Entanglement transformation at absorbing and amplifying four-port devices

Abstract

Quantum communication schemes widely use dielectric four-port devices as basic elements for constructing optical quantum channels. A typical example of such a device is a beam splitter as a basic element not only for classical interference experiments but also for implementing quantum interferences. Another example is an optical fiber, which can be regarded as a dielectric fourport device that essentially realizes transmission of light over longer distances. Dielectric matter is commonly described in terms of the (spatially varying) permittivity as a complex function of frequency, whose real and imaginary parts are related to each other by the Kramers–Kronig relations. Since the appearance of the imaginary part (responsible for absorption and/or amplification) is unavoidably associated with additional noise, dielectric devices are typical examples of noisy quantum channels. Using them for generating or processing entangled quantum states of light, e.g., in quantum teleportation or quantum cryptography, the question of quantum decoherence arises. In order to study the problem, quantization of the electromagnetic field in the presence of dielectric media is needed. For absorbing bulk material, a consistent formalism is given in [1], using the Hopfield model of a dielectric [2]. A method of direct quantization of Maxwell’s equations with a phenomenologically introduced permittivity is given in [3]. It replaces the familiar mode decomposition of the electromagnetic field with a source-quantity representation, expressing the field in terms of the classical Green function and the fundamental variables of the composed system. The method has the benefit of being independent of microscopic models of the medium and can be extended to arbitrary inhomogeneous dielectrics [4,5]. All relevant information about the medium are contained in the permittivity (and the resulting Green function), and quantization is performed by the association of bosonic quantum excitations with the fundamental variables. Quantization of the phenomenological Maxwell field is especially well suited for deriving the input-output relations of the field [6–9] on the basis of the really observed transmission and absorption coefficients. In particular, there is no need to introduce artificial replacement schemes. Applications to low-order correlations in two-photon interference effects have been given [8,10,11]. The formalism has also been extended to amplifying media [5,12]. The resulting input–output relations for amplifying beam splitters have been used to compute firstand second-order moments of photo counts [13] and normally ordered Poynting vectors [14]. Further, propagation of squeezed radiation through amplifying or absorbing multi-port devices has been considered [15]. For the study of entanglement, however, knowledge of some moments and correlations is not enough. In particular, to answer the question as to whether or not a bipartite quantum state is separable and to calculate the degree of entanglement of a non-separable state, the complete information on the state is required in general. Recently we have presented closed formulas for calculating the output quantum state from the input quantum state [16], using the input–output relations for the field at an absorbing four-port device of given complex refractive-index profile. In this paper we apply these results to study the entanglement properties influenced by propagation in real dielectrics and extend the theory also to amplifying four-port devices. Enlarging the system by introducing appropriately chosen auxiliary degrees of freedom, we first construct the unitary transformation in the enlarged Hilbert space. Taking the trace with regard to the auxiliary variables, we then obtain the sought formulas for the transformation of arbitrary input quantum states. Finally, we discuss some applications, with special emphasis on the dependence of entanglement on absorption and amplification. The paper is organized as follows. In Sec. II the basic equations are reviewed and the general transformation formulas are derived. Examples of possible applications

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