Entanglement transformation at absorbing and amplifying four-port devices

Abstract Quantum communication schemes widely use dielec-tric four-port devices as basic elements for construct-ing optical quantum channels. Since for causalityrea-sons the permittivity is necessarily a complex func-tion of frequency, dielectrics are typical examples ofnoisy quantum channels in which quantum coherencewill not be preserved. Basing on quantization of thephenomenological electrodynamics, we construct thetransformation relating the output quantum state tothe input quantum state without placing frequencyrestrictions. Knowledge of the full transformed quan-tum state enables us to compute the entanglementcontained in the output quantum state. We applythe formalism to some typical examples in quantumcommunication. 1 Introduction Quantum communication experiments widely use di-electricfour-portdevices suchasbeam splitters orop-tical fibers as basic elements for constructing opticalquantum channels. Since any frequency-dependentdielectric function describing an optical element, byvirtue of the Kramers-Kronig relations, is necessar-ily a complex function of frequency, absorption is al-ways present which leads to well-known phenomenaas decoherence and entanglement degradation. In or-der to study the problem, quantization of the elec-tromagnetic field in the presence of dielectric mediais needed. A consistent formalism of quantum elec-trodynamics in absorbing media is reviewed in [1].It is based on the Green function expansion of theelectromagnetic field with respect to the fundamen-tal variables of the system composed of the field, thedielectric matter and the reservoir. All relevant infor-mation about the dielectric and geometric propertiesare contained in the classical Green function of thecorresponding scattering problem.The formalism is especially suited for derivinginput-output relations of the field at dielectric slabs[2] on the basis of measurable quantities as trans-mission and absorption coefficients. From the input-output relations we can then derive closed formu-las for calculating the output quantum state fromthe (known) input quantum state [3]. That is, thecomplete density matrix after the transformation isknown, which makes the theory most suitable forstudying entanglement properties of quantum statesof light. The theory has also been extended to coveramplifying media.In this article we will proceed as follows. The quan-tum-state transformation at dielectric four-port de-vices is shortly reviewed in Sec. 2. An application toentanglement degradation of Bell states as well as thederivation of separability criteria for the two-modesqueezed vacuum state are given in Sec. 3 followedby a summary in Sec. 4.1