Determining the correlation functions of the polarization states of light is a common experimental task in quantum optics and quantum information. Often, it is simpler and more convenient to work with fiber optics and phase transformations rather than manipulating polarization states in free-space optical setups, due to the greater operational convenience and stability of fiber-optic systems. Therefore, establishing a correspondence between measurable quantities for polarization and phase states, including correlators, becomes essential.

This correspondence is also employed in quantum cryptography. For example, the Reference Frame Independent (RFI) protocol is implemented in both free-space systems, where it was originally proposed, and fiber-optic systems.

Since in experiments the correspondence between phase and polarization transformations is often applied intuitively, leading to questions about the validity of such substitutions, it is necessary to establish a rigorous correspondence between polarization and phase transformations.

In this study, we have demonstrated that the sum of the squares of the correlator operators in the + and × bases is independent of the misalignment angle β between the coordinate systems during the preparation and measurement of both polarization and phase states.

Finally, we have established a correspondence between polarization transformations and phase transformations. We have also shown that there exists an invariant—the correlators of the field states—that does not depend on the interferometer's balancing. This finding allows for the use of fiber-optic implementations in experiments, thereby achieving greater convenience and stability in optical setups. We have provided phase transformations for single-photon states in terms of operator transformations, which remain valid for coherent states as well.

Figure. The transformations of quantum phase states during the passage through the Mach-Zehnder-Interferometer.

Molotkov S.N. and Sushchev I.S.
JETP Letters 120, issue 7 (2024)