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Absolutely maximally entangled state

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Quantum Information Science

The absolutely maximally entangled (AME) state is a concept in quantum information science, which has many applications in quantum error-correcting code, discrete AdS/CFT correspondence, AdS/CMT correspondence, and more. It is the multipartite generalization of the bipartite maximally entangled state.

Definition

The bipartite maximally entangled state | ψ ⟩ A B {\displaystyle |\psi \rangle _{AB}} is the one for which the reduced density operators are maximally mixed, i.e., ρ A = ρ B = I / d {\displaystyle \rho _{A}=\rho _{B}=I/d} . Typical examples are Bell states.

A multipartite state | ψ ⟩ {\displaystyle |\psi \rangle } of a system S {\displaystyle S} is called absolutely maximally entangled if for any bipartition A | B {\displaystyle A|B} of S {\displaystyle S} , the reduced density operator is maximally mixed ρ A = ρ B = I / d {\displaystyle \rho _{A}=\rho _{B}=I/d} , where d = min { d A , d B } {\displaystyle d=\min\{d_{A},d_{B}\}} .

Property

The AME state does not always exist; in some given local dimension and number of parties, there is no AME state. There is a list of AME states in low dimensions created by Huber and Wyderka.

The existence of the AME state can be transformed into the existence of the solution for a specific quantum marginal problem.

The AME can also be used to build a kind of quantum error-correcting code called holographic error-correcting code.

References

  1. Goyeneche, Dardo; Alsina, Daniel; Latorre, José I.; Riera, Arnau; Życzkowski, Karol (2015-09-15). "Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices". Physical Review A. 92 (3): 032316. arXiv:1506.08857. Bibcode:2015PhRvA..92c2316G. doi:10.1103/PhysRevA.92.032316. hdl:1721.1/98529. S2CID 13948915.
  2. ^ Pastawski, Fernando; Yoshida, Beni; Harlow, Daniel; Preskill, John (2015-06-23). "Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence". Journal of High Energy Physics. 2015 (6): 149. arXiv:1503.06237. Bibcode:2015JHEP...06..149P. doi:10.1007/JHEP06(2015)149. ISSN 1029-8479. S2CID 256004738.
  3. Huber, F.; Wyderka, N. "Table of AME states".
  4. Huber, Felix; Eltschka, Christopher; Siewert, Jens; Gühne, Otfried (2018-04-27). "Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity". Journal of Physics A: Mathematical and Theoretical. 51 (17): 175301. arXiv:1708.06298. Bibcode:2018JPhA...51q5301H. doi:10.1088/1751-8121/aaade5. ISSN 1751-8113. S2CID 12071276.
  5. Yu, Xiao-Dong; Simnacher, Timo; Wyderka, Nikolai; Nguyen, H. Chau; Gühne, Otfried (2021-02-12). "A complete hierarchy for the pure state marginal problem in quantum mechanics". Nature Communications. 12 (1): 1012. arXiv:2008.02124. Bibcode:2021NatCo..12.1012Y. doi:10.1038/s41467-020-20799-5. ISSN 2041-1723. PMC 7881147. PMID 33579935.
  6. "Holographic code". "Holographic code", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. 2022.
  7. Pastawski, Fernando; Preskill, John (2017-05-15). "Code Properties from Holographic Geometries". Physical Review X. 7 (2): 021022. arXiv:1612.00017. Bibcode:2017PhRvX...7b1022P. doi:10.1103/PhysRevX.7.021022. S2CID 44236798.
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