Single biphoton ququarts as either pure or mixed states

When only polarization measurements are used (or equivalently, when the ququart's state is averaged over photon frequencies), the originally pure biphoton ququart state appears as a mixed two-qubit biphoton polarization state.

“If in experiments one uses only polarization measurements, the originally pure states of ququarts can be seen as mixed biphoton polarization states.”

The Schmidt parameter K and concurrence C are two correlation parameters found in general form from the density matrices (12), (14), (15).

“Two correlation parameters found in the general form from the density matrices (12), (14), (15) are the Schmidt parameter K and concurrence C [3]:”

In biphoton polarization-frequency ququarts, photons have two degrees of freedom: polarization and frequency.

“In such states photons have two degrees of freedom: polarization and frequency.”

The symmetry of the total wave function under transposition of photon variables is an obligatory feature of two-boson pure states.

“This is the obligatory feature of two-boson pure states, which often is not taken seriously but which manifests itself, e.g., in the existence of MPS discussed below and in Ref. [3].”

For MPS with C1=C4=0, the relative entropy is less than concurrence for any values of the remaining nonzero parameters |B-| and |B+|=1-|B-|.

“For MPS with C 1 = C 4 = 0 the relative entropy was found in [3] and shown to be less than concurrence at any values of the remaining nonzero parameters | B -| and | B + | = 1 - | B -| .”

The density matrix of MPS can be reduced over polarization variables of one of the two photons to give the mixed-state reduced density matrix of Eq (15).

“The density matrix of MPS can be further reduced over polarization variables of one of two photons to give rise to the mixed-state reduced density matrix of the form [3]”

Averaging ρ(4) over frequency variables turns it into the density matrix of a mixed two-qubit polarization state.

“But being averaged over frequency variables, ρ (4) turns into the density matrix of a mixed two-qubit polarization state [3].”

The only exceptions to S_rel < C occur at |B-|=0, 1, and 1/√2, where concurrence and relative entropy are equal.

“The only exceptions occur at | B -| = 0 , 1 and 1 / √ 2, where the concurrence and relative entropy are equal.”

Averaging of states of biphoton ququarts over frequencies ω1,2 gives rise to the MPS considered here and in Ref. [3].

“Averaging of states of bophoton ququarts over frequencies ω 1 , 2 gives rise to”