Scheme for the generation of hybrid entanglement between time-bin and wavelike encodings

Élie Gouzien, Floriane Brunel, Sébastien Tanzilli, and Virginia D'Auria

Phys. Rev. A 102, 012603 – Published 2 July 2020

Abstract

We propose a scheme for the generation of hybrid states entangling a single-photon time-bin qubit with a coherent-state qubit encoded on phases. Compared to other reported solutions, time-bin encoding makes hybrid entanglement particularly well adapted to applications involving long-distance propagation in optical fibers. This makes our proposal a promising resource for future out-of-the-laboratory quantum communication. In this perspective, we analyze our scheme by taking into account realistic experimental resources and discuss the impact of their imperfections on the quality of the obtained hybrid state.

Received 11 February 2020
Revised 20 May 2020
Accepted 8 June 2020

DOI:https://doi.org/10.1103/PhysRevA.102.012603

Physics Subject Headings (PhySH)

Quantum communication Quantum entanglement Quantum information architectures & platforms

Quantum Information, Science & Technology

Authors & Affiliations

Élie Gouzien , Floriane Brunel, Sébastien Tanzilli, and Virginia D'Auria^*

Université Côte d'Azur, CNRS, Institut de Physique de Nice (INPHYNI), Parc Valrose, 06108 Nice Cedex 2, France

^*virginia.dauria@inphyni.cnrs.fr

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Vol. 102, Iss. 1 — July 2020

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Images

Figure 1
Scheme for hybrid entanglement generation with time-bin encoding on the discrete variable part.
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Figure 2
Fidelity with the target state of a CV qubit remotely prepared as a function of the distance traveled by the DV part of the hybrid state. The target state is an odd Schrödinger cat state. The cases of hybrid entangled states with time-bin, single-rail, and polarization DV encodings are compared. For the polarization dispersion, we considered the value reported in a recent out-of-the-laboratory experiment [30]. We assumed a DV mode projected onto $\frac{{| 1 〉}_{A, l} + {| 1 〉}_{A, e}}{\sqrt{2}}$ for hybrid entanglement with time-bin DV encoding and similar projections for single-rail and polarization DV encodings (see Appendix pp1). For the initial amplitude of the CV part of the hybrid states, we set $α_{f} = 2$ . The distance traveled by the CV part is taken as negligible for the three states.
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Figure 3
Fidelity of the heralded hybrid state to the target state $| φ 〉 〈 φ |$ (top) and heralding probability (bottom) as a function of $r α$ , for the two proposed heralding protocols: ${\hat{Π}}^{id}$ (dashed line) and $\hat{Π}$ (plain lines). The ideal case is discussed in Sec. 2 and refers to perfect photon number resolving detectors with unitary efficiency. All other cases refer to on-off detector, whose quantum efficiencies, $η$ , have been chosen according to typical experimental values [25, 33, 34]. For the input CV state, we set $α = 2$ . The fidelity and the product $r α$ are adimensional quantities.
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Figure 4
Negativity of the Wigner function obtained after having projected the DV part of ${\hat{ρ}}^{(1)}$ onto the state $\frac{{| 1 〉}_{A, e} + {| 1 〉}_{A, l}}{\sqrt{2}}$ (left). In the ideal case discussed in Sec. 2, the negativity of the Wigner function of the conditional state is constant with $r α$ . The optimal value of $\frac{2}{π} \approx 0.64$ corresponds to the Wigner function definition $W (x, p) = \frac{1}{π} \int 〈 x + \frac{u}{2} | {\hat{ρ}}^{(1)} 〉 x - \frac{u}{2} e^{- 2 i p u} d u$ . Negativity of the partial transpose (NPT) as a function of $r α$ for the state ${\hat{ρ}}^{(1)}$ (right). The ideal case refers to a hybrid state of Eq. (8) and to a perfect projective measurement. For the nonideal case, quantum efficiencies, $η$ , of on-off detectors have been chosen according to typical experimental values [25, 33, 34]. Wigner negativity, NPT, and the product $r α$ are adimensional quantities.
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Figure 5
Fidelities with respect to $| φ 〉$ of the states obtained when considering multiple pairs on the discrete variable input as functions of the excitation parameter $λ^{2}$ of a realistic source based on parametric down conversion and providing the DV input state. Cat and squeezed states were considered for the continuous variable input. Detection efficiency $η = 0.95$ . We chose $\frac{r α}{\sqrt{2}} = 0.075$ , with a value of $α = 0.25$ . Fidelity and $λ^{2}$ are adimensional quantities.
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