Properties of biphoton qutrits and ququarts are characterized with allowance for the demands of the symmetry of wave functions of two-boson states and, in the case of ququarts, for the presence of two independent degrees of freedom (e.g., polarization and frequency) of the photons.
We investigate, in a general form, entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of, correspondingly, degenerate and non-degenerate spontaneous parametric down-conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and polarization entanglement are fully taken into account. In the case of qutrits, the most general three-parametric families of maximally entangled and non-entangled states are found, and anticorrelation 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 differentiates 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.
We investigate features of biphoton qutrits and ququarts with the symmetry of biphoton wavefunctions fully taken into account. Entanglement quantifiers of such states are found as functions of qutrit and ququart parameters. For biphoton qutrits, a general relation between degrees of entanglement and polarization is established. In the case of biphoton ququarts, photon frequencies are considered as variables independent of polarization variables, with both polarization and frequency degrees of freedom equally contributing to the total ququart entanglement. Averaging over one of these two degrees of freedom is shown to reduce pure ququart states to mixed two-qubit states. The degree of polarization and degrees of correlations in mixed polarization states are investigated.
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.
In this paper we discuss physical aspects of entanglement in biphoton-based qutrits and ququarts, taking into account the bosonic correlations due to the indistinguishability of photons. We show that two apparently different types of entanglement in these systems, 'entanglement of modes' and 'entanglement of photons', are in fact closely related to each other. The latter is to a remarkable extent connected with the degree of polarization of two-photon light. Specifically, we demonstrate that a beamsplitter and subsequent post-selection using a coincidence circuit converts 'entanglement of photons' of a qutrit into 'entanglement of modes' of a ququart.
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, degree of polarization P, as well as the von Neumann mutual information I. In some simple cases, we also find the relative entropy Srel and the degree of classical correlations Ccl. In mixed states, the Schmidt parameter does not characterize the degree of entanglement anymore, 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 and mixed states (averaged over frequencies). Differences between these results can be seen in experiments with and without frequency filters in front of a detector.