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.
Revealing the topology of quantum states via Kirkwood-Dirac quasiprobabilities
Istituto Nazionale di Ottica (INO), Consiglio Nazionale delle Ricerche（意大利国家光学研究所，意大利国家研究委员会）；European Laboratory for Non-linear Spectroscopy (LENS), Università di Firenze（欧洲非线性光谱学实验室，佛罗伦萨大学）；Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma（帕尔马大学数学、物理与信息科学系）；Gruppo Collegato di Parma, INFN-Sezione Milano-Bicocca（帕尔马联合小组，INFN米兰比可卡分部）；UdR Parma, INSTM（帕尔马研究组，意大利材料科学与技术研究中心）
Joint Center for Quantum Information and Computer Science, University of Maryland and NIST（马里兰大学与NIST联合量子信息与计算中心）；Department of Physics, University of Maryland（马里兰大学物理系）；QAT team, DIENS, École Normale Supérieure, PSL University, CNRS, INRIA（巴黎高等师范学院、PSL大学、CNRS、INRIA量子算法团队）；Department of Basic Science, The University of Tokyo（东京大学基础科学系）；Duke Quantum Center and Departments of Physics and Electrical and Computer Engineering, Duke University（杜克大学杜克量子中心及物理与电气计算机工程系）
Extending Covariant Fluctuation Theorems into Quantum Regime through Quasiprobability Approach
School of Physics, Peking University（北京大学物理学院）；Department of Physics and Astronomy, KU Leuven（荷语鲁汶大学物理与天文学系）；Instituut-Lorentz, Universiteit Leiden（莱顿大学洛伦兹研究所）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Frontiers Science Center for Nano-optoelectronics, Peking University（北京大学纳米光电前沿科学中心）
Path-Integral Formulation of Bosonic Markovian Open Quantum Dynamics with Monte Carlo stochastic trajectories using the Glauber-Sudarshan P, Wigner, and Husimi Q Functions and Hybrids
Nagoya University（名古屋大学）；Institute of Science Tokyo（东京科学大学）
Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography
Institute of Quantum Physics, School of Physics, Central South University（中南大学物理学院量子物理研究所）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Science, Mathematics and Technology Cluster, Singapore University of Technology and Design（新加坡科技设计大学科学、数学与技术集群）
Full Quantum Work Statistics for Non-Homogeneous Many-Body Systems
Università della Calabria（卡拉布里亚大学）；INFN, Gruppo Collegato di Cosenza（意大利国家核物理研究所，科森扎联合小组）；University of York（约克大学）；York Center for Quantum Technologies, The University of York（约克大学约克量子技术中心）
Semidefinite optimization as many-body thermodynamics: Boltzmann, Fermi-Dirac, and Bose-Einstein frameworks
School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）；Institute of Natural Sciences, School of Mathematical Sciences, Ministry of Education Key Laboratory in Scientific and Engineering Computing, and Global College, Shanghai Jiao Tong University（上海交通大学数学科学学院、自然研究院、教育部科学与工程计算重点实验室、全球学院）；School of Computer Science, Cornell University（康奈尔大学计算机科学学院）
Second law of thermodynamics in closed quantum many-body systems
RIKEN（理化学研究所）；Center for Quantum Computing, RIKEN（理科大学量子计算中心）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学所交叉理论与数学科学中心）；Institute for Photon Science and Technology, The University of Tokyo（东京大学光子科学技术研究所）；Center for Quantum Information and Quantum Biology, The University of Osaka（大阪大学量子信息与量子生物学中心）
