivially upper bounded.

12 pages, 2 figures. Material rearranged, typos corrected, uncertainty relation-1 made more general, applications added to match the published version

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• Heisenberg's Uncertainty Principle
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• Heisenberg Uncertainty Relation for Three Canonical Observables
• Tight state-independent uncertainty relations for qubits
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• Discrete coherent and squeezed states of many-qudit systems
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• Strong unitary and overlap uncertainty relations: theory and experiment
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• State-independent preparation uncertainty relations
• Improving sum uncertainty relations with the quantum Fisher information
• Strong unitary uncertainty relations
• Uncertainty of quantum channels via modified generalized variance and modified generalized Wigner-Yanase-Dyson skew information
• State independent uncertainty relations from eigenvalue minimization
• Geometry of Uncertainty Relations for Linear Combinations of Position and Momentum
• State-independent Uncertainty Relations and Entanglement Detection
• Complementarity-Entanglement Tradeoff in Quantum Gravity
• Experimental investigation of uncertainty relations for non-Hermitian operators
• On the discrete Wigner function for SU(N)
• One-particle and two-particle visibilities in bipartite entangled Gaussian states
• Probing Uncertainty Relations in Non-Commutative Space
• An improved bound for strong unitary uncertainty relations with refined sequence
• Improved unitary uncertainty relations
• Inferred-variance uncertainty relations in the presence of quantum entanglement
• Tighter uncertainty relations based on $(\alpha ,\beta ,\gamma )$ modified weighted Wigner-Yanase-Dyson skew information of quantum channels
• Unextendible product operator basis
• Entropic Bounds as Uncertainty Measure of Unitary Operators
• Error-disturbance uncertainty relations in Faraday measurements
• Quantum Uncertainty Equalities and Inequalities for Unitary Operators
• Stronger sum uncertainty relations for non-Hermitian operators
• Local sum uncertainty relations for angular momentum operators of bipartite permutation symmetric systems
• Multi-Operator Quantum Uncertainty Relations from New Cauchy-Schwarz Inequalities