Optimal sensing of photon addition and subtraction on quantized light

Photon addition to, and subtraction from quantum states of light is of immense current interest, both experimentally and theoretically. We identify a set of markers of photon addition to coherent light and nonclassical single mode states of light such as the squeezed vacuum and the cat states, which are directly computable from relevant optical tomograms. These markers quantify the ‘distance’ between probability density functions corresponding to different states. Examples of such markers are the Wasserstein distance, the Kullback Leibler divergence and the Bhattacharyya distance.
Although the details in the manner in which these markers vary with changes in relevant parameters, depends on the state concerned, they are, in general, sensitive to the underlying interference structures that arise in the tomogram after photon addition/subtraction. Our procedure is universally applicable to the cat and squeezed states, the former displaying the characteristic negativity in its Wigner function, while the latter does not do so. We explicate this in the case of the squeezed vacuum and even coherent states and show that photon addition (or subtraction) is mirrored in the shift in the intensity of specific regions in the tomogram.
In the case of coherent light, the amplification gain due to photon addition, and the dependence of quadrature variances on relevant parameters, are calculated from the tomograms (circumventing state reconstruction), and compared with results obtained (after state reconstruction), in a recent experiment reported in the literature. Our results match well with the fidelity and variance plots obtained by the experimenters.