Bounding Kirkwood-Dirac negativity of Gaussian processes

This paper investigates the Kirkwood-Dirac (KD) quasiprobability distribution within the framework of continuous-variable (CV) Gaussian processes. The researchers aim to characterize the range of KD negativity—a key measure of nonclassicality—attainable by Gaussian states and to determine whether non-Gaussian states are required to reach the maximum possible negativity in these systems.

The authors define the KD distribution for an arbitrary $M$-mode quantum state undergoing $N$ sequential Gaussian measurements. By utilizing the phase-space representation of Gaussian operators and unitaries, they derive a general upper bound on the KD negativity that depends solely on the covariance matrices of the measurements and the symplectic unitaries defining the dynamics. To identify the states that saturate this bound, they map the single-mode, two-measurement problem onto the kinematics of a particle in a $(2+1)$-dimensional Minkowski spacetime, allowing for a geometric optimization of the negativity.

The study establishes that the KD negativity for any Gaussian process is upper-bounded by a function of the covariance matrices of the sequential measurements. For the specific case of a single-mode, two-measurement process, the authors prove that the maximum negativity is achieved by quadrature eigenstates. Furthermore, by comparing the negativity of non-Gaussian states (such as Fock states and cat states) against this bound, the researchers demonstrate that these states do not exceed the maximum negativity attainable by Gaussian states. This indicates that Gaussian states are sufficient to achieve extreme values of nonclassicality in this framework.