Kirkwood-Dirac distributions in classical optics

F.J. Duarte

Alfredo Luis, Lorena Ballesteros Ferraz

We develop a comprehensive analysis of the Kirkwood-Dirac distributions in classical optics, revealing their deep connection with optical coherence as fundamental concept in optics. From their very definition, the Kirkwood-Dirac distributions emerge as generalized mutual coherence functions involving two different bases instead of just one. This perspective provides a unified interpretation of the so-called anomalous values, that are complex and negative values, as direct manifestations of coherence. We show that this interpretation consistently applies across all field variables considered in this work, including polarization, interference and wave propagation. Furthermore, we propose diverse methods of experimental determination of these distributions based on interference, in full agreement with their coherence-based interpretation.

Jiayu He, Shuangshuang Fu

The Kirkwood-Dirac (KD) quasiprobability is a quantum analog expression of the phase-space probability in classical statistical mechanics, yet it can behave nonclassically by taking negative or nonreal values. The nonclassicality of the KD quasiprobability distribution has lately been shown to provide advantages in some quantum information tasks and shed light on a number of aspects of quantum physics. Characterization and quantification of KD nonclassicality are crucial both theoretically and experimentally. Most of the existing measures of KD nonclassicality concentrate merely on the negative or/and imaginary part(s) of the KD distribution. Yet it is not clear whether these measures can account for all nonclassicality in the distribution. In this paper, we suggest a KD nonclassicality measure based on the two quantum modification terms in the quasiprobability distribution. The present paper is mainly motivated by the following two considerations: First, the two quantum modification terms are actually the sources of the negative and imaginary parts of the KD quasiprobability respectively. Secondly, these quantum modification terms deviate the KD quasiprobability from the classical joint probability distribution function of two successive projective measurements (termed as the Wigner formula), which can be obtained from the multiplication principle in classical probability theory. Our measure refines the quantification of KD nonclassicality in the sense that even for real and nonnegative KD distribution, its quantum modification terms can still exist, resulting in a nonvanishing KD nonclassicality. As an illustration, we analyze the KD nonclassicality of the positive KD distribution in the postselected metrology scheme [S. Das, S. Modak, and M. N. Bera, Phys. Rev. A 107, 042413 (2023)], which provides interpretation on the source of quantum advantages. We also give an upper bound on the KD nonclassicality measure and offer an easily verifiable sufficient condition for nonvanishing KD nonclassicality.

Robert F. O’Connell

Thomas Konrad, Sandeep K. Goyal

Ziheng Ding, Si-Qi Zhou

The Kirkwood–Dirac (KD) quasiprobability distribution describes any quantum state with respect to the eigenbases of two incompatible observables. While the KD quasiprobability distribution behaves similarly to a classical probability distribution, it can take on negative or nonreal values. Recently, the framework of the temporal Kirkwood–Dirac quasiprobability distribution has been proposed, generalizing the KD quasiprobability distribution to arbitrary multi-time quantum processes. In this work, we specifically focus on the temporal KD quasiprobability distribution within the context of two-time dynamics. We begin by constructing a nonclassicality measure derived from the real and imaginary parts of the temporal KD quasiprobability distribution. Next, we establish two uncertainty relations closely linked to this nonclassicality measure, one of which shows that the nonclassicality measure is bounded below by the measurement disturbance caused by the first measurement. Finally, we elucidate the relationships among temporal KD nonclassicality, the spatiotemporal Born rule, and spatiotemporal compatibility.