Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables

Matteo Lostaglio1, Alessio Belenchia2,3, Amikam Levy4, Santiago Hernández-Gómez5,6,7, Nicole Fabbri5,7, and Stefano Gherardini8,5

1Korteweg-de Vries Institute for Mathematics and QuSoft, University of Amsterdam, The Netherlands
2Institut für Theoretische Physik, Eberhard-Karls-Universität Tübingen, 72076 Tübingen, Germany
3Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
4Department of Chemistry and Center for Quantum Entanglement Science and Technology, Bar-Ilan University, Ramat-Gan 52900, Israel
5European Laboratory for Non-linear Spectroscopy (LENS), Università di Firenze, I-50019 Sesto Fiorentino, Italy
6Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino, Italy
7Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), I-50019 Sesto Fiorentino, Italy
8Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), Area Science Park, Basovizza, I-34149 Trieste, Italy

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Recent work has revealed the central role played by the Kirkwood-Dirac quasiprobability (KDQ) as a tool to properly account for non-classical features in the context of condensed matter physics (scrambling, dynamical phase transitions) metrology (standard and post-selected), thermodynamics (power output and fluctuation theorems), foundations (contextuality, anomalous weak values) and more. Given the growing relevance of the KDQ across the quantum sciences, our aim is two-fold: First, we highlight the role played by quasiprobabilities in characterizing the statistics of quantum observables and processes in the presence of measurement incompatibility. In this way, we show how the KDQ naturally underpins and unifies quantum correlators, quantum currents, Loschmidt echoes, and weak values. Second, we provide novel theoretical and experimental perspectives by discussing a wide variety of schemes to access the KDQ and its non-classicality features.

The study explores the manifold ways in which a quasiprobability introduced by Kirkwood and Dirac enters disparate physical concepts and fields. The main focus of the work is twofold: First, to show how this quasiprobability can be used to characterize the statistics of incompatible quantum observables during quantum dynamical processes and, second, to provide schemes to access the non-classicality encoded in this quasiprobability.

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