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Displacement-enhanced continuous-variable entanglement concentration

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We study entanglement concentration of continuous variable Gaussian states by local photon subtractions enhanced by coherent displacements. Instead of the previously considered symmetric two-mode squeezed vacuum states, we investigate the protocol for input states in the form of split single-mode squeezed vacuum, i.e., states obtained by mixing a single-mode squeezed vacuum with a vacuum state on a beam splitter, which is an experimentally highly relevant configuration. We analyze two scenarios in which the displacement-enhanced photon subtraction is performed either only on one, or on both of the modes and show that local displacements can lead to improved performance of the concentration protocol.

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Displacement-enhanced continuous-variable entanglement concentration

  1. 1. Displacement-enhanced continuous-variable entanglement concentration Ond°ej ƒernotík and Jaromír Fiurá²ek Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic Introduction Gaussian states and transformations represent an interesting counterpart to two-level photonic systems, due to both ecient theoretical de- scription and easy experimental manipulation. Gaussian operations are not, however, sucient to distill Gaussian entanglement and a non- Gaussian operation, such as photon subtraction, is required [1]. Gaussian operations can, nev- ertheless, enhance entanglement concentration based on photon subtraction [2, 3]. Here, we in- vestigate enhancing entanglement concentration based on photon subtraction by coherent dis- placements [4]. The input state we consider is obtained by mixing a single-mode squeezed vac- uum state mixed with vacuum on a beam split- ter, making the protocol experimentally more feasible than using two-mode squeezed vacuum state. The protocol Photon subtraction can be achieved by reecting a small part of the incoming light onto a single photon detector whose click heralds a successful subtraction. In our protocol, photon subtrac- tion is combined with two displacements, ˆD(α), ˆD(−α), leading to the operation ˆF = ˆa + α. The lter can be applied to only one of the modes, leading to the operation ˆF1 = (ˆa + α) ⊗ ˆIB, which is experimentally easier. Alterna- tively, displacement-enhanced photon subtrac- tion can be performed on both modes, resulting in the operation ˆF2 = (ˆa + α) ⊗ (ˆb + β), which brings another degree of freedom for protocol optimization. Input state As input, we consider split single-mode squeezed vacuum, i.e., single-mode squeezed vacuum state mixed with vacuum on a beam splitter which is easier to prepare experimentally than two-mode squeezed vacuum, |ψin = N ∞ n=0 2n k=0 λn 2nn! (2n)!t2n−k rk k!(2n − k)! |2n − k, k . Entanglement concentration We analyze the protocol in two dierent regimes. In the weak-squeezing approximation, only vac- uum and two-photon contributions are con- sidered, |ψin ≈ |00 + λrt|11 + λ(t2 |20 + r2 |02 )/ √ 2, and it can be seen how destructive quantum interference can lead to an enhance- ment of entanglement. Assuming arbitrary input squeezing, we consider a realistic scenario with bi- nary single-photon detectors with limited detec- tion eciency and limited transmittance of tap- o beam splitters. For details, see Ref. [4]. Single-mode subtraction For single-mode subtraction in the weak- squeezing regime, the output state reads |ψ1 = λt(t|10 + r|01 ) + α|ψin . The best choice is to use a balanced beam splitter together with no displacement as this gives the Bell state (|01 + |10 )/ √ 2. 0.0 0.2 0.4 0.6 0.8 1.0 ES -0.05 0.0 0.05 (a) 0.0 0.2 0.4 0.6 0.8 1.0 ES -0.05 0.0 0.05 (b) Two-mode subtraction With photon subtraction on both modes, de- structive quantum interference leads to an en- hancement of entanglement. Choosing αβ = −λrt [dark hyperbole in (a)], one can elimi- nate the dominant vacuum term. One-photon terms can be eliminated by choosing α = √ λt, β = − √ λr, resulting in a two-qutrit state (b) |ψ2 = √ 2rt|11 + t2 |20 + r2 |02 . -0.15 0.0 0.15 -0.15 0.0 0.15 (a) 0.8 1.0 1.2 1.4 1.6 ES 0.0 0.05 0.1 0.15 (b) Single-mode subtraction In a realistic scenario with photon subtrac- tion and displacements on one mode, zero displacement is optimal (a). While loga- rithmic negativity is not aected much by the input squeezing (b), success probability grows rapidly with increased squeezing (c). 0.0 0.2 0.4 0.6 0.8 1.0 1.2 EN -0.5 0.0 0.5 (a) 0.9 1.0 1.1 1.2 EN 0.0 0.1 0.2 0.3 0.4 (b) 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 P1 0.0 0.1 0.2 0.3 0.4 (c) Two-mode subtraction With photon subtraction performed on both modes, the results are similar to the weak squeezing limit (a). We get about 0.4-0.5 ebit of entanglement more than with single-mode subtraction (b). The cost for this is smaller success probability (c) which is further reduced by limited detection eciency. This is the main drawback of the two-mode subtraction strategy compared to the one-mode subtraction protocol. -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 (a) 1.3 1.4 1.5 EN 0.0 0.1 0.2 0.3 0.4 (b) 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 P2 0.0 0.1 0.2 0.3 0.4 (c) Conclusions and extensions We studied entanglement concentration of split single-mode squeezed vacuum states by photon sub- traction and local displacements. In the weak-squeezing limit, we can see the core of the protocol, destructive quantum interference of dominant terms, while with arbitrary squeezing, we analyze a realistic experimental scenario. While with single-mode photon subtraction, local displacements do not constitute an advantage, subtraction from both modes can be largely improved by displacmenets. While we considered only pure input states, the eect of loss can be also investigated [5]. It turns out that local displacements improve entanglement concentration only for small values of loss and after a certain threshold loss, depending on the input squeezing, simple photon subtraction from both modes is optimal. References [1] J. Fiurá²ek, Phys. Rev. Lett. 89, 137903 (2002). [2] J. Fiurá²ek, Phys. Rev. A 84, 012335 (2011). [3] S. L. Zhang and P. van Loock, Phys. Rev. A 84, 062309 (2011). [4] O. ƒernotík and J. Fiurá²ek, Phys. Rev. A 86, 052339 (2012). [5] A. Tipsmark, J. S. Neergaard-Nielsen, and U. L. Andersen, Opt. Express 21, 6670 (2013). Acknowledgements This work was supported by Project No. P205/12/0577 of the Czech Science Foundation and by Palacký Univer- sity (Grant No. PrF-2012-019).

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