Quasiprobability

Jan Sperling, W. Vogel

Any bipartite quantum state has quasiprobability representations in terms of separable states. For entangled states these quasiprobabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one may reconstruct such quasiprobabilities from experimental data. Because of ambiguity, the quasiprobabilities obtained by the bare reconstruction are insufficient to identify entanglement. An optimization procedure is introduced to derive quasiprobabilities with a minimal amount of negativity. Negativities of optimized quasiprobabilities are necessary and sufficient for entanglement; their positivity proves separability.

Gianluca Francica

How to give a statistical description of the thermodynamics in quantum systems is an open fundamental question. Concerning the work, the presence of initial quantum coherence in the energy basis can give rise to a quasiprobability of work, which can take negative values. Our aim is to identify the most general quasiprobability of work satisfying some fundamental conditions. By doing so, we introduce a general notion of quasiprobability in analogy to Gleason's theorem. Then, we use these quasiprobabilities to define the quasiprobability of work, and finally we discuss the contextuality of the protocol.

Perry R. Rice

Jan Sperling, Laura Ares, Elizabeth Agudelo

We discuss the (re-)construction of quasiprobability representations from generic measurements, including noisy ones. Based on the measurement under study, quasiprobabilities and the associated concept of nonclassicality are introduced. A practical concern that we address is the treatment of informationally incomplete and overcomplete measurement scenarios, which can significantly alter the assessment of which states are deemed classical. Notions, such as Kirkwood-Dirac quasiprobabilities and s-parametrized quasiprobabilities in quantum optics, are generalized by our approach. Single-qubit systems are used to exemplify and to compare different measurement schemes, together with the resulting quasiprobabilities and set of nonclassical states.

Nicole Yunger Halpern, Brian G. Swingle, Justin Dressel

Two topics, evolving rapidly in separate fields, were combined recently: the out-of-time-ordered correlator (OTOC) signals quantum-information scrambling in many-body systems. The Kirkwood-Dirac (KD) quasiprobability represents operators in quantum optics. The OTOC was shown to equal a moment of a summed quasiprobability [Yunger Halpern, Phys. Rev. A 95, 012120 (2017)]. That quasiprobability, we argue, is an extension of the KD distribution. We explore the quasiprobability's structure from experimental, numerical, and theoretical perspectives. First, we simplify and analyze Yunger Halpern's weak-measurement and interference protocols for measuring the OTOC and its quasiprobability. We decrease, exponentially in system size, the number of trials required to infer the OTOC from weak measurements. We also construct a circuit for implementing the weak-measurement scheme. Next, we calculate the quasiprobability (after coarse graining) numerically and analytically: we simulate a transverse-field Ising model first. Then, we calculate the quasiprobability averaged over random circuits, which model chaotic dynamics. The quasiprobability, we find, distinguishes chaotic from integrable regimes. We observe nonclassical behaviors: the quasiprobability typically has negative components. It becomes nonreal in some regimes. The onset of scrambling breaks a symmetry that bifurcates the quasiprobability, as in classical-chaos pitchforks. Finally, we present mathematical properties. We define an extended KD quasiprobability that generalizes the KD distribution. The quasiprobability obeys a Bayes-type theorem, for example, that exponentially decreases the memory required to calculate weak values, in certain cases. A time-ordered correlator analogous to the OTOC, insensitive to quantum-information scrambling, depends on a quasiprobability closer to a classical probability. This work not only illuminates the OTOC's underpinnings, but also generalizes quasiprobability theory and motivates immediate-future weak-measurement challenges.