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arXiv:quant-ph/0004003 · Physical Review A

Entanglement transformation at absorbing and amplifying four-port devices

Stefan Scheel, L. Knöll, Tomáš Opatrný, D.‐G. Welsch

Published Sep 11, 2000• 52 citations
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Abstract

Dielectric four-port devices play an important role in optical quantum information processing. Since for causality reasons the permittivity is a complex function of frequency, dielectrics are typical examples of noisy quantum channels, which cannot preserve quantum coherence. To study the effects of quantum decoherence, we start from the quantized electromagnetic field in an arbitrary Kramers-Kronig dielectric of given complex permittivity and construct the transformation relating the output quantum state to the input quantum state, without placing restrictions on the frequency. We apply the formalism to some typical examples in quantum communication. In particular we show that for Fock-entangled qubits the Bell-basis states $|{\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}〉$ are more robust against decoherence than the states $|{\ensuremath{\Phi}}^{\ifmmode\pm\else\textpm\fi{}}〉.$

DOI
10.1103/physreva.62.043803

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Dielectric four-port devices play an important role in optical quantum information processing. Since for causality reasons the permittivity is a complex function of frequency, dielectrics are typical examples of noisy quantum channels, which cannot preserve quantum coherence. To study the effects of quantum decoherence, we start from the quantized electromagnetic field in an arbitrary Kramers-Kronig dielectric of given complex permittivity and construct the transformation relating the output quantum state to the input quantum state, without placing restrictions on the frequency. We apply the formalism to some typical examples in quantum communication. In particular we show that for Fock-entangled qubits the Bell-basis states ∣\ensuremathΨ\ifmmode±\else\textpm\fi〉|{\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}〉∣\ensuremathΨ\ifmmode±\else\textpm\fi〉 are more robust against decoherence than the states ∣\ensuremathΦ\ifmmode±\else\textpm\fi〉.|{\ensuremath{\Phi}}^{\ifmmode\pm\else\textpm\fi{}}〉.∣\ensuremathΦ\ifmmode±\else\textpm\fi〉.
Quantum communication schemes widely use dielectric four-port devices as basic elements for constructing optical quantum channels. Dielectrics with complex permittivity are typical examples of noisy quantum channels in which quantum coherence will not be preserved. Basing on quantization of phenomenological electrodynamics, we construct the transformation relating the output quantum state to the input quantum state without placing frequency restrictions. Knowledge of the full transformed quantum state enables us to compute the entanglement contained in the output state.
The recently derived input-output relations for radiation at a dispersive and absorbing four-port device [T. Gruner and D.-G. Welsch, Phys. Rev. A 54, 1661 (1996)] are used to derive the unitary transformation that relates the output quantum state to the input quantum state, including radiation and matter and without placing frequency restrictions. It is shown that for each frequency component the transformation can be regarded as a U(4) group transformation, which can directly be calculated from the underlying complex refractive-index profile of the device without additional postulates. If for narrow-bandwidth radiation far from the medium resonances the absorption matrix of the four-port device can be disregarded, the well-known U(2) group transformation for a lossless device is recognized. Explicit formulas for the transformation of Fock states, coherent states, and phase-space functions are given.
Applying the recently developed formalism of quantum-state transformation of light at absorbing dielectric four-port devices [5], we calculate the quantum state of the outgoing modes of a two-mode squeezed vacuum transmitted through optical fibers of given extinction coefficients. Using the Peres-Horodecki separability criterion for continuous variable systems [4], we then compute both the maximal length of transmission of a two-mode squeezed vacuum through an absorbing system for which the transmitted state is still inseparable and the maximal gain for which inseparability can be observed in an amplifying setup. Finally, we estimate an upper bound of the entanglement preserved after transmission through an absorbing system. The results show that the characteristic length of entanglement degradation drastically decreases with increasing strength of squeezing.