# Optimal Multi-port-based Teleportation Schemes

Marek Mozrzymas1, Michał Studziński2, and Piotr Kopszak1

1Institute for Theoretical Physics, University of Wrocław, 50-204 Wrocław, Poland
2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, University of Gdańsk, 80-952 Gdańsk, Poland

### Abstract

In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on global parameters describing the problem, such as the number of ports $N$, the number of teleported systems $k$, and local dimension $d$. It allows us to show square improvement in the number of ports with respect to the non-optimal case. We also show that the number of teleported systems can grow when the number $N$ of ports increases as $o(N)$ still giving high efficiency. In the deterministic case, we connect entanglement fidelity with the maximal eigenvalue of a generalised teleportation matrix. In both cases the optimal set of measurements and the optimal state shared between sender and receiver is presented. All the results are obtained by formulating and solving primal and dual SDP problems, which due to existing symmetries can be solved analytically. We use extensively tools from representation theory and formulate new results that could be of the separate interest for the potential readers.

### ► References

[1] Ron M. Adin and Yuval Roichman. Enumeration of standard young tableaux, 2014.

[2] A. C. Aitken. Xxvi.—the monomial expansion of determinantal symmetric functions. Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences, 61 (3): 300–310, 1943. 10.1017/​S0080454100006312.
https:/​/​doi.org/​10.1017/​S0080454100006312

[3] Leonardo Banchi, Jason Pereira, Seth Lloyd, and Stefano Pirandola. Convex optimization of programmable quantum computers. npj Quantum Information, 6 (1): 42, May 2020. ISSN 2056-6387. 10.1038/​s41534-020-0268-2. URL https:/​/​doi.org/​10.1038/​s41534-020-0268-2.
https:/​/​doi.org/​10.1038/​s41534-020-0268-2

[4] Salman Beigi and Robert König. Simplified instantaneous non-local quantum computation with applications to position-based cryptography. New Journal of Physics, 13 (9): 093036, 2011. ISSN 1367-2630. 10.1088/​1367-2630/​13/​9/​093036. URL http:/​/​stacks.iop.org/​1367-2630/​13/​i=9/​a=093036.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​9/​093036
http:/​/​stacks.iop.org/​1367-2630/​13/​i=9/​a=093036

[5] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70 (13): 1895–1899, March 1993. 10.1103/​PhysRevLett.70.1895. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.70.1895.
https:/​/​doi.org/​10.1103/​PhysRevLett.70.1895

[6] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu. Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels. Physical Review Letters, 80 (6): 1121–1125, February 1998. 10.1103/​PhysRevLett.80.1121. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.80.1121.
https:/​/​doi.org/​10.1103/​PhysRevLett.80.1121

[7] Harry Buhrman, Łukasz Czekaj, Andrzej Grudka, Michał Horodecki, Paweł Horodecki, Marcin Markiewicz, Florian Speelman, and Sergii Strelchuk. Quantum communication complexity advantage implies violation of a Bell inequality. Proceedings of the National Academy of Sciences, 113 (12): 3191–3196, March 2016. ISSN 0027-8424, 1091-6490. 10.1073/​pnas.1507647113. URL http:/​/​www.pnas.org/​content/​113/​12/​3191.
https:/​/​doi.org/​10.1073/​pnas.1507647113
http:/​/​www.pnas.org/​content/​113/​12/​3191

[8] Matthias Christandl, Felix Leditzky, Christian Majenz, Graeme Smith, Florian Speelman, and Michael Walter. Asymptotic performance of port-based teleportation. Communications in Mathematical Physics, Nov 2020. ISSN 1432-0916. 10.1007/​s00220-020-03884-0. URL https:/​/​doi.org/​10.1007/​s00220-020-03884-0.
https:/​/​doi.org/​10.1007/​s00220-020-03884-0

[9] W. Feit. The degree formula for the skew-representations of the symmetric group. Proceedings of the American Mathematical Society, 4 (5): 740–744, 1953. ISSN 00029939, 10886826. 10.2307/​2032406. URL http:/​/​www.jstor.org/​stable/​2032406.
https:/​/​doi.org/​10.2307/​2032406
http:/​/​www.jstor.org/​stable/​2032406

[10] W. Fulton and J. Harris. Representation Theory - A first Course. Springer-Verlag, New York, 1991.

[11] Daniel Gottesman and Isaac L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390–393, November 1999. ISSN 0028-0836. 10.1038/​46503. URL http:/​/​www.nature.com/​nature/​journal/​v402/​n6760/​abs/​402390a0.html.
https:/​/​doi.org/​10.1038/​46503
http:/​/​www.nature.com/​nature/​journal/​v402/​n6760/​abs/​402390a0.html

[12] D. Gross and J. Eisert. Novel Schemes for Measurement-Based Quantum Computation. Physical Review Letters, 98 (22): 220503, May 2007. 10.1103/​PhysRevLett.98.220503. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.98.220503.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.220503

[13] Satoshi Ishizaka and Tohya Hiroshima. Asymptotic Teleportation Scheme as a Universal Programmable Quantum Processor. Physical Review Letters, 101 (24): 240501, December 2008. 10.1103/​PhysRevLett.101.240501. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.101.240501.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.240501

[14] Satoshi Ishizaka and Tohya Hiroshima. Quantum teleportation scheme by selecting one of multiple output ports. Physical Review A, 79 (4): 042306, April 2009. 10.1103/​PhysRevA.79.042306. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.79.042306.
https:/​/​doi.org/​10.1103/​PhysRevA.79.042306

[15] Richard Jozsa. An introduction to measurement based quantum computation, 2005. URL https:/​/​arxiv.org/​abs/​quant-ph/​0508124.
arXiv:quant-ph/0508124

[16] Piotr Kopszak, Marek Mozrzymas, Michał Studziński, and Michał Horodecki. Multiport based teleportation – transmission of a large amount of quantum information, 2021. URL https:/​/​arxiv.org/​abs/​2008.00856.
arXiv:2008.00856

[17] Felix Leditzky. Optimality of the pretty good measurement for port-based teleportation, 2020. URL https:/​/​arxiv.org/​abs/​2008.11194.
arXiv:2008.11194

[18] Maciej Lewenstein and Anna Sanpera. Separability and entanglement of composite quantum systems. Phys. Rev. Lett., 80: 2261–2264, Mar 1998. 10.1103/​PhysRevLett.80.2261. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.80.2261.
https:/​/​doi.org/​10.1103/​PhysRevLett.80.2261

[19] Marek Mozrzymas, Michał Horodecki, and Michał Studziński. Structure and properties of the algebra of partially transposed permutation operators. Journal of Mathematical Physics, 55 (3): 032202, March 2014. ISSN 0022-2488, 1089-7658. 10.1063/​1.4869027. URL http:/​/​scitation.aip.org/​content/​aip/​journal/​jmp/​55/​3/​10.1063/​1.4869027.
https:/​/​doi.org/​10.1063/​1.4869027

[20] Marek Mozrzymas, Michał Studziński, and Michał Horodecki. A simplified formalism of the algebra of partially transposed permutation operators with applications. Journal of Physics A Mathematical General, 51 (12): 125202, Mar 2018a. 10.1088/​1751-8121/​aaad15.

[21] Marek Mozrzymas, Michał Studziński, Sergii Strelchuk, and Michał Horodecki. Optimal port-based teleportation. New Journal of Physics, 20 (5): 053006, May 2018b. 10.1088/​1367-2630/​aab8e7.
https:/​/​doi.org/​10.1088/​1367-2630/​aab8e7

[22] M. A. Nielsen and Isaac L. Chuang. Programmable quantum gate arrays. Phys. Rev. Lett., 79: 321–324, Jul 1997. 10.1103/​PhysRevLett.79.321. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.79.321.
https:/​/​doi.org/​10.1103/​PhysRevLett.79.321

[23] S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein. Advances in quantum teleportation. Nature Photonics, 9 (10): 641–652, October 2015. ISSN 1749-4885. 10.1038/​nphoton.2015.154. URL http:/​/​www.nature.com/​nphoton/​journal/​v9/​n10/​full/​nphoton.2015.154.html.
https:/​/​doi.org/​10.1038/​nphoton.2015.154
http:/​/​www.nature.com/​nphoton/​journal/​v9/​n10/​full/​nphoton.2015.154.html

[24] Stefano Pirandola, Riccardo Laurenza, Cosmo Lupo, and Jason L. Pereira. Fundamental limits to quantum channel discrimination. npj Quantum Information, 5: 50, Jun 2019. 10.1038/​s41534-019-0162-y.
https:/​/​doi.org/​10.1038/​s41534-019-0162-y

[25] Robert Raussendorf and Hans J. Briegel. A One-Way Quantum Computer. Physical Review Letters, 86 (22): 5188–5191, May 2001. 10.1103/​PhysRevLett.86.5188. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.86.5188.
https:/​/​doi.org/​10.1103/​PhysRevLett.86.5188

[26] Sergii Strelchuk, Michał Horodecki, and Jonathan Oppenheim. Generalized Teleportation and Entanglement Recycling. Physical Review Letters, 110 (1): 010505, January 2013. 10.1103/​PhysRevLett.110.010505. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.110.010505.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.010505

[27] Michał Studziński, Michał Horodecki, and Marek Mozrzymas. Commutant structure of $u^{\otimes(n- 1)}\otimes u^{\ast}$ transformations. Journal of Physics A: Mathematical and Theoretical, 46 (39): 395303, sep 2013. 10.1088/​1751-8113/​46/​39/​395303. URL https:/​/​doi.org/​10.1088/​1751-8113/​46/​39/​395303.
https:/​/​doi.org/​10.1088/​1751-8113/​46/​39/​395303

[28] Michał Studziński, Sergii Strelchuk, Marek Mozrzymas, and Michał Horodecki. Port-based teleportation in arbitrary dimension. Scientific Reports, 7: 10871, Sep 2017. 10.1038/​s41598-017-10051-4.
https:/​/​doi.org/​10.1038/​s41598-017-10051-4

[29] Michał Studziński, Marek Mozrzymas, Piotr Kopszak, and Michał Horodecki. Efficient multi-port teleportation schemes. 2020. URL https:/​/​arxiv.org/​abs/​2008.00984.
arXiv:2008.00984

[30] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. Event-ready-detectors'' Bell experiment via entanglement swapping. Physical Review Letters, 71 (26): 4287–4290, December 1993. 10.1103/​PhysRevLett.71.4287. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.71.4287.
https:/​/​doi.org/​10.1103/​PhysRevLett.71.4287

### Cited by

[1] Piotr Kopszak, Marek Mozrzymas, Michał Studziński, and Michał Horodecki, "Multiport based teleportation – transmission of a large amount of quantum information", Quantum 5, 576 (2021).

[2] Marco Túlio Quintino, "Quantum teleportation beyond its standard form: Multi-Port-Based Teleportation", Quantum Views 5, 56 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-20 06:49:06). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2022-05-20 06:49:06).