Single biphoton ququarts as either pure or mixed states

Abstract

We analyze features of mixed biphoton polarization states, which arise from pure states of polarizationfrequency 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 S rel and the degree of classical correlations C cl . 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.

Biphoton ququarts as either pure or mixed states, features and reconstruction from coincidence measurements

“…Averaging of states of bophoton ququarts over frequencies ω 1,2 gives rise to MPS considered here and in Ref. [3]. There is no way to get such states in a two-qubit model.…”

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

“…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]. Besides, owing to the symmetry, both terms in the ququart's wave function (3) obey the entanglement criterion (2) and, hence, all biphoton ququarts are entangled.…”

“…This density matrix characterizes pure states. But being averaged over frequency variables, ρ (4) turns into the density matrix of a mixed two-qubit polarization state [3]. Written down in the basis Ψ HH , Ψ pol…”

“…(24) vanishes the term, proportional to Ψ − , and reduces the ququart's wave function Ψ (4) 2qb to that of a qutrit Ψ (3) (4). To get a correct quqaurt's wave function (3), in addition to symmetrization, one has to give freedom to photon frequencies ω 1,2 by considering them as variables which can take one of two values each: either ω 1 = ω h and ω 2 = ω l or ω 1 = ω l and ω 2 = ω h . Actually, this means that we never know which photon has which frequency.…”