Polarization Coherence and Entanglement

Abstract

We consider parametric downconversion (PDC)—a nonlinear optical process in which a pump photon gets annihilated to generate a pair of photons, termed as signal and idler, which are entangled in various degrees of freedom. In this paper, we consider the polarization degree of freedom and study how the polarization coherence properties of the pump photon get transferred to the signal-idler photons to manifest as two-qubit polarization entanglement. We show that for any generation process that is non-postselective and entropy-nondecreasing, the entanglement concurrence $C(\rho )$ of the generated two-qubit signal-idler state satisfies an intrinsic upper limit with $C(\rho ) \le (1 + P)/2$, where P is the degree of polarization of the pump photon. We further show that for the restricted class of two-qubit states having only two nonzero diagonal elements, the upper limit on the concurrence is the degree of polarization itself, that is, $C(\rho ) \le P$. This study shows that the manifestation of the pump correlations as entanglement in the signal-idler state is dependent on the dimensionality of the computational subspace on which the state has support. This formalism could pave the way to inferring bounds on high-dimensional entanglement, for which no universally accepted measure exists.