Absolutely maximally entangled state
The absolutely maximally entangled (AME) state is a concept in quantum information science, which has many applications in quantum error-correcting code,[1] discrete AdS/CFT correspondence,[2] AdS/CMT correspondence,[2] and more. It is the multipartite generalization of the bipartite maximally entangled state.
Definition
The bipartite maximally entangled state [math]\displaystyle{ |\psi\rangle_{AB} }[/math] is the one for which the reduced density operators are maximally mixed, i.e., [math]\displaystyle{ \rho_A=\rho_B=I/d }[/math]. Typical examples are Bell states.
A multipartite state [math]\displaystyle{ |\psi\rangle }[/math] of a system [math]\displaystyle{ S }[/math] is called absolutely maximally entangled if for any bipartition [math]\displaystyle{ A|B }[/math] of [math]\displaystyle{ S }[/math], the reduced density operator is maximally mixed [math]\displaystyle{ \rho_A=\rho_B=I/d }[/math], where [math]\displaystyle{ d=\min\{d_A,d_B\} }[/math].
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.[3][4]
The existence of the AME state can be transformed into the existence of the solution for a specific quantum marginal problem.[5]
The AME can also be used to build a kind of quantum error-correcting code called holographic error-correcting code.[2][6][7]
References
- ↑ 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. doi:10.1103/PhysRevA.92.032316. Bibcode: 2015PhRvA..92c2316G. https://link.aps.org/doi/10.1103/PhysRevA.92.032316.
- ↑ 2.0 2.1 2.2 Pastawski, Fernando; Yoshida, Beni; Harlow, Daniel; Preskill, John (2015-06-23). "Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence" (in en). Journal of High Energy Physics 2015 (6): 149. doi:10.1007/JHEP06(2015)149. ISSN 1029-8479. Bibcode: 2015JHEP...06..149P. https://doi.org/10.1007/JHEP06(2015)149.
- ↑ Huber, F.; Wyderka, N.. "Table of AME states". https://www.tp.nt.uni-siegen.de/+fhuber/ame.html.
- ↑ 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. doi:10.1088/1751-8121/aaade5. ISSN 1751-8113. Bibcode: 2018JPhA...51q5301H. https://iopscience.iop.org/article/10.1088/1751-8121/aaade5.
- ↑ 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" (in en). Nature Communications 12 (1): 1012. doi:10.1038/s41467-020-20799-5. ISSN 2041-1723. PMID 33579935. Bibcode: 2021NatCo..12.1012Y.
- ↑ "Holographic code". "Holographic code", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.. 2022. https://errorcorrectionzoo.org/c/holographic.
- ↑ Pastawski, Fernando; Preskill, John (2017-05-15). "Code Properties from Holographic Geometries". Physical Review X 7 (2): 021022. doi:10.1103/PhysRevX.7.021022. Bibcode: 2017PhRvX...7b1022P. https://link.aps.org/doi/10.1103/PhysRevX.7.021022.
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