Skip to main content
SpringerLink
Log in

Some Remarks on Super-Gram Operators for General Bipartite Quantum States

  • Conference paper
  • First Online:
Parallel Processing and Applied Mathematics (PPAM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13827))

Abstract

The Gramian matrices approach to study certain aspects of quantum entanglement contained in the bipartite pure quantum states is being extended to the level of a general quantum bipartite states. The corresponding Gram matrices, called here super-gram matrices are being constructed over the Hilbert-Schmidt structure build on the Hilbert space of pure states. The main result is the extension of the widely known realignment criterion to the level of super-operators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Dalcin, L., Fang, Y.-L.L.: mpi4py: status update after 12 years of development. Comput. Sci. Eng. 23(4), 47–54 (2021). https://doi.org/10.1109/MCSE.2021.3083216

    Article  Google Scholar 

  2. Decheng, F., Jon, S., Pang, C., Dong, W., Won, C.: Improved quantum clustering analysis based on the weighted distance and its application. Heliyon 4(11), e00984 (2018). https://doi.org/10.1016/j.heliyon.2018.e00984

    Article  Google Scholar 

  3. Devidas, S., Subba Rao, Y.V., Rukma Rekha, N.: A decentralized group signature scheme for privacy protection in a blockchain. Int. J. Appl. Math. Comput. Sci. 31(2), 353–364 (2021). https://doi.org/10.34768/amcs-2021-0024

    Article  MathSciNet  MATH  Google Scholar 

  4. Farahbakhsh, A., Feng, C.: Opportunistic routing in quantum networks. arXiv preprint (2022). https://doi.org/10.48550/arXiv.2205.08479

  5. Faridi, A.R., Masood, F., Shamsan, A.H.T., Luqman, M., Salmony, M.Y.: Blockchain in the quantum world. Int. J. Adv. Comput. Sci. Appl. 13(1), 542–552 (2022). https://doi.org/10.14569/IJACSA.2022.0130167

    Article  Google Scholar 

  6. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC), pp. 212–219 (1996). https://doi.org/10.1145/237814.237866

  7. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865

    Article  MathSciNet  MATH  Google Scholar 

  8. Gielerak, R., Sawerwain, M.: A Gramian approach to entanglement in bipartite finite dimensional systems: the case of pure states. Quantum Inf. Comput. 20(13 &14), 1081–1108 (2020). https://doi.org/10.26421/QIC20.13-1

    Article  MathSciNet  Google Scholar 

  9. Gielerak, R., Sawerwain, M., Wiśniewska, J., Wróblewski, M.: EntDetector: entanglement detecting toolbox for bipartite quantum states. In: Paszynski, M., Kranzlmüller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds.) ICCS 2021. LNCS, vol. 12747, pp. 113–126. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77980-1_9

    Chapter  Google Scholar 

  10. Gielerak, R., Sawerwain, M.: Gramian and super-gramian approach to infinite-dimensional quantum states. In preparation (2022)

    Google Scholar 

  11. Gielerak, R., Wiśniewska, J., Sawerwain, M., Wróblewski, M., Korbicz, J.: Classical computer assisted analysis of small multiqudit systems. IEEE Access 10, 82636–82655 (2022). https://doi.org/10.1109/ACCESS.2022.3196656

    Article  Google Scholar 

  12. Kuptsov, L.P. : Gram matrix entry. In: Hazewinkel, M. (ed.) Encyclopaedia of Mathematics: Coproduct - Hausdorff - Young Inequalities, p. 861. Springer, New York (1995). https://doi.org/10.1007/978-1-4899-3795-7

  13. Kaliszewska, A., Syga, M.: A comprehensive study of clustering a class of 2D shapes. Int. J. Appl. Math. Comput. Sci. 32(1), 95–109 (2022). https://doi.org/10.34768/amcs-2022-0008

    Article  MATH  Google Scholar 

  14. Klyachko, A.: Quantum marginal problem and representations of the symmetric group. arXiv preprint (2004). https://doi.org/10.48550/arXiv.quant-ph/0409113

  15. Kopszak, P., Mozrzymas, M., Studziński, M., Horodecki, M.: Multiport based teleportation - transmission of a large amount of quantum information. Quantum 5, 576 (2021). https://doi.org/10.22331/q-2021-11-11-576

    Article  Google Scholar 

  16. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  17. van Meter, R.: Quantum Networking. Wiley, Hoboken (2014). https://doi.org/10.1002/9781118648919

    Book  MATH  Google Scholar 

  18. Rajan, D., Visser, M.: Quantum blockchain using entanglement in time. Quantum Rep. 1(1), 3–11 (2019). https://doi.org/10.3390/quantum1010002

    Article  Google Scholar 

  19. Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4, 219–239 (2005). https://doi.org/10.1007/s11128-005-5664-1

    Article  MathSciNet  MATH  Google Scholar 

  20. Sawerwain, M., Wiśniewska, J., Wróblewski, M., Gielerak, R.: GitHub repository for EntDectector package (2022). https://github.com/qMSUZ/EntDetector

  21. Schuld, M., Petruccione, F.: Prospects for near-term quantum machine learning. In: Supervised Learning with Quantum Computers. QST, pp. 273–279. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96424-9_9

    Chapter  MATH  Google Scholar 

  22. Schuld, M., Sinayskiy, I., Petruccione, F.: An introduction to quantum machine learning. Contemp. Phys. 56(2), 172–185 (2014)

    Article  MATH  Google Scholar 

  23. Schuld, M., Sinayskiy, I., Petruccione, F.: Quantum computing for pattern classification. In: Pham, D.-N., Park, S.-B. (eds.) PRICAI 2014. LNCS (LNAI), vol. 8862, pp. 208–220. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13560-1_17

    Chapter  Google Scholar 

  24. da Silva, A.J., Ludermir, T.B., de Oliveira, W.R.: Quantum perceptron over a field and neural network architecture selection in a quantum computer. Neural Netw. 76, 55–64 (2016). https://doi.org/10.1016/j.neunet.2016.01.002

    Article  MATH  Google Scholar 

  25. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172

    Article  MathSciNet  MATH  Google Scholar 

  26. Wiebe, N., Kapoor, A., Svore, K.M.: Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning. Quantum Inf. Comput. 15(3 &4), 318–358 (2015). https://doi.org/10.26421/QIC15.3-4-7

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank for useful discussions with the Q-INFO group at the Institute of Control and Computation Engineering (ISSI) of the University of Zielona Góra, Poland. We would like also to thank to anonymous referees for useful comments on the preliminary version of the chapter. The numerical results were done using the hardware and software available at the “GPU \(\mu \)-Lab” located at the Institute of Control and Computation Engineering of the University of Zielona Góra, Poland.

Author information

Authors and Affiliations

  1. Institute of Control and Computation Engineering, University of Zielona Góra, Licealna 9, 65-417, Zielona Góra, Poland

    Roman Gielerak & Marek Sawerwain

Authors
  1. Roman Gielerak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marek Sawerwain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marek Sawerwain .

Editor information

Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Knoxville, TN, USA

    Jack Dongarra

  3. University of Southern California, Marina del Rey, CA, USA

    Ewa Deelman

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gielerak, R., Sawerwain, M. (2023). Some Remarks on Super-Gram Operators for General Bipartite Quantum States. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2022. Lecture Notes in Computer Science, vol 13827. Springer, Cham. https://doi.org/10.1007/978-3-031-30445-3_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-30445-3_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-30444-6

  • Online ISBN: 978-3-031-30445-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation