# Efficient Quantum Teleportation of Unknown Qubit Based on DV-CV Interaction Mechanism

## Abstract

**:**

## 1. Introduction

## 2. DV-CV Quantum Teleportation of Unknown Qubit via Hybrid Non-Maximally Entangled State

## 3. Methods to Increase the Success Probabilities of the DV-CV Quantum Teleportation

## 4. DV-CV Quantum Teleportation of Unknown initially Amplitude-Distorting Qubit

## 5. Results

## Funding

## Conflicts of Interest

## References

- Bell, J. Speakable and Unspeakable in Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Aspect, A.; Grangier, P.; Roger, G. Experimental realization of Einstein-Podoslsky-Rosen-Bohm gedankenexperiment: A new violation of Bell’s inequalities. Phys. Rev. Lett.
**1982**, 49, 91–94. [Google Scholar] [CrossRef] - Rowe, M.A.; Kielpinski, D.; Meyer, V.; Sackett, C.A.; Itano, W.M.; Monroe, C.; Wineland, D.J. Experimental violation of a Bell’s inequality with efficient detection. Nature
**2001**, 409, 791–794. [Google Scholar] [CrossRef] [PubMed] - Bennett, C.H.; Brassard, G.; Crepeau, C.; Jozsa, R.; Peres, A.; Wootters, W.K. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett.
**1993**, 70, 1895–1899. [Google Scholar] [CrossRef] [PubMed] - Ghirardi, G.C.; Rimini, A.; Weber, T. A general argument against superluminal transmission through the quantum mechanical measurement process. Lettere Al Nuovo Climento
**1980**, 27, 293–298. [Google Scholar] [CrossRef] - Briegel, H.-J.; Dür, W.; Cirac, J.I.; Zoller, P. Quantum repeaters: The role of imperfect local operations in quantum communication. Phys. Rev. Lett.
**1998**, 81, 5932–5935. [Google Scholar] [CrossRef] - Gottesman, D.; Chuang, I.L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature
**1999**, 402, 390–393. [Google Scholar] [CrossRef] - Rausendorf, R.; Briegel, H.J. A one-way quantum computer. Phys. Rev. Lett.
**2001**, 86, 5188–5191. [Google Scholar] [CrossRef] [PubMed] - Bouwmeester, D.; Pan, J.W.; Mattle, K.; Eible, M.; Weinfurter, H.; Zeilinger, A. Experimental quantum teleportation. Nature
**1997**, 390, 575–579. [Google Scholar] [CrossRef] - Furusawa, A.; Sørensen, J.L.; Braunstein, S.L.; Fuchs, C.A.; Kimble, H.J.; Polzik, E.S. Unconditional quantum teleportation. Science
**1998**, 252, 706–707. [Google Scholar] [CrossRef] - Lee, N.; Benichi, H.; Takeno, Y.; Takeda, S.; Webb, J.; Huntington, E.; Furusawa, A. Teleportation of nonclassical wave packets of light. Science
**2011**, 352, 330–333. [Google Scholar] [CrossRef] - Takeda, S.; Mizuta, T.; Fuwa, M.; vby van Loock, P.; Furusawa, A. Deterministic quantum teleportation of photonic quantum bits by a hybrid technique. Nature
**2013**, 500, 315–318. [Google Scholar] [CrossRef] [PubMed] - Nielsen, M.A.; Knill, E.; Laflamme, R. Complete quantum teleportation using nuclear magnetic resonance. Nature
**1998**, 396, 52–55. [Google Scholar] [CrossRef] - Krauter, H.; Salart, D.; Muschik, C.A.; Petersen, J.M.; Shen, H.; Fernholz, T.; Polzik, E.S. Deterministic quantum teleportation between distant atomic objects. Nat. Phys.
**2013**, 9, 400–404. [Google Scholar] [CrossRef] - Barrett, M.D.; Chiaverini, J.; Schaetz, T.; Britton, J.; Itano, W.M.; Jost, J.D.; Knill, E.; Lager, C.; Leibfried, D.; Ozeri, R.; Wineland, D.J. Deterministic quantum teleportation of atomic qubits. Nature
**2004**, 429, 737–739. [Google Scholar] [CrossRef] [PubMed] - Gao, W.B.; Fallahi, P.; Togan, E.; Delteil, A.; Chin, Y.S.; Miguel-Sanchez, J.; Imamoğlu, A. Quantum teleportation from a photon to a solid-state spin qubit. Nat. Commun.
**2013**, 4, 1–8. [Google Scholar] [CrossRef] [PubMed] - Lutkenhaus, N.; Calsamiglia, J.; Suominen, K.A. Bell measurements for teleportation. Phys. Rev. A
**2000**, 59, 3295–3300. [Google Scholar] [CrossRef] - Knill, E.; Laflamme, L.; Milburn, G.J. A scheme for efficient quantum computation with linear optics. Nature
**2001**, 409, 46–52. [Google Scholar] [CrossRef] [PubMed] - Grice, W.P. Arbitrarily complete Bell-state measurement using only linear optical elements. Phys. Rev. A
**2011**, 84, 042331. [Google Scholar] [CrossRef] - Zaidi, H.A.; van Loock, P. Beating the one-half limit of ancilla-free linear optics Bell measurements. Phys. Rev. Lett.
**2013**, 110, 260501. [Google Scholar] [CrossRef] - Vaidman, L. Teleportation of quantum states. Phys. Rev. A
**1994**, 49, 1473–1476. [Google Scholar] [CrossRef] - Braunstein, S.L.; Kimble, H.J. Teleportation of continuous quantum variables. Phys. Rev. Lett.
**1998**, 80, 869–872. [Google Scholar] [CrossRef] - Braunstein, S.; van Loock, P. Quantum information with continuous variable states. Rev. Mod. Phys.
**2005**, 77, 513–577. [Google Scholar] [CrossRef] - Vaidman, L.; Yoran, N. Methods for reliable teleportation. Phys. Rev. A
**1999**, 59, 116–125. [Google Scholar] [CrossRef] - van Loock, P. Optical hybrid approaches to quantum information. Laser Photonics. Rev.
**2011**, 5, 167–200. [Google Scholar] [CrossRef] - Morin, O.; Haung, K.; Liu, J.; Jeannic, H.L.; Fabre, C.; Laurat, J. Remote creation of hybrid entanglement between particle-like and wave-like optical qubits. Nature Photonics
**2014**, 8, 570–574. [Google Scholar] [CrossRef] - Le Jeannic, H.; Cavailles, A.; Raskop, J.; Huang, K.; Laurat, J. Remote preparation of continuous-variable qubits using loss-tolerant hybrid entanglement of light. Optica
**2018**, 5, 1012–1015. [Google Scholar] [CrossRef] - Podoshvedov, S.A. Generation of displaced squeezed superpositions of coherent states. J. Exp. Theor. Phys.
**2012**, 114, 451–464. [Google Scholar] [CrossRef] - Podoshvedov, S.A.; Kim, J.; Kim, K. Elementary quantum gates with Gaussian states. Quantum. Inf. Proc.
**2014**, 13, 1723–1749. [Google Scholar] [CrossRef] - Lvovsky, A.I.; Ghobadi, R.; Chandra, A.; Prasad, A.S.; Simon, C. Observation of micro-macro entanglement of light. Nature Phys.
**2013**, 9, 541–544. [Google Scholar] [CrossRef] - Sekatski, P.; Sangouard, N.; Stobinska, M.; Bussieres, F.; Afzelius, M.; Gisin, N. Proposal for exploring macroscopic entanglement with a single photon and coherent states. Phys. Rev. A
**2012**, 86, 060301. [Google Scholar] [CrossRef] - Bruno, S.; Martin, A.; Sekatski, P.; Sangouard, N.; Thew, R.; Gisin, N. Displacement of entanglement back and forth between the micro and macro domains. Nat. Phys.
**2013**, 9, 545–550. [Google Scholar] [CrossRef] - Podoshvedov, S.A. Building of one-way Hadamard gate for squeezed coherent states. Phys. Rev. A
**2013**, 87, 012307. [Google Scholar] [CrossRef] - Podoshvedov, S.A. Elementary quantum gates in different bases. Quantum. Inf. Proc.
**2016**, 15, 3967–3993. [Google Scholar] [CrossRef] - Vidal, G.; Werner, R.F. Computable measure of entanglement. Phys. Rev. A
**2002**, 65, 032314. [Google Scholar] [CrossRef] - Peres, A. Separability criterion for density matrices. Phys. Rev. Lett.
**1996**, 77, 1413–1415. [Google Scholar] [CrossRef] [PubMed] - Paris, M.G.A. Displacement operator by beam splitter. Phys. Lett. A
**1996**, 217, 78–80. [Google Scholar] [CrossRef] - Podoshvedov, S.A. Extraction of displaced number state. JOSA B
**2014**, 31, 2491–2503. [Google Scholar] [CrossRef] - Reck, M.; Zeilinger, A.; Bernstein, H.J.; Bertani, P. Experimental realization of any discrete unitary operator. Phys. Rev. Lett.
**1994**, 73, 58–61. [Google Scholar] [CrossRef] - Podoshvedov, S.A.; Kim, J.; Lee, J. Generation of a displaced qubit and entangled displaced photon state via conditional measurement and their properties. Opt. Commun.
**2008**, 281, 3748–3754. [Google Scholar] [CrossRef] - Pan, J.-W.; Bouwmeester, D.; Weinfurter, H.; Zeilinger, A. Experimental entanglement swapping: Entangling photons that never interacted. Phys. Rev. Lett.
**1998**, 80, 3891–3894. [Google Scholar] [CrossRef]

**Figure 1.**A schematic representation of implementation of DV-CV quantum teleportation with help of the hybrid non-maximally entangled state (1). Coherent components interact with unknown qubit in an indistinguishable manner on the HTBS. Measurements made at a microscopic level allow for Bob to obtain (after the corresponding unitary transformations initiated by the classical information (CI) from Alice) set of the states depending on Alice’s measurement outcomes due to quantum nonlocality. Part of states is the original unknown states, while the others acquire additional amplitude known factors. DV-CV quantum teleportation can be performed in various interpretations in order to influence which part of the teleported qubits is original unknown state and which are (amplitude-modulated) AM states. Different implementation schemes also determine the amplitude-distorting factors of the output states. So, if the scheme involves additional HTBS for interaction of coherent state |0,−β〉

_{2}with the original state (4), then we deal with amplitude-distorting factors (16). If the scheme without additional HTBS is used, then recipient obtains the states (37) with corresponding known amplitude-distorting factors. Another interpretation includes the third party that initially generates AM unknown qubit. The considered schemes should also include a demodulation procedure (DP) in order to get rid of amplitude-distorting factors. Commercially achievable avalanche photodiode (APD), being highly sensitive semiconductor electronic devices, are used for registration of the measurement outcomes. Photon number resolving detector is used in first (coherent) mode to determine the parity of the (superposition of coherent states) SCS. S means a source of the hybrid entangled state (6).

**Figure 2.**(

**a**) Plots of the success probabilities ${P}_{T}^{\left(01\right)}$ and ${P}_{nm}^{\left(01\right)}$ for different n and m in dependency on the displacement amplitude α. (

**b**) Only three graphs of probabilities ${P}_{T}^{\left(01\right)}$, ${P}_{01}^{\left(01\right)S}$, giving the maximum contribution, and ${P}_{S}={P}_{T}^{\left(01\right)}+{P}_{01}^{\left(01\right)S}$ are shown. The value ${P}_{S}=0.4418$ corresponds to quantum teleportation of unknown qubit without amplitude-distorting factor.

**Figure 3.**(

**a**) Plots of the success probabilities ${P}_{T}^{\left(12\right)}$ and ${P}_{nm}^{\left(12\right)}$ for different $n$ and $m$ in dependency on the displacement amplitude $\alpha $. (

**b**) Only three graphs of probabilities ${P}_{T}^{\left(12\right)}$, ${P}_{12}^{\left(12\right)S}$, giving the maximum contribution, and ${P}_{S}={P}_{T}^{\left(12\right)}+{P}_{12}^{\left(12\right)S}$ are shown. The value ${P}_{S}=0.401$ corresponds to quantum teleportation of unknown qubit without amplitude-distorting factor.

**Figure 4.**Plots of the success probabilities $\delta {P}_{T}^{\left(01\right)S}$ and $\delta {P}_{T}^{\left(01\right)D}$ to teleport and restore (get rid of amplitude-distorting factor by one two proposed methods) an unknown qubit in dependency on the displacement amplitude $\alpha $.

**Figure 5.**Plots of the success probabilities ${P}_{T}^{\left(01\right)}$ to teleport and restore (get rid of amplitude-distorting factor) unknown initially AM qubit in dependency on $|{a}_{1}|$ for the different values of the displacement amplitude $\alpha $. (

**a**) The plot shows the success probability when using the initially modulated unknown qubit (56) with amplitude-distorting factor ${A}_{01}^{\left(01\right)-1}$. The quantum swapping method (Equations (51) and (52)) is used to get rid of amplitude-distorting factors (Equation (60)). (

**b**) The plot shows the success probability of teleporting AM unknown qubit (61), where original state is restored with help of mixing AM unknown qubit with coherent state (Equations (48) and (49)).

**Table 1.**Values of amplitude-distorting factors ${A}_{nm}^{\left(01\right)}\left(\alpha =1/\sqrt{2}\right)$ for different values of n and m.

$\mathit{n}$ | $0$ | $0$ | $1$ | $0$ | $1$ | $0$ | $1$ | $0$ |
---|---|---|---|---|---|---|---|---|

$m$ | $2$ | $3$ | $2$ | $4$ | $3$ | $5$ | $4$ | $5$ |

${A}_{nm}^{\left(01\right)}$ | $-1/3$ | $-0.2$ | $1/3$ | $-1/7$ | $0.2$ | $-1/9$ | $1/7$ | $3/5$ |

${A}_{mn}^{\left(01\right)}$ | $-3$ | $-5$ | $3$ | $-7$ | $5$ | $-9$ | $7$ | $5/3$ |

**Table 2.**Values of amplitude-distorting factors ${A}_{nm}^{\left(12\right)}\left(\alpha =0.5053\right)$ for different values of n and m.

$\mathit{n}$ | $0$ | $0$ | $0$ | $0$ | $1$ |
---|---|---|---|---|---|

$m$ | $1$ | $2$ | $3$ | $4$ | $3$ |

${A}_{nm}^{\left(12\right)}$ | $0.427$ | $-0.427$ | $-0.155$ | $-0.0954$ | $-0.362$ |

${A}_{mn}^{\left(12\right)}$ | $2.343$ | $-2.343$ | $-6.468$ | $-10.481$ | $-2.76$ |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Podoshvedov, S.A.
Efficient Quantum Teleportation of Unknown Qubit Based on DV-CV Interaction Mechanism. *Entropy* **2019**, *21*, 150.
https://doi.org/10.3390/e21020150

**AMA Style**

Podoshvedov SA.
Efficient Quantum Teleportation of Unknown Qubit Based on DV-CV Interaction Mechanism. *Entropy*. 2019; 21(2):150.
https://doi.org/10.3390/e21020150

**Chicago/Turabian Style**

Podoshvedov, Sergey A.
2019. "Efficient Quantum Teleportation of Unknown Qubit Based on DV-CV Interaction Mechanism" *Entropy* 21, no. 2: 150.
https://doi.org/10.3390/e21020150