Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements

Abstract

We address the problem of characterising the compatible tuples of measurements that admit a unique joint measurement. We derive a uniqueness criterion based on the method of perturbations and apply it to show that extremal points of the set of compatible tuples admit a unique joint measurement, while all tuples that admit a unique joint measurement lie in the boundary of such a set. We also provide counter-examples showing that none of these properties are both necessary and sufficient, thus completely describing the relation between joint measurement uniqueness and the structure of the compatible set. As a by-product of our investigations, we completely characterise the extremal and boundary points of the set of general tuples of measurements and of the subset of compatible tuples.

9 pages, 4 figures. v2, v3: minor corrections. v4: new proof of Thm. 3 and Cor. 4. The authors are thankful to Alessandro Toigo and Claudio Carmeli for pointing a flaw in the previous proof that restricted the theorem to pairs of POVMs

References in corpus (4)

• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Joint measurability of generalized measurements implies classicality
• Notes on Joint Measurability of Quantum Observables
• Approximating incompatible von Neumann measurements simultaneously

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• Incompatibility robustness of quantum measurements: a unified framework
• Distributed quantum incompatibility
• The quantum switch is uniquely defined by its action on unitary operations
• The complexity of compatible measurements
• Constructing extremal compatible quantum observables by means of two mutually unbiased bases
• Quantum incompatibility from the viewpoint of entanglement theory
• Post-processing minimal joint observables
• Joint measurability meets Birkhoff-von Neumann's theorem