Complementary observables in quantum mechanics
arXiv:1905.06254 · doi:10.1007/s10701-019-00261-3
Abstract
We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity - some of which new - in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul's work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.
References in corpus (22)
- Heisenberg's Uncertainty Principle
- Proof of Heisenberg's error-disturbance relation
- Quantum root-mean-square error and measurement uncertainty relations
- Heisenberg uncertainty for qubit measurements
- Measurement uncertainty relations
- Noise Robustness of the Incompatibility of Quantum Measurements
- Comparing the degrees of incompatibility inherent in probabilistic physical theories
- Coexistence of qubit effects
- Effects as functions on projective Hilbert space
- Maximally incompatible quantum observables
- Radon-Nikodym derivatives of quantum operations
- Uncertainty from Heisenberg to Today
- On the coexistence of position and momentum observables
- The complementarity of quantum observables: theory and experiment
- Direct tests of measurement uncertainty relations: what it takes
- Measurement uncertainty relations: characterising optimal error bounds for qubits
- Uncertainty in the context of multislit interferometry
- A proof for the informational completeness of the rotated quadrature observables
- On Quantum Uncertainty Relations and Uncertainty Regions
- Multi-slit interferometry and commuting functions of position and momentum
- Number and phase: complementarity and joint measurement uncertainties
- On the complementarity of the quadrature observables