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Teleportation of a Schr¨Odinger's-Cat State Via Satellite-Based

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Teleportation of a Schr¨Odinger's-Cat State Via Satellite-Based Teleportation of a Schrodinger’s-Cat¨ State via Satellite-based Quantum Communications Hung Do, Robert Malaney and Jonathan Green Abstract—The Schrodinger’s-cat¨ state is created from the The CV teleportation macroscopic superposition of coherent states and is well-known channel CV to be a useful resource for quantum information processing Entanglement protocols. In order to extend such protocols to a global scale, The state to be # % we study the continuous variable (CV) teleportation of the teleported # $ cat state via a satellite in low-Earth-orbit. Past studies have shown that the quantum character of the cat state can be % Dis place preserved after CV teleportation, even when taking into account )* $ the detector efficiency. However, the channel transmission loss has not been taken into consideration. Traditionally, optical ' ( fibers with fixed attenuation are used as teleportation channels. Our results show that in such a setup, the quantum character !" of the cat state is lost after 5dB of channel loss. We then Alice Bob investigate the free-space channel between the Earth and a satellite, where the loss is caused by atmospheric turbulence. For Fig. 1: In the hybrid scheme, the CV entanglement A − B is used as a a down-link channel of less than 500km and 30dB of loss, we teleportation channel. The channels have transmissivity TA and TB . The find that the teleported state preserves higher fidelity relative input state Win is teleported from Alice to Bob, resulting in the output to a fixed attenuation channel. The results in this work will Wout. be important for deployments over Earth-Satellite channels of protocols dependent on cat-state qubits. Index Terms—Schrodinger’s-cat¨ state, quantum teleportation. around 30dB [5]. In this work, we use CV entanglement distributed from a satellite as a CV teleportation channel. We then use this channel to distribute a Schrodinger-cat¨ state. Such a protocol has the potential to extend quantum I. INTRODUCTION information protocols over a global scale. The Schrodinger’s-cat¨ state is of fundamental interest be- The satellite-Earth channel is a free-space fading channel, cause it represents quantum superposition between two macro- where the loss is caused by diffraction, absorption/scattering, scopic states. Experimentally, the cat state can be created from and atmospheric turbulence [6]. The most significant loss the macroscopic superposition of coherent states. Such a su- comes from the turbulence, which could be characterized by perposition is known to be useful for universal quantum com- the Rytov parameter σR. This parameter is a function of puting using continuous variables (CV), where the quantum the atmospheric refraction index structure constant parameter 2 information is carried by the quadratures of the optical field Cn, which can be calculated from the wind velocity and the [1]. CV protocols use homodyne (or heterodyne) detectors altitude. Large turbulent eddies can cause beam-wandering, which are faster and more efficient (as compared to discrete while smaller eddies can induce beam broadening. The chan- variable (DV) protocols, where the quantum information is nel attenuation has a probabilistic nature and can be simulated carried in the polarization or the photon-number of single by different mathematical models. In the beam-wandering arXiv:1911.04613v1 [quant-ph] 11 Nov 2019 photons). model, the beam profile is modeled by a circular Gaussian Among the different quantum communication protocols, shape. The turbulence causes the beam center to fluctuate teleportation is one of the most useful. In many cases, in- around the center of the receiver aperture, resulting in a formation is transferred through optical fibers, which exhibits misalignment which reduces the transmissivity of the beam fixed attenuation. Since the loss in optical fiber scales ex- [7]. Recently, a new fading model was introduced taking into ponentially with distance, terrestrial quantum communication account additional deformation and broadening effects. In this has been limited to a few hundred kilometer. Quantum signal paper, we call this model the elliptic model. In the elliptic cannot be further amplified due to the no-cloning theorem [2]– model, the beam profile at the receiver aperture is broadened [4]. In contrast, for a satellite in low orbit (400 to 600km from and has an elliptic shape. There is fluctuations not only in the Earth), the channel is only affected by atmospheric turbulence. beam center but also in the widths of the elliptic profile [8]. In fact, for a down-link channel of less than 500km above Past studies have shown that the quantum character of the ground, the average attenuation was demonstrated to be the cat state can be preserved after CV teleportation [9], however, the channel transmission loss has not been taken Hung Do and Robert Malaney are at the University of New South into consideration. Our main contributions in this paper are: Wales, Sydney, NSW, Australia. Jonathan Green is at Northrop Grumman Corporation, San Diego, California, USA. • We derive a mathematical model detailing how channel Distribution Statement A: Approved for Public Release, Distribution is Unlimited, #19-1418; Dated 08/12/19. attenuation affects the quality of the CV teleportation ζout ! ζout=g in phase space. A general Gaussian function channel, as well as the quality of the teleportation out- with variance V can be written as come. 1 ZZ x2 + p2 G (ζ) = dxdp exp − ; (4) • For fixed attenuation we determine the loss conditions V πV V under which the cat state can be preserved. where ζ is (x; p). We have • We adopt the free-space fading channel between the Earth and a satellite for a typical down-link channel and 1 ζ W (ζ ) = [W ∗ G ]( out ) determine whether the teleported cat state retains higher out out g2 in V g fidelity in such channels relative to a fixed attenuation 1 ZZ ζ = dxdpW (ζ)G out − ζ ; (5) channel. g2 in V g The structure of the remainder of this paper is as follows. with variance V given by Section II studies the mathematical model for CV telepor- 1 tation. Section III studies different models of atmospheric +2r p p 2 V = 2 [ e (g TA − TB) turbulence. Section IV shows our simulation, while section 4g −2r p p 2 V summarizes our findings and discusses future work. + e (g TA + TB) 2 + 2g (1 − TA) + 2(1 − TB)]: (6) II. TELEPORTATION BY AN ATTENUATED CV When TA = TB = 1, we have [11] TELEPORTATION CHANNEL e2r(1 − g)2 + e−2r(1 + g)2 A. Attenuation on a CV teleportation channel V = : (7) 4g2 Among different CV entanglement resources, the two-mode When we further set g = 1, the variance is simplified to squeezed vacuum (TMSV) state is widely used, in part, −2r V = e , so Wout(ζout) = Win ∗ Ge−2r (ζout) [10], [12], because it has a Gaussian form. The Wigner function of a [13]. TMSV state can be written as [10] 4 C. Teleportation of a Schrodinger¨ cat state WT MSV (ζA; ζB) = expf π2 In this section, we will study the specific case where the −2r 2 2 − e (xA + xB) + (pA − pB) input is a Schrodinger’s-cat¨ state. The cat state is formed by +2r 2 2 − e (xA − xB) + (pA + pB) g; (1) a superposition of coherent states [14] −1 iφ where r is the squeezing parameter, ζi = (xi; pi), xi and pi jαic = N jα0i + e j − α0i ; (8) are the quadrature values, with subscript i 2 fA; Bg denoting where α0 is a complex number, φ is the phase and N is the the corresponding modes. When r ! 1, WT MSV (ζA; ζB) / normalization constant δ(xA − xB)δ(pA + pB), which represents a maximally entan- p 2 gled EPR state. N = 2 + 2e−2jα0j cos φ. (9) For a TMSV state, when the modes are sent through two Let α = x + ip, then the Wigner function of the cat state is channels with different transmissivities TA and TB (see Fig. 2N −2 1), our calculation shows that the state becomes W (α) = f exp −2jα − α j2 c π 0 (T ;T ) 4 2 A B + exp −2jα + α0j WT MSV (ζA; ζB) = 2 expf π τ −2jαj2 e−2r p p 2 p p 2 + 2e cos [φ + 4Im(α0 ∗ α)]g: (10) − xA TB + xB TA + pA TB − pB TA τ When Wc(α) is used as the input state Win(ζ), with ζ = e+2r p p 2 p p 2 (x; p), the output of the CV teleportation is calculated by Eq. − x T − x T + p T + p T τ A B B A A B B A (5). When the CV teleportation channels has transmissivities 2 TA = TB = T and the optimal gain is unity, the Gaussian − (1 − T ) x2 + p2 + (1 − T ) x2 + p2 g; (2) τ B A A A B B variance V in Eq. (6) becomes where τ is given by V = T e−2r + (1 − T ): (11) τ = 1 + [cosh(2r) − 1] (TA + TB − 2TATB): (3) When taking into account the detector amplitude-efficiency η, 0 1−η2 the Gaussian variance is modified to V = V + η2 . In this work, an attenuated CV entangled state is used as the In order to quantify the quality of the output state as CV teleportation channel (see Fig. 1). compared to input state, we use the entanglement fidelity [15]. For the Schrodinger’s-cat¨ state defined in Eq. (8), the fidelity B. Teleportation of an arbitrary mode was given in [13] 2 0 2 2 −4jα0j −4V jα0j −4jα0j Let Win(ζ) be the input state (Fig. 1). After CV telepor- 1 1 + e − exp 1+V 0 − exp 1+V 0 F = − 2 : tation with gain g, the output Wigner function becomes a 1 + V 0 2(1 + V 0)(1 + e−2jα0j cos φ)2 Gaussian convolution of the input, followed by a rescaling of (12) Distribution Statement A: Approved for Public Release, Distribution is Unlimited, #19-1418; Dated 08/12/19.
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