Measuring incompatibility and clustering quantum observables with a quantum switch

Abstract

The existence of incompatible observables is a cornerstone of quantum mechanics and a valuable resource in quantum technologies. Here we introduce a measure of incompatibility, called the mutual eigenspace disturbance (MED), which quantifies the amount of disturbance induced by the measurement of a sharp observable on the eigenspaces of another. The MED provides a metric on the space of von Neumann measurements, and can be efficiently estimated by letting the measurement processes act in an indefinite order, using a setup known as the quantum switch, which also allows one to quantify the noncommutativity of arbitrary quantum processes. Thanks to these features, the MED can be used in quantum machine learning tasks. We demonstrate this application by providing an unsupervised algorithm that clusters unknown von Neumann measurements. Our algorithm is robust to noise can be used to identify groups of observers that share approximately the same measurement context.

13 pages, 2 figures

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• Dimension-independent weak value estimation via controlled SWAP operations
• Activation of thermal states by coherently controlled thermalization processes
• Quantum information theoretic approach to the hard problem of consciousness
• Tsirelson bounds for quantum correlations with indefinite causal order
• An Elementary Characterization of Bargmann Invariants
• Correlations in a quantum switch-based heat engine with measurements: A proof-of-principle demonstration
• Estimation of multivariate traces of states given partial classical information