SPDC: once again on the parameters of transverse entanglement outside the near zone

TL;DR

This paper revisits the parameters used to quantify transverse entanglement of biphoton states, proposing a modification to the R parameter that makes it consistent across different representations and propagation zones.

Contribution

The authors introduce a modified definition of the R parameter that aligns with the K parameter in all representations and zones, improving entanglement measurement consistency.

Findings

Modified R parameter matches K in all zones and representations

Original R parameter deviates from K beyond the near zone

Proposed measurement methods for the new R parameter

Abstract

As known, the degree of entanglement of biphoton states with respect to the transverse components of photon wave vectors (momenta) or coordinates can be characterized either by the parameter K associated with the Schmidt decompositions, or by the parameter R, defined as the ratio of the widths of the unconditional and conditional single-particle distributions of biphoton states. As entanglement is a fundamental characteristics of a state, it must be independent of a choice of its different representations, i.e. its entanglement parameters must be identical in the coordinate and momentum representations. Likewise, entanglement itself and its characterization parameters must remain constant along the photon propagation length. It's known, however, that only the parameter K obeys these requirements, whereas the parameter R in the coordinate representation deviates from K beyond the near zone and, thus, becomes inapplicable as a measure of the degree of entablement. But, as shown below, there is a way of saving the parameter R by means of modifying its definition in such a way that the modified parameter R turns out to be as good as the parameter K both in the momentum and coordinate representations, as well as in the near and far zones and everywhere between them. We will discuss also possible ways of measuring this newly defined parameter R experimentally.