# Nontraditional Deterministic Remote State Preparation Using a Non-Maximally Entangled Channel without Additional Quantum Resources

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Deterministic RSP via a Generally Entangled Quantum Channel

#### 3.1. Deterministic RSP of a 2-Dimensional Quantum State via a Generally Entangled Quantum Channel

#### 3.2. Deterministic RSP of a d-Dimensional Quantum State via a Generally Entangled Quantum Channel

## 4. Realization

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

QT | Quantum Teleportation |

QKD | Quantum Key Distribution |

QDC | Quantum Dense Coding |

QSDC | Quantum Secure Direct Communication |

RSP | Remote State Preparation |

## References

- Yurke, B.; Denker, J.S. Quantum network theory. Phys. Rev. A
**1984**, 29, 1419–1437. [Google Scholar] [CrossRef] - Cirac, J.I.; Zoller, P.; Kimble, H.J.; Mabuchi, H. Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network. Phys. Rev. Lett.
**1997**, 78, 3221–3224. [Google Scholar] [CrossRef] - Ritter, S.; Nölleke, C.; Hahn, C.; Reiserer, A.; Neuzner, A.; Uphoff, M.; Mücke, M.; Figueroa, E.; Bochmann, J.; Rempe, G. An elementary quantum network of single atoms in optical cavities. Nature
**2012**, 484, 195–200. [Google Scholar] [CrossRef] - Simon, C. Towards a global quantum network. Nat. Photonics
**2017**, 11, 678–680. [Google Scholar] [CrossRef] - Liao, S.K.; Cai, W.Q.; Handsteiner, J.; Liu, B.; Yin, J.; Zhang, L.; Rauch, D.; Fink, M.; Ren, J.G.; Liu, W.Y.; et al. Satellite-Relayed Intercontinental Quantum Network. Phys. Rev. Lett.
**2018**, 120, 030501. [Google Scholar] [CrossRef] [PubMed] - Wei, S.; Chen, Y.; Zhou, Z.; Long, G. A quantum convolutional neural network on NISQ devices. AAPPS Bull.
**2022**, 32, 2. [Google Scholar] [CrossRef] - Ekert, A.K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.
**1991**, 67, 661–663. [Google Scholar] [CrossRef] [PubMed] - Duan, L.M.; Lukin, M.D.; Cirac, J.I.; Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature
**2001**, 414, 413–418. [Google Scholar] [CrossRef] - Wang, M.; Wang, X.; Ruan, D.; Long, G. Quantum direct portation. Acta Phys. Sin.
**2001**, 70, 190301. [Google Scholar] [CrossRef] - Ursin, R.; Tiefenbacher, F.; Schmitt-Manderbach, T.; Weier, H.; Scheidl, T.; Lindenthal, M.; Blauensteiner, B.; Jennewein, T.; Perdigues, J.; Trojek, P.; et al. Entanglement-based quantum communication over 144 km. Nat. Phys.
**2007**, 3, 481–486. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G.; Crépeau, 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] - Karlsson, A.; Bourennane, M. Quantum teleportation using three-particle entanglement. Phys. Rev. A
**1998**, 58, 4394–4400. [Google Scholar] [CrossRef] - Stenholm, S.; Bardroff, P.J. Teleportation of N-dimensional states. Phys. Rev. A
**1998**, 58, 4373–4376. [Google Scholar] [CrossRef] - Cabello, A. Quantum Key Distribution in the Holevo Limit. Phys. Rev. Lett.
**2000**, 85, 5635–5638. [Google Scholar] [CrossRef] - Shor, P.W.; Preskill, J. Simple Proof of Security of the BB84 Quantum Key Distribution Protocol. Phys. Rev. Lett.
**2000**, 85, 441–444. [Google Scholar] [CrossRef] [PubMed] - Lo, H.K.; Ma, X.; Chen, K. Decoy State Quantum Key Distribution. Phys. Rev. Lett.
**2005**, 94, 230504. [Google Scholar] [CrossRef] [PubMed] - Barrett, J.; Hardy, L.; Kent, A. No Signaling and Quantum Key Distribution. Phys. Rev. Lett.
**2005**, 95, 010503. [Google Scholar] [CrossRef] - Qi, R.; Zhang, H.; Gao, J.; Yin, L.; Long, G.L. Loophole-free plug-and-play quantum key distribution. N. J. Phys.
**2021**, 23, 063058. [Google Scholar] [CrossRef] - Kwek, L.C.; Cao, L.; Luo, W.; Wang, Y.; Sun, S.; Wang, X.; Liu, A.Q. Chip-based quantum key distribution. AAPPS Bull.
**2021**, 31, 15. [Google Scholar] [CrossRef] - Bennett, C.H.; Wiesner, S.J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett.
**1992**, 69, 2881–2884. [Google Scholar] [CrossRef] - Mattle, K.; Weinfurter, H.; Kwiat, P.G.; Zeilinger, A. Dense Coding in Experimental Quantum Communication. Phys. Rev. Lett.
**1996**, 76, 4656–4659. [Google Scholar] [CrossRef] - Bruß, D.; D’Ariano, G.M.; Lewenstein, M.; Macchiavello, C.; Sen, A.; Sen, U. Distributed Quantum Dense Coding. Phys. Rev. Lett.
**2004**, 93, 210501. [Google Scholar] [CrossRef] - Guo, Y.; Liu, B.H.; Li, C.F.; Guo, G.C. Advances in Quantum Dense Coding. Adv. Quantum Technol.
**2019**, 2, 1900011. [Google Scholar] [CrossRef] - Long, G.L.; Liu, X.S. Theoretically efficient high-capacity quantum-key-distribution scheme. Phys. Rev. A
**2002**, 65, 032302. [Google Scholar] [CrossRef] - Deng, F.G.; Long, G.L.; Liu, X.S. Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block. Phys. Rev. A
**2003**, 68, 042317. [Google Scholar] [CrossRef] - Deng, F.G.; Long, G.L. Secure direct communication with a quantum one-time pad. Phys. Rev. A
**2004**, 69, 052319. [Google Scholar] [CrossRef] - Zhang, W.; Ding, D.S.; Sheng, Y.B.; Zhou, L.; Shi, B.S.; Guo, G.C. Quantum Secure Direct Communication with Quantum Memory. Phys. Rev. Lett.
**2017**, 118, 220501. [Google Scholar] [CrossRef] [PubMed] - Shor, P.W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A
**1995**, 52, R2493–R2496. [Google Scholar] [CrossRef] [PubMed] - Zurek, W.H.; Habib, S.; Paz, J.P. Coherent states via decoherence. Phys. Rev. Lett.
**1993**, 70, 1187–1190. [Google Scholar] [CrossRef] - Lidar, D.A.; Chuang, I.L.; Whaley, K.B. Decoherence-Free Subspaces for Quantum Computation. Phys. Rev. Lett.
**1998**, 81, 2594–2597. [Google Scholar] [CrossRef] - Schlosshauer, M. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys.
**2005**, 76, 1267–1305. [Google Scholar] [CrossRef] - Xiang, G.Y.; Li, J.; Yu, B.; Guo, G.C. Remote preparation of mixed states via noisy entanglement. Phys. Rev. A
**2005**, 72, 012315. [Google Scholar] [CrossRef] - Banaszek, K. Optimal quantum teleportation with an arbitrary pure state. Phys. Rev. A
**2000**, 62, 024301. [Google Scholar] [CrossRef] - Li, W.L.; Li, C.F.; Guo, G.C. Probabilistic teleportation and entanglement matching. Phys. Rev. A
**2000**, 61, 034301. [Google Scholar] [CrossRef] - Roa, L.; Delgado, A.; Fuentes-Guridi, I. Optimal conclusive teleportation of quantum states. Phys. Rev. A
**2003**, 68, 022310. [Google Scholar] [CrossRef] - Kimble, H.J. The quantum internet. Nature
**2008**, 453, 1023–1030. [Google Scholar] [CrossRef] [PubMed] - Wang, D.; Hoehn, R.D.; Ye, L.; Kais, S. Generalized Remote Preparation of Arbitrary m-qubit Entangled States via Genuine Entanglements. Entropy
**2015**, 17, 1755–1774. [Google Scholar] [CrossRef] - Pirandola, S.; Braunstein, S.L. Physics: Unite to build a quantum Internet. Nat. News
**2016**, 532, 169. [Google Scholar] [CrossRef] - Wehner, S.; Elkouss, D.; Hanson, R. Quantum internet: A vision for the road ahead. Science
**2018**, 362. [Google Scholar] [CrossRef] - Yang, F.; Chen, X.; Zhao, D.; Wei, S.; Wen, J.; Wang, H.; Xin, T.; Long, G. Quantum Multi-Round Resonant Transition Algorithm. Entropy
**2023**, 25, 61. [Google Scholar] [CrossRef] - Lu, B.; Liu, L.; Song, J.Y.; Wen, K.; Wang, C. Recent progress on coherent computation based on quantum squeezing. AAPPS Bull.
**2023**, 33, 7. [Google Scholar] [CrossRef] - Xu, G.F.; Tong, D.M. Realizing multi-qubit controlled nonadiabatic holonomic gates with connecting systems. AAPPS Bull.
**2022**, 32, 13. [Google Scholar] [CrossRef] - Dakić, B.; Lipp, Y.O.; Ma, X.; Ringbauer, M.; Kropatschek, S.; Barz, S.; Paterek, T.; Vedral, V.; Zeilinger, A.; Brukner, Č.; et al. Quantum discord as resource for remote state preparation. Nat. Phys.
**2012**, 8, 666–670. [Google Scholar] [CrossRef] - Monz, T.; Schindler, P.; Barreiro, J.T.; Chwalla, M.; Nigg, D.; Coish, W.A.; Harlander, M.; Hänsel, W.; Hennrich, M.; Blatt, R. 14-Qubit Entanglement: Creation and Coherence. Phys. Rev. Lett.
**2011**, 106, 130506. [Google Scholar] [CrossRef] - Yao, X.C.; Wang, T.X.; Xu, P.; Lu, H.; Pan, G.S.; Bao, X.H.; Peng, C.Z.; Lu, C.Y.; Chen, Y.A.; Pan, J.W. Observation of eight-photon entanglement. Nat. Photonics
**2012**, 6, 225–228. [Google Scholar] [CrossRef] - Huang, Y.F.; Liu, B.H.; Peng, L.; Li, Y.H.; Li, L.; Li, C.F.; Guo, G.C. Experimental generation of an eight-photon Greenberger–Horne–Zeilinger state. Nat. Commun.
**2011**, 2, 1–6. [Google Scholar] [CrossRef] - Zhang, F.; Xing, J.; Hu, X.; Pan, X.; Long, G. Coupling-selective quantum optimal control in weak-coupling NV-
^{13}C system. AAPPS Bull.**2023**, 33, 2. [Google Scholar] [CrossRef] - Wagenknecht, C.; Li, C.M.; Reingruber, A.; Bao, X.H.; Goebel, A.; Chen, Y.A.; Zhang, Q.; Chen, K.; Pan, J.W. Experimental demonstration of a heralded entanglement source. Nat. Photonics
**2010**, 4, 549–552. [Google Scholar] [CrossRef] - Zhang, H.; Jin, X.M.; Yang, J.; Dai, H.N.; Yang, S.J.; Zhao, T.M.; Rui, J.; He, Y.; Jiang, X.; Yang, F.; et al. Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion. Nat. Photonics
**2011**, 5, 628–632. [Google Scholar] [CrossRef] - Yin, J.; Li, Y.H.; Liao, S.K.; Yang, M.; Cao, Y.; Zhang, L.; Ren, J.G.; Cai, W.Q.; Liu, W.Y.; Li, S.L.; et al. Entanglement-based secure quantum cryptography over 1,120 kilometres. Nature
**2020**, 582, 501–505. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G.; Popescu, S.; Schumacher, B.; Smolin, J.A.; Wootters, W.K. Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels. Phys. Rev. Lett.
**1996**, 76, 722–725. [Google Scholar] [CrossRef] - Deutsch, D.; Ekert, A.; Jozsa, R.; Macchiavello, C.; Popescu, S.; Sanpera, A. Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels. Phys. Rev. Lett.
**1996**, 77, 2818–2821. [Google Scholar] [CrossRef] - Bennett, C.H.; DiVincenzo, D.P.; Smolin, J.A.; Wootters, W.K. Mixed-state entanglement and quantum error correction. Phys. Rev. A
**1996**, 54, 3824–3851. [Google Scholar] [CrossRef] - Bombin, H.; Martin-Delgado, M.A. Topological Quantum Distillation. Phys. Rev. Lett.
**2006**, 97, 180501. [Google Scholar] [CrossRef] - Zwerger, M.; Briegel, H.J.; Dür, W. Universal and Optimal Error Thresholds for Measurement-Based Entanglement Purification. Phys. Rev. Lett.
**2013**, 110, 260503. [Google Scholar] [CrossRef] [PubMed] - Hu, X.M.; Huang, C.X.; Sheng, Y.B.; Zhou, L.; Liu, B.H.; Guo, Y.; Zhang, C.; Xing, W.B.; Huang, Y.F.; Li, C.F.; et al. Long-Distance Entanglement Purification for Quantum Communication. Phys. Rev. Lett.
**2021**, 126, 010503. [Google Scholar] [CrossRef] [PubMed] - Jonathan, D.; Plenio, M.B. Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett.
**1999**, 83, 3566. [Google Scholar] [CrossRef] - Daftuar, S.; Klimesh, M. Mathematical structure of entanglement catalysis. Phys. Rev. A
**2001**, 64, 042314. [Google Scholar] [CrossRef] - Van Dam, W.; Hayden, P. Universal entanglement transformations without communication. Phys. Rev. A
**2003**, 67, 060302. [Google Scholar] [CrossRef] - Sanders, Y.R.; Gour, G. Necessary conditions for entanglement catalysts. Phys. Rev. A
**2009**, 79, 054302. [Google Scholar] [CrossRef] - Popescu, S. Bell’s Inequalities and Density Matrices: Revealing “Hidden” Nonlocality. Phys. Rev. Lett.
**1995**, 74, 2619–2622. [Google Scholar] [CrossRef] - Peres, A. Collective tests for quantum nonlocality. Phys. Rev. A
**1996**, 54, 2685–2689. [Google Scholar] [CrossRef] - Masanes, L. All Bipartite Entangled States Are Useful for Information Processing. Phys. Rev. Lett.
**2006**, 96, 150501. [Google Scholar] [CrossRef] - Masanes, L.; Liang, Y.C.; Doherty, A.C. All Bipartite Entangled States Display Some Hidden Nonlocality. Phys. Rev. Lett.
**2008**, 100, 090403. [Google Scholar] [CrossRef] - Liang, Y.C.; Masanes, L.; Rosset, D. All entangled states display some hidden nonlocality. Phys. Rev. A
**2012**, 86, 052115. [Google Scholar] [CrossRef] - Riera-Sàbat, F.; Sekatski, P.; Pirker, A.; Dür, W. Entanglement-Assisted Entanglement Purification. Phys. Rev. Lett.
**2021**, 127, 040502. [Google Scholar] [CrossRef] - Lipka-Bartosik, P.; Skrzypczyk, P. Catalytic Quantum Teleportation. Phys. Rev. Lett.
**2021**, 127, 080502. [Google Scholar] [CrossRef] - Li, J.Y.; Fang, X.X.; Zhang, T.; Tabia, G.N.M.; Lu, H.; Liang, Y.C. Activating hidden teleportation power: Theory and experiment. Phys. Rev. Res.
**2021**, 3, 023045. [Google Scholar] [CrossRef] - Lo, H.K. Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity. Phys. Rev. A
**2000**, 62, 012313. [Google Scholar] [CrossRef] - Bennett, C.H.; DiVincenzo, D.P.; Shor, P.W.; Smolin, J.A.; Terhal, B.M.; Wootters, W.K. Remote State Preparation. Phys. Rev. Lett.
**2001**, 87, 077902. [Google Scholar] [CrossRef] [PubMed] - Pati, A.K. Minimum classical bit for remote preparation and measurement of a qubit. Phys. Rev. A
**2000**, 63, 014302. [Google Scholar] [CrossRef] - Nguyen, B.A.; Cao, T.B.; Nung, V.D.; Kim, J. Remote state preparation with unit success probability. Adv. Nat. Sci.
**2011**, 2, 035009. [Google Scholar] [CrossRef] - Xu, G.; Chen, X.B.; Dou, Z.; Li, J.; Liu, X.; Li, Z. Novel Criteria for Deterministic Remote State Preparation via the Entangled Six-Qubit State. Entropy
**2016**, 18, 267. [Google Scholar] [CrossRef] - An, N.B.; Dat, L.T.; Kim, J. Nonstandard protocols for joint remote preparation of a general quantum state and hybrid entanglement of any dimension. Phys. Rev. A
**2018**, 98, 042329. [Google Scholar] [CrossRef] - Du, Z.; Li, X. Deterministic joint remote state preparation of four-qubit cluster type with tripartite involvement. Quantum Inf. Process.
**2019**, 19, 39. [Google Scholar] [CrossRef] - DiVincenzo, D.P. Two-bit gates are universal for quantum computation. Phys. Rev. A
**1995**, 51, 1015–1022. [Google Scholar] [CrossRef] - Sleator, T.; Weinfurter, H. Realizable Universal Quantum Logic Gates. Phys. Rev. Lett.
**1995**, 74, 4087–4090. [Google Scholar] [CrossRef] - Barenco, A.; Bennett, C.H.; Cleve, R.; DiVincenzo, D.P.; Margolus, N.; Shor, P.; Sleator, T.; Smolin, J.A.; Weinfurter, H. Elementary gates for quantum computation. Phys. Rev. A
**1995**, 52, 3457–3467. [Google Scholar] [CrossRef] - Kim, Y.H. Single-photon two-qubit entangled states: Preparation and measurement. Phys. Rev. A
**2003**, 67, 040301. [Google Scholar] [CrossRef] - U’Ren, A.B.; Silberhorn, C.; Banaszek, K.; Walmsley, I.A. Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks. Phys. Rev. Lett.
**2004**, 93, 093601. [Google Scholar] [CrossRef] - Mosley, P.J.; Lundeen, J.S.; Smith, B.J.; Wasylczyk, P.; U’Ren, A.B.; Silberhorn, C.; Walmsley, I.A. Heralded Generation of Ultrafast Single Photons in Pure Quantum States. Phys. Rev. Lett.
**2008**, 100, 133601. [Google Scholar] [CrossRef] - Barreiro, J.T.; Wei, T.C.; Kwiat, P.G. Remote Preparation of Single-Photon “Hybrid” Entangled and Vector-Polarization States. Phys. Rev. Lett.
**2010**, 105, 030407. [Google Scholar] [CrossRef] - Wang, X.L.; Cai, X.D.; Su, Z.E.; Chen, M.C.; Wu, D.; Li, L.; Liu, N.L.; Lu, C.Y.; Pan, J.W. Quantum teleportation of multiple degrees of freedom of a single photon. Nature
**2015**, 518, 516–519. [Google Scholar] [CrossRef] [PubMed] - Burnham, D.C.; Weinberg, D.L. Observation of Simultaneity in Parametric Production of Optical Photon Pairs. Phys. Rev. Lett.
**1970**, 25, 84–87. [Google Scholar] [CrossRef] - Kwiat, P.G.; Mattle, K.; Weinfurter, H.; Zeilinger, A.; Sergienko, A.V.; Shih, Y. New High-Intensity Source of Polarization-Entangled Photon Pairs. Phys. Rev. Lett.
**1995**, 75, 4337–4341. [Google Scholar] [CrossRef] - Kwiat, P.G.; Waks, E.; White, A.G.; Appelbaum, I.; Eberhard, P.H. Ultrabright source of polarization-entangled photons. Phys. Rev. A
**1999**, 60, R773–R776. [Google Scholar] [CrossRef] - De Caro, L.; Garuccio, A. Reliability of Bell-inequality measurements using polarization correlations in parametric-down-conversion photon sources. Phys. Rev. A
**1994**, 50, R2803–R2805. [Google Scholar] [CrossRef] - Bouwmeester, D.; Pan, J.W.; Mattle, K.; Eibl, M.; Weinfurter, H.; Zeilinger, A. Experimental quantum teleportation. Nature
**1997**, 390, 575–579. [Google Scholar] [CrossRef] - Pan, J.W.; Bouwmeester, D.; Daniell, M.; Weinfurter, H.; Zeilinger, A. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature
**2000**, 403, 515–519. [Google Scholar] [CrossRef] - Yin, J.; Ren, J.G.; Lu, H.; Cao, Y.; Yong, H.L.; Wu, Y.P.; Liu, C.; Liao, S.K.; Zhou, F.; Jiang, Y.; et al. Quantum teleportation and entanglement distribution over 100-kilometre free-space channels. Nature
**2012**, 488, 185–188. [Google Scholar] [CrossRef] - Ren, J.G.; Xu, P.; Yong, H.L.; Zhang, L.; Liao, S.K.; Yin, J.; Liu, W.Y.; Cai, W.Q.; Yang, M.; Li, L.; et al. Ground-to-satellite quantum teleportation. Nature
**2017**, 549, 70–73. [Google Scholar] [CrossRef] [PubMed] - Hu, X.M.; Chen, J.S.; Liu, B.H.; Guo, Y.; Huang, Y.F.; Zhou, Z.Q.; Han, Y.J.; Li, C.F.; Guo, G.C. Experimental Test of Compatibility-Loophole-Free Contextuality with Spatially Separated Entangled Qutrits. Phys. Rev. Lett.
**2016**, 117, 170403. [Google Scholar] [CrossRef] - Hu, X.M.; Zhang, C.; Liu, B.H.; Cai, Y.; Ye, X.J.; Guo, Y.; Xing, W.B.; Huang, C.X.; Huang, Y.F.; Li, C.F.; et al. Experimental High-Dimensional Quantum Teleportation. Phys. Rev. Lett.
**2020**, 125, 230501. [Google Scholar] [CrossRef] - Zhang, D.; Chen, Y.; Gong, S.; Wu, W.; Cai, W.; Ren, M.; Ren, X.; Zhang, S.; Guo, G.; Xu, J. All-optical modulation of quantum states by nonlinear metasurface. Light Sci. Appl.
**2022**, 11, 58. [Google Scholar] [CrossRef] - Nemoto, K.; Munro, W.J. Nearly Deterministic Linear Optical Controlled-NOT Gate. Phys. Rev. Lett.
**2004**, 93, 250502. [Google Scholar] [CrossRef] - Zhao, Z.; Zhang, A.N.; Chen, Y.A.; Zhang, H.; Du, J.F.; Yang, T.; Pan, J.W. Experimental Demonstration of a Nondestructive Controlled-NOT Quantum Gate for Two Independent Photon Qubits. Phys. Rev. Lett.
**2005**, 94, 030501. [Google Scholar] [CrossRef] - Yamamoto, T.; Pashkin, Y.A.; Astafiev, O.; Nakamura, Y.; Tsai, J.S. Demonstration of conditional gate operation using superconducting charge qubits. Nature
**2003**, 425, 941–944. [Google Scholar] [CrossRef] - Tipsmark, A.; Dong, R.; Laghaout, A.; Marek, P.; Ježek, M.; Andersen, U.L. Experimental demonstration of a Hadamard gate for coherent state qubits. Phys. Rev. A
**2011**, 84, 050301. [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] - Larsen, M.V.; Guo, X.; Breum, C.R.; Neergaard-Nielsen, J.S.; Andersen, U.L. Deterministic multi-mode gates on a scalable photonic quantum computing platform. Nat. Phys.
**2021**, 17, 1018–1023. [Google Scholar] [CrossRef] - Rauschenbeutel, A.; Nogues, G.; Osnaghi, S.; Bertet, P.; Brune, M.; Raimond, J.M.; Haroche, S. Coherent Operation of a Tunable Quantum Phase Gate in Cavity QED. Phys. Rev. Lett.
**1999**, 83, 5166–5169. [Google Scholar] [CrossRef] - Lemr, K.; Černoch, A.; Soubusta, J.; Kieling, K.; Eisert, J.; Dušek, M. Experimental Implementation of the Optimal Linear-Optical Controlled Phase Gate. Phys. Rev. Lett.
**2011**, 106, 013602. [Google Scholar] [CrossRef] - Heuck, M.; Jacobs, K.; Englund, D.R. Controlled-Phase Gate Using Dynamically Coupled Cavities and Optical Nonlinearities. Phys. Rev. Lett.
**2020**, 124, 160501. [Google Scholar] [CrossRef] - Leonhardt, U. Quantum-State Tomography and Discrete Wigner Function. Phys. Rev. Lett.
**1995**, 74, 4101–4105. [Google Scholar] [CrossRef] - Gross, D.; Liu, Y.K.; Flammia, S.T.; Becker, S.; Eisert, J. Quantum State Tomography via Compressed Sensing. Phys. Rev. Lett.
**2010**, 105, 150401. [Google Scholar] [CrossRef] - Rambach, M.; Qaryan, M.; Kewming, M.; Ferrie, C.; White, A.G.; Romero, J. Robust and Efficient High-Dimensional Quantum State Tomography. Phys. Rev. Lett.
**2021**, 126, 100402. [Google Scholar] [CrossRef] [PubMed] - Roa, L.; Groiseau, C. Probabilistic teleportation without loss of information. Phys. Rev. A
**2015**, 91, 012344. [Google Scholar] [CrossRef] - Yu, C.S.; Song, H.S.; Wang, Y.H. Remote preparation of a qudit using maximally entangled states of qubits. Phys. Rev. A
**2006**, 73, 022340. [Google Scholar] [CrossRef] - Zeng, B.; Zhang, P. Remote-state preparation in higher dimension and the parallelizable manifold S
^{n−1}. Phys. Rev. A**2002**, 65, 022316. [Google Scholar] [CrossRef] - Dada, A.C.; Leach, J.; Buller, G.S.; Padgett, M.J.; Andersson, E. Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities. Nat. Phys.
**2011**, 7, 677–680. [Google Scholar] [CrossRef] - Erhard, M.; Fickler, R.; Krenn, M.; Zeilinger, A. Twisted photons: New quantum perspectives in high dimensions. Light Sci. Appl.
**2018**, 7, 17146. [Google Scholar] [CrossRef] [PubMed] - Erhard, M.; Krenn, M.; Zeilinger, A. Advances in high-dimensional quantum entanglement. Nat. Rev. Phys.
**2020**, 2, 365–381. [Google Scholar] [CrossRef]

**Figure 1.**The success probability as a function of the entanglement coefficient $\theta $ in conventional quantum communication protocols. It is presented that the probability of success can reach 1 only when $\theta =\frac{\pi}{4}$, i.e., the quantum channel is an ideal maximally entangled quantum state ($\left|\alpha \right|=\left|\beta \right|=\frac{1}{\sqrt{2}}$).

**Figure 2.**A schematic of deterministic RSP using a non-maximally entangled channel. The entanglement resource (ES) is employed to produce the polarization-entangled state of photons A and B. The polarization state of photon C is set on ${|H\rangle}_{C}$ initially. Alice first performs a C-NOT gate ${C}_{AC}^{\left(d\right)}$ on photons A and C. One extra ancilla photon $|H\rangle +|V\rangle $ is needed beyond the control and target photons to realize a deterministic C-NOT operation. Secondly, Alice applies a single-photon unitary transformation ${U}_{A}^{\left(d\right)}$ on photon A. Thirdly, she performs C-NOT gates ${C}_{AB}^{\left(d\right)}$ and ${C}_{BA}^{\left(d\right)}$ on photons A and B in succession. After that, Alice distributes photon B to Bob. Finally, the projective measurement is implemented on photons A and C, and the measurement results are sent to Bob. Bob performs the corresponding operations on photon B according to the measurement results. The quantum state of photon B is analyzed via QST to test the quality of this communication scheme. Note that CNG represents the C-NOT gate, PBS represents the polarizing beam splitter, QWP represents the quarter-wave plate, and HWP represents the half-wave plate.

**Figure 3.**The success probability is a function of entanglement coefficients of the quantum channel. As the quantum channel degenerates from a maximally entangled state $(|\alpha |=|\beta |=\frac{1}{\sqrt{2}})$ to a partially entangled one $\left(\right|\alpha |\ne |\beta \left|\right)$, the success probability of conventional schemes is decayed accordingly, but the success probability of our perfect transport scheme remains 1 and constant.

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Xin, X.; He, S.; Li, Y.; Li, C.
Nontraditional Deterministic Remote State Preparation Using a Non-Maximally Entangled Channel without Additional Quantum Resources. *Entropy* **2023**, *25*, 768.
https://doi.org/10.3390/e25050768

**AMA Style**

Xin X, He S, Li Y, Li C.
Nontraditional Deterministic Remote State Preparation Using a Non-Maximally Entangled Channel without Additional Quantum Resources. *Entropy*. 2023; 25(5):768.
https://doi.org/10.3390/e25050768

**Chicago/Turabian Style**

Xin, Xuanxuan, Shiwen He, Yongxing Li, and Chong Li.
2023. "Nontraditional Deterministic Remote State Preparation Using a Non-Maximally Entangled Channel without Additional Quantum Resources" *Entropy* 25, no. 5: 768.
https://doi.org/10.3390/e25050768