Implications and applications of the variance-based uncertainty equalities

Abstract

In quantum mechanics, the variance-based Heisenberg-type uncertainty relations are a series of mathematical inequalities posing the fundamental limits on the achievable accuracy of the state preparations. In contrast, we construct and formulate two quantum uncertainty equalities, which hold for all pairs of incompatible observables and indicate the new uncertainty relations recently introduced by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113, 260401 (2014)]. Furthermore, we present an explicit interpretation lying behind the derivations and relate these relations to the so-called intelligent states. As an illustration, we investigate the properties of these uncertainty inequalities in the qubit system and a state-independent bound is obtained for the sum of variances. Finally, we apply these inequalities to the spin squeezing scenario and its implication in interferometric sensitivity is also discussed.

arXiv comment9 pages, 4 figures. Any comments will be appreciated

References in corpus (34)

• Advances in Quantum Metrology
• Quantum-enhanced measurements: beating the standard quantum limit
• Quantum metrology
• Entanglement, Non-linear Dynamics, and the Heisenberg Limit
• Quantum spin squeezing
• Heisenberg's Uncertainty Principle
• Fisher Information and entanglement of non-Gaussian spin states
• Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement
• Violation of local uncertainty relations as a signature of entanglement
• Entropic uncertainty relations - A survey
• Stronger uncertainty relations for the sum of variances
• Proof of Heisenberg's error-disturbance relation
• Quantum root-mean-square error and measurement uncertainty relations
• How well can one jointly measure two incompatible observables on a given quantum state?
• Prior information: how to circumvent the standard joint-measurement uncertainty relation
• Noise and disturbance in quantum measurements: an information-theoretic approach
• Entanglement and sensitivity in precision measurements with states of a fluctuating number of particles
• Heisenberg uncertainty for qubit measurements
• Uncertainty Relations for Joint Measurements of Noncommuting Observables
• Translation of Lueders' "Uber die Zustandsanderung durch den Messprozess"
• Experimental test of universal complementarity relations
• Certainty in Heisenberg's uncertainty principle: Revisiting definitions for estimation errors and disturbance
• An optimal entropic uncertainty relation in a two-dimensional Hilbert space
• Collision entropy and optimal uncertainty
• Deriving tight error-trade-off relations for approximate joint measurements of incompatible quantum observables
• Detecting mixedness of qutrit systems using the uncertainty relation
• Quantum Estimation by Local Observables
• Optimized entropic uncertainty relation for successive measurement
• Robertson-Schroedinger type formulation of Ozawa's noise-disturbance uncertainty principle
• Improved Error-Tradeoff and Error-Disturbance Relations
• Optimal condition for measurement observable via error-propagation
• A disturbance tradeoff principle for quantum measurements
• Unitary equivalence between ordinary intelligent states and generalized intelligent states
• Uncertainty and Complementarity Relations in Weak Measurement

Cited by in corpus (22)

• Tighter Uncertainty and Reverse Uncertainty Relations
• Stronger Schrödinger-like Uncertainty Relations
• A Stronger Multi-observable Uncertainty Relation
• Weighted Uncertainty Relations
• Sum uncertainty relations based on Wigner-Yanase skew information
• Uncertainty relations for general unitary operators
• Variance uncertainty relations without covariances for three and four observables
• Tight $N$-observable uncertainty relations and their experimental demonstrations
• Experimental investigation of multi-observable uncertainty relations
• Wave-particle duality employing quantum coherence in superposition with non-orthogonal pointers
• Quantum state engineering via coherent-state superpositions in traveling optical fields
• Direct Experimental Test of Forward and Reverse Uncertainty Relations
• Experimental investigation of the uncertainty relations with coherent light
• Single step quantum state engineering in traveling optical fields
• Uncertainty from the Aharonov-Vaidman Identity
• Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation
• Uncertainty Relations in Pre- and Post-Selected Systems
• Quantum Uncertainty Equalities and Inequalities for Unitary Operators
• Uncertainty Relations of Variances in View of the Weak Value
• Variance-based uncertainty relation for incompatible observers
• Multi-observable Uncertainty Equality based on the sum of standard deviations in the qubit system
• Uncertainty relations in the product form