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arXiv:quant-ph/0103167 · Physical Review A

Entanglement generation and degradation by passive optical devices

Stefan Scheel, D.‐G. Welsch

Published Nov 16, 2001• 99 citations
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Abstract

The influence of losses in the interferometric generation and the transmission of continuous-variable entangled light is studied, with special emphasis on Gaussian states. Based on the theory of quantum-state transformation at absorbing dielectric devices, the amount of entanglement is quantified by means of the relative-entropy measure. Upper bounds of entanglement and the distance to the set of separable Gaussian states are calculated. Compared with the distance measure, the bounds can substantially overestimate the entanglement. In particular, they do not show the drastic decrease of entanglement with increasing mean photon number, as does the distance measure.

DOI
10.1103/physreva.64.063811

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The influence of losses in the interferometric generation and the transmission of continuous-variable entangled light is studied, with special emphasis on Gaussian states. Based on the theory of quantum-state transformation at absorbing dielectric devices, the amount of entanglement is quantified by means of the relative-entropy measure. Upper bounds of entanglement and the distance to the set of separable Gaussian states are calculated. Compared with the distance measure, the bounds can substantially overestimate the entanglement. In particular, they do not show the drastic decrease of entanglement with increasing mean photon number, as does the distance measure.
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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.
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.
We study the effects of losses on the entanglement created between two separate atomic gases by optical probing and homodyne detection of the transmitted light. The system is well-described in the Gaussian state formulation. Analytical results quantifying the degree of entanglement between the two gases are derived and compared with the entanglement in a pair of light pulses generated by an Einstein-Podolsky-Rosen (EPR) source. For low (high) transmission losses, the highest degree of entanglement is obtained by probing with squeezed (antisqueezed) light. In an asymmetric setup where light is only sent one way through the atomic samples, we find that the logarithmic negativity of entanglement attains a constant value $\ensuremath{-}{\mathrm{log}}_{2}(N)$ with N=1∕3N=1∕3N=1∕3, irrespective of the loss along the transmission line.
The problem of decorrelation of entangled (squeezed-vacuum-type) light pulses of arbitrary shape passing through dispersive and absorbing four-port devices of arbitrary frequency response is studied, applying recently obtained results on quantum-state transformation (L Knöll et al 1999 Phys. Rev. A 59 4716). The fidelity and indices of (quantum) correlation based on the von Neumann entropy are calculated, with special emphasis on the dependence on the mean photon number, the pulse shape and the frequency response of the devices. In particular, it is shown that the quantum correlations can decay very rapidly due to dispersion and absorption, and the degree of degradation intensifies with increasing mean photon number.
For entangled states of light both the amount of entanglement and the sensitivity to noise generally increase with the number of photons in the state. The entanglement-sensitivity tradeoff is investigated for a particular set of states, multidimensional entangled coherent states. Those states possess an arbitrarily large amount of entanglement EEE provided the number of photons is at least of order 22E{2}^{2E}22E. We calculate how fast that entanglement decays due to photon absorption losses and how much entanglement is left. We find that for very small losses the amount of entanglement lost is equal to $2∕\mathrm{ln}(2)\ensuremath{\approx}2.89$ ebits per absorbed photon, irrespective of the amount of pure-state entanglement EEE one started with. In contrast, for larger losses it tends to be the remaining amount of entanglement that is independent of EEE. This may provide a useful strategy for creating states with a fixed amount of entanglement.