Quasiprobabilities in quantum thermodynamics and many-body systems

In this tutorial, we present the definition, interpretation and properties of some of the main quasiprobabilities that can describe the statistics of measurement outcomes evaluated at two or more times. Such statistics incorporate the incompatibility of the measurement observables and the state of the measured quantum system. We particularly focus on Kirkwood-Dirac quasiprobabilities and related distributions. We also discuss techniques to experimentally access a quasiprobability distribution, ranging from the weak two-point measurement scheme, to a Ramsey-like interferometric scheme and procedures assisted by an external detector. We illustrate the use of quasiprobabilities in quantum thermodynamics to describe the quantum statistics of work and heat, and to explain anomalies in the energy exchanges entailed by a given thermodynamic transformation. On the one hand, in work protocols, we show how absorbed energy can be converted to extractable work and vice versa due to Hamiltonian incompatibility at distinct times. On the other hand, in exchange processes between two quantum systems initially at different temperatures, we explain how quantum correlations in their initial state may induce cold-to-hot energy exchanges, which are unnatural between any pair of equilibrium nondriven systems. We conclude the tutorial by giving simple examples where quasiprobabilities are applied to many-body systems: scrambling of quantum information, sensitivity to local perturbations, and quantum work statistics in the quenched dynamics of models that can be mapped onto systems of free fermions, for instance the Ising model with a transverse field. We meticulously present derivations of essential concepts alongside straightforward examples, aiming to enhance comprehension and facilitate learning.