Entanglement of qutrits and ququarts

Abstract

We investigate in a general form entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of collinear and, correspondingly, degenerate and non-degenerate Spontaneous Parametric Down-Conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and and polarization entanglement are fully taken into account. In the case of qutrits the most general 3-parametric families of maximally entangled and non-entangled states are found, and anti-correlation of the degree of entanglement and polarization is shown to occur and to be characterized by a rather simple formula. Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality $d=4$, which differs them significantly from the often used two-qubit model ($d=2$). New expressions for entanglement quantifiers of biphoton ququarts are derived and discussed. Rather simple procedures for a direct measurement of the degree of entanglement are described for both qutrits and ququarts.