Cartesian and polar Schmidt bases for down-converted photons

TL;DR

This paper derives analytical Schmidt modes for SPDC biphotons in Cartesian and polar coordinates, enabling optimized measurement bases that enhance information extraction and account for experimental parameters affecting entanglement dimensionality.

Contribution

It provides explicit forms of Schmidt modes in both coordinate systems and links them to experimental parameters, facilitating optimized measurements using LG modes for high-dimensional entanglement.

Findings

Schmidt modes correspond to HG or LG modes at specific widths

Optimized measurement bases can maximize mutual information

Quantifies the impact of non-ideal measurements on shared bits

Abstract

We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite-Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties of orbital angular momentum eigenmodes. We derive a general entropic entanglement measure using the R\'enyi entropy as a function of the Schmidt number, K, and then retrieve the von Neumann entropy, S. Using the relation between S and K we show that, for highly entangled states, a non-ideal measurement basis does not degrade the number of shared bits by a large extent. More specifically, given a non-ideal measurement which corresponds to the loss of a fraction of the total number of modes, we can quantify the experimental parameters needed to generate an entangled SPDC state with a sufficiently high dimensionality to retain any given fraction of shared bits.