  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 has been shown to equal a moment of a
summed quasiprobability. 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 the 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. 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.
