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arXiv:1011.0824 · Physical Review A

Distillation and purification of symmetric entangled Gaussian states

Jaromı́r Fiurášek

Published Oct 26, 2010• 44 citations
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Abstract

We propose an entanglement distillation and purification scheme for symmetric two-mode entangled Gaussian states that allows to asymptotically extract a pure entangled Gaussian state from any input entangled symmetric Gaussian state. The proposed scheme is a modified and extended version of the entanglement distillation protocol originally developed by Browne et al. [Phys. Rev. A 67, 062320 (2003)]. A key feature of the present protocol is that it utilizes a two-copy degaussification procedure that involves a Mach-Zehnder interferometer with single-mode non-Gaussian filters inserted in its two arms. The required non-Gaussian filtering operations can be implemented by coherently combining two sequences of single-photon addition and subtraction operations.

DOI
10.1103/physreva.82.042331

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We propose an entanglement distillation and purification scheme for symmetric two-mode entangled Gaussian states that allows to asymptotically extract a pure entangled Gaussian state from any input entangled symmetric Gaussian state. The proposed scheme is a modified and extended version of the entanglement distillation protocol originally developed by Browne et al. [Phys. Rev. A 67, 062320 (2003)]. A key feature of the present protocol is that it utilizes a two-copy degaussification procedure that involves a Mach-Zehnder interferometer with single-mode non-Gaussian filters inserted in its two arms. The required non-Gaussian filtering operations can be implemented by coherently combining two sequences of single-photon addition and subtraction operations.
We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via local Gaussian operations and classical communication. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol considered by Eisert et al. [preceding Letter, Phys. Rev. Lett. 89, 137903 ()]] is the optimum Gaussian distillation protocol for two copies of entangled Gaussian states.
Abstract Entanglement of continuous-variable Gaussian states can be distilled by combination of de-Gaussifying operation such as single-photon subtraction and iterative heralded Gaussification. Here we present and analyze a simplified equivalent version of such entanglement distillation protocol, where the Gaussian measurements utilized in heralded Gaussification are eliminated and are absorbed into the preparation of suitable input Gaussian states of the simplified protocol. The simplified scheme contains less detectors and its overall success probability increases in comparison with the original scheme, while producing completely equivalent outputs. Our simplification of the entanglement distillation protocol closely parallels the recently proposed simplification of a scheme for breeding optical single-mode Gottesman–Kitaev–Preskill states (Aghaee Rad et al 2025 Nature 638 912 ). We investigate operation of the simplified entanglement distillation scheme for both pure and mixed input states and clarify how multicopy distillation of Gaussian entanglement emerges in a setup without any heralding Gaussian measurements.
The entanglement between Gaussian entangled states can be increased by non-Gaussian operations. We design a new scheme, named coherent photon addition, which can coherently add one photon generated by a spontaneous parametric down-conversation process to Gaussian quadrature-entangled light pulses created by a non-degenerate optical parametric amplifier. This operation can increase the entanglement of input two-mode Gaussian states as an entanglement distillation, and provides us with a new method of non-Gaussian operation. This scheme can also help us to study the decoherence of adding one- to two-mode Gaussian states from coherent photon addition to normal photon addition.
Photon subtraction (PS), due to its feasibility in physical implementation, has been widely utilized in Gaussian entanglement distillation. However, in this paper, we will show that photon addition (PA) can be employed to provide an additional enhancement in both the quality of distilled entanglement and the success probability if the input state is a weak two-mode squeezed vacuum state. Moreover, compared with biside PS, the distillation probability gains a $O(\frac{1}{{\ensuremath{\lambda}}^{2}})$ improvement, which increases rapidly when $\ensuremath{\lambda}\ensuremath{\ll}1$. Our PA-based scheme is obtained first by single-photon injection and later by vacuum state projection. The performance of PA in the presence of both an approximated single photon and amplitude-damping noise has been analyzed. Our result sheds more light on the indispensable non-Gaussian ingredients in Gaussian-state distillation and could be of great use in developing more efficient distillation schemes. Finally, the sensitivity of the PA scheme to inefficient photon detection has also been investigated.
It has been shown that entanglement distillation of Gaussian entangled states by means of local photon subtraction can be improved by local Gaussian transformations. Here we show that a similar effect can be expected for the distillation of an asymmetric Gaussian entangled state that is produced by a single squeezed beam. We show that for low initial entanglement, our largely simplified protocol generates more entanglement than previous proposed protocols. Furthermore, we show that the distillation scheme also works efficiently on decohered entangled states as well as with a practical photon subtraction setup.
Summary form only given. One of the key elements of quantum communication is the generation of maximally entangled states between distant nodes of a quantum network. We present an entanglement purification scheme which generates maximally entangled states in finite Hilbert spaces from the continuous two-mode squeezed states. With high probabilities, we get a more entangled stale, and the entanglement in the continuous partially entangled state is transferred to the maximally entangled states with high efficiency. Furthermore, in several practical cases this purification protocol can be readily extended to distill maximally entangled states from the mixed Gaussian continuous states.