Uncertainties in Successive Measurements

Abstract

When you measure an observable, A, in Quantum Mechanics, the state of the system changes. This, in turn, affects the quantum-mechanical uncertainty in some non-commuting observable, B. The standard Uncertainty Relation puts a lower bound on the uncertainty of B in the initial state. What is relevant for a subsequent measurement of B, however, is the uncertainty of B in the post-measurement state. We re-examine this problem, both in the case where A has a pure point spectrum and in the case where A has a continuous spectrum. In the latter case, the need to include a finite detector resolution, as part of what it means to measure such an observable, has dramatic implications for the result of successive measurements. Ozawa proposed an inequality satisfied in the case of successive measurements. Among our results, we show that his inequality is ineffective (can never come close to being saturated). For the cases of interest, we compute a sharper lower bound.

Improvements in the prose (thanks to the referee). Version to appear in Phys. Rev. A. 23 pages, utarticle.cls

References in corpus (3)

• Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements
• Measuring measurement--disturbance relationships with weak values
• Correlations between detectors allow violation of the Heisenberg noise-disturbance principle for position and momentum measurements

Cited by in corpus (21)

• Multi-observable Uncertainty Relations in Product Form of Variances
• Sequential measurements of non-commuting observables with quantum controlled interactions
• Phase-space noncommutative extension of the Robertson-Schroedinger formulation of Ozawa's uncertainty principle
• Entropic uncertainty relations for successive measurements of canonically conjugate observables
• A note on uncertainty relations of arbitrary N quantum channels
• On entropic uncertainty relations in the presence of a minimal length
• Rényi formulation of uncertainty relations for POVMs assigned to a quantum design
• Tighter sum uncertainty relations via variance and Wigner-Yanase skew information for N incompatible observables
• A note on uncertainty relations of metric-adjusted skew information
• Entropic uncertainty relations for successive generalized measurements
• Sum uncertainty relations based on $(\alpha ,\beta ,\gamma )$ weighted Wigner-Yanase-Dyson skew information
• Uncertainty and certainty relations for successive projective measurements of a qubit in terms of Tsallis' entropies
• Conditionally valid uncertainty relations
• Entropic uncertainty relations for successive measurements in the presence of a minimal length
• Wigner-Yanase skew information-based uncertainty relations for quantum channels
• Parameterized Multi-observable Sum Uncertainty Relations
• Heisenberg Uncertainty Relation Revisited
• Stronger sum uncertainty relations for non-Hermitian operators
• von Neumann's Formula, Measurements and the Lindblad Equation
• Sum uncertainty relations for arbitrary $N$ incompatible observables
• The High-Energy Interpretation of Quantum Mechanics