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 correspondenc e ,[ 2] and more. It is the multipartite generalization of the bipartite maximally entangled state .
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}\))
.
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]
^ 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 .
^ a b c 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 .
^ Huber, F.; Wyderka, N. "Table of AME states" .
^ 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 .
^ 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 .
^ "Holographic code" . "Holographic code", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023 . 2022.
^ 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 .