Spatio-temporal properties of multipartite entanglement

Abstract

Quantum multipartite entanglement is a striking phenomenon predicted by quantum mechanics when several parts of a physical system share the same quantum state that cannot be factorized into the states of individual subsystems. The Gaussian quantum states are usually characterized by the covariance matrix of the quadrature components. A powerful formalism for treating the Gaussian states is that of the symplectic eigenvalues. In particular, a quantitative measure of multipartite entanglement is the so-called logarithmic negativity, related to the symplectic eigenvalues of the partially transposed covariance matrix. Considering only global variances of the field quadratures one completely neglects the spatiotemporal properties of the electromagnetic field. We propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous variable Gaussian states by considering the local correlation matrix at two different spatiotemporal points [see manuscript for characters] and [see manuscript for characters] with [see manuscript for characters] being the transverse coordinate. For stationary and homogeneous systems one can also introduce the spatiotemporal Fourier components of the correlation matrix. The formalism of the global symplectic eigenvalues can be straightforwardly generalized to the frequency-dependent symplectic eigenvalues. This generalized theory allows, in particular, to introduce the characteristic spatial area and time of the multipartite entanglement, which in general depend on the number of parties in the system. As an example we consider multipartite entanglement in spontaneous parametric down-conversion with spatially-structured pump. We investigate spatial properties of such entanglement and calculate its characteristic spatial length.