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Updated: Oct 12, 2025

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Published on: September 5, 2019
Complete Incompatibility, Support Uncertainty, and Kirkwood-Dirac Nonclassicality
1Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
We introduce complete incompatibility for quantum observables, linking it to state uncertainty. This finding helps classify quantum states as classical or nonclassical using the Kirkwood-Dirac distribution.
Area of Science:
- Quantum mechanics
- Quantum information theory
Background:
- The Kirkwood-Dirac (KD) quasiprobability distribution is a tool for assessing nonclassical features of quantum states.
- These nonclassical features can serve as a resource in quantum information protocols.
Purpose of the Study:
- To introduce and define the concept of complete incompatibility for two observables in quantum systems.
- To establish the equivalence between complete incompatibility and large support uncertainty for all states.
- To apply this equivalence to classify quantum states as KD classical or nonclassical.
Main Methods:
- Defining complete incompatibility for pairs of observables in finite-dimensional Hilbert spaces.
- Proving the equivalence between complete incompatibility and large support uncertainty.
- Analyzing the Kirkwood-Dirac distribution for states under completely incompatible observables.
Main Results:
- Complete incompatibility of two observables is equivalent to the large support uncertainty of all quantum states.
- When observables are completely incompatible, only states with minimal support uncertainty are KD classical.
- Other states with completely incompatible observables are KD nonclassical.
Conclusions:
- The concept of complete incompatibility provides a new perspective on quantum state characterization.
- Support uncertainty is a key factor in determining the KD classicality of quantum states.
- This work offers insights into the nonclassical properties of quantum states and their resources.
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