Biphoton Double-Bell States and Ququarts with "Invisible" Variables

TL;DR

This paper investigates the properties of mixed biphoton polarization states derived from pure polarization-frequency ququart states, analyzing their entanglement, correlations, and polarization characteristics, and comparing different experimental configurations.

Contribution

It introduces a detailed analysis of mixed biphoton states, revealing how entanglement measures differ from pure states and how the Schmidt parameter relates to polarization in mixed states.

Findings

Schmidt parameter no longer characterizes entanglement in mixed states

The Schmidt parameter remains related to polarization degree in mixed states

Differences observed in experimental setups with frequency filters and beam-splitters

Abstract

We analyze features of mixed biphoton polarization states which arise from pure states of polarization-frequency biphoton ququarts after averaging over frequencies of photons. For mixed states we find their concurrence C, Schmidt parameter K and degree of polarization P, as well as the von Neumann mutual information I. In some simple cases we find also the relative entropy S_{rel} and the degree of classical correlations C_{cl}. We show that in mixed states the Schmidt parameter does not characterize anymore the degree of entanglement, as it does in pure states. Nevertheless, the Schmidt parameter remains useful even in the case of mixed states because it remains directly related to the degree of polarization. We compare results occurring in the cases of full pure polarization-frequency states of ququarts, mixed states (averaged over frequencies) and states with separated high- and low-frequency parts. Differences between these results can be seen in experiments with and without a dichroic beam-splitter, as well as with and without frequency filters in front of detector.