Measurement uncertainty relations for position and momentum: Relative entropy formulation

Abstract

Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true' probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) $n$-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension $n$; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.

arXiv comment33 pages

References in corpus (27)

• Quantum information with continuous variables
• Gaussian Quantum Information
• Entropic Uncertainty Relations and their Applications
• Heisenberg's Uncertainty Principle
• Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement
• Entropic uncertainty relations - A survey
• Proof of Heisenberg's error-disturbance relation
• An Invitation to Quantum Incompatibility
• Quantum root-mean-square error and measurement uncertainty relations
• Noise and disturbance in quantum measurements: an information-theoretic approach
• Physical content of Heisenberg's uncertainty relation: Limitation and reformulation
• Measurement uncertainty relations
• Comparing the degrees of incompatibility inherent in probabilistic physical theories
• Approximate reversibility in the context of entropy gain, information gain, and complete positivity
• Maximally incompatible quantum observables
• Entropic information-disturbance tradeoff
• Position measuring interactions and the Heisenberg uncertainty principle
• State-Dependent Approach to Entropic Measurement-Disturbance Relations
• Positive operator valued measures covariant with respect to an irreducible representation
• Entropic uncertainty relations in multidimensional position and momentum spaces
• Position and momentum observables on R and on R^3
• Instruments and channels in quantum information theory
• Sequential measurements of conjugate observables
• Breaking Gaussian incompatibility on continuous variable quantum systems
• Measurement uncertainty relations for position and momentum: Relative entropy formulation
• Informationally complete sets of Gaussian measurements
• Classical Coding and the Cauchy-Schwarz Inequality

Cited by in corpus (8)

• Steering criteria from general entropic uncertainty relations
• Measurement uncertainty relations for discrete observables: Relative entropy formulation
• Relative entropic uncertainty relation
• Measurement uncertainty relations for position and momentum: Relative entropy formulation
• Geometry of Uncertainty Relations for Linear Combinations of Position and Momentum
• Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state
• Number and phase: complementarity and joint measurement uncertainties
• Entropic measurement uncertainty relations for all the infinite components of a spin vector