Abstract
Entanglement is one of the important resources in many quantum tasks. And the issue of high-dimensional entangled systems is intriguing. Here we consider the entanglement distribution of higher-dimensional multipartite systems. Specifically, we show that the n-qudit pure states satisfy the entanglement polygon inequality (EPI) in terms of geometrical entanglement measure, then we offer an entanglement indicator for three-qubit pure states based on the geometrical entanglement measure. At last, we show that the EPI is not generally valid for pure states in higher-dimensional systems in terms of negativity. Nevertheless, the above inequality is valid for higher-dimensional systems in terms of concurrence.
Similar content being viewed by others
Data availability statement
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)
C.H. Bennett, S.J. Wiesner, Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69(20), 2881 (1992)
C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70(13), 1895 (1993)
Y. Shimoni, D. Shapira, O. Biham, Entangled quantum states generated by Shor’s factoring algorithm. Phys. Rev. A 72(6), 062308 (2005)
V. Coffman, J. Kundu, W.K. Wootters, Distributed entanglement. Phys. Rev. A 61(5), 052306 (2000)
T.J. Osborne, F. Verstraete, General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96(22), 220503 (2006)
M. Christandl, A. Winter, “squashed entanglement’’: an additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)
Y.-K. Bai, Y.-F. Xu, Z. Wang, General monogamy relation for the entanglement of formation in multiqubit systems. Phys. Rev. Lett. 113(10), 100503 (2014)
Y. Luo, T. Tian, L.-H. Shao, Y. Li, General monogamy of tsallis q-entropy entanglement in multiqubit systems. Phys. Rev. A 93(6), 062340 (2016)
W. Song, Y.-K. Bai, M. Yang, M. Yang, Z.-L. Cao, General monogamy relation of multiqubit systems in terms of aared rényi-\(\alpha \) entanglement. Phys. Rev. A 93(2), 022306 (2016)
Y.-C. Ou, Violation of monogamy inequality for higher-dimensional objects. Phys. Rev. A 75(3), 034305 (2007)
N. Friis, G. Vitagliano, M. Malik, M. Huber, Entanglement certification from theory to experiment. Nat. Rev. Phys. 1(1), 72–87 (2019)
N.J. Cerf, M. Bourennane, A. Karlsson, N. Gisin, Security of quantum key distribution using d-level systems. Phys. Rev. Lett. 88(12), 127902 (2002)
M. Mirhosseini, O.S. Magana-Loaiza, M.N.O. Sullivan, B. Rodenburg, M. Malik, M.P. Lavery, M.J. Padgett, D.J. Gauthier, R.W. Boyd, High-dimensional quantum cryptography with twisted light. New J. Phys. 17(3), 033033 (2015)
D. Cozzolino, B. Da Lio, D. Bacco, L.K. Oxenlowe, High-dimensional quantum communication: benefits, progress, and future challenges. Adv. Quantum Technol. 2(12), 1900038 (2019)
Y. Wang, Z. Hu, B.C. Sanders, S. Kais, Qudits and high-dimensional quantum computing. Front. Phys. 8, 589504 (2020)
M. Erhard, M. Krenn, A. Zeilinger, Advances in high-dimensional quantum entanglement. Nat. Rev. Phys. 2(7), 365–381 (2020)
M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, T. Monz, A universal qudit quantum processor with trapped ions. Nat. Phys. 18(9), 1053–1057 (2022)
G. Gour, Y. Guo, Monogamy of entanglement without inequalities. Quantum 2, 81 (2018)
X. Shi, L. Chen, M. Hu, Multilinear monogamy relations for multiqubit states. Phys. Rev. A 104(1), 012426 (2021)
X.-N. Zhu, G. Bao, Z.-X. **, S.-M. Fei, Monogamy of entanglement for tripartite systems. Phys. Rev. A 107(5), 052404 (2023)
C. Eltschka, J. Siewert, Distribution of entanglement and correlations in all finite dimensions. Quantum 2, 64 (2018)
S. Imai, N. Wyderka, A. Ketterer, O. Gühne, Bound entanglement from randomized measurements. Phys. Rev. Lett. 126(15), 150501 (2021)
A.A. Klyachko, Quantum marginal problem and n-representability, in Journal of Physics: Conference Series, vol. 36, no. 1 (IOP Publishing, 2006), p. 72
E. Haapasalo, T. Kraft, N. Miklin, R. Uola, Quantum marginal problem and incompatibility. Quantum 5, 476 (2021)
X.-D. Yu, T. Simnacher, N. Wyderka, H.C. Nguyen, O. Gühne, A complete hierarchy for the pure state marginal problem in quantum mechanics. Nat. Commun. 12(1), 1012 (2021)
X.-F. Qian, M.A. Alonso, J.H. Eberly, Entanglement polygon inequality in qubit systems. New J. Phys. 20(6), 063012 (2018)
X. Yang, Y.-H. Yang, M.-X. Luo, Entanglement polygon inequality in qudit systems. Phys. Rev. A 105(6), 062402 (2022)
T.-C. Wei, P.M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68(4), 042307 (2003)
G. Vidal, R.F. Werner, Computable measure of entanglement. Phys. Rev. A 65(3), 032314 (2002)
W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245 (1998)
X. Yang, M.-X. Luo, Y.-H. Yang, S.-M. Fei, Parametrized entanglement monotone. Phys. Rev. A 103(5), 052423 (2021)
K.M. Audenaert, Subadditivity of q-entropies for q> 1. J. Math. Phys. 48(8), 083507 (2007)
X.-N. Zhu, S.-M. Fei, Entanglement monogamy relations of qubit systems. Phys. Rev. A 90(2), 024304 (2014)
A. Acin, A. Andrianov, L. Costa, E. Jane, J. Latorre, R. Tarrach, Generalized schmidt decomposition and classification of three-quantum-bit states. Phys. Rev. Lett. 85(7), 1560 (2000)
J. San Kim, B.C. Sanders, Generalized w-class state and its monogamy relation, in Journal of Physics A: Mathematical and Theoretical, vol. 41, no. 49 (2008) p. 495301
X. Shi, L. Chen, Monogamy relations for the generalized w-class states beyond qubits. Phys. Rev. A 101(3), 032344 (2020)
L.-M. Lai, S.-M. Fei, Z.-X. Wang, Tighter monogamy and polygamy relations for a superposition of the generalized w-class state and vacuum. J. Phys. A: Math. Theor. 54(42), 425301 (2021)
Acknowledgements
X. S. was supported by the Fundamental Research Funds for the Central Universities (Grant No. ZY2306), and Funds of College of Information Science and Technology, Bei**g University of Chemical Technology (Grant No. 0104/11170044115)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shi, X. Entanglement polygon inequalities for pure states in qudit systems. Eur. Phys. J. Plus 138, 768 (2023). https://doi.org/10.1140/epjp/s13360-023-04399-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-023-04399-y