Absolutely maximally entangled state explained

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, and more. It is the multipartite generalization of the bipartite maximally entangled state.

Definition

The bipartite maximally entangled state

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

A multipartite state

|\psi\rangle of a system S is called absolutely maximally entangled if for any bipartition A|B of S , the reduced density operator is maximally mixed \rho _{A=\rho} _{B=I/d} , where 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.^{[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.^{[6] [7]}

Notes and 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 . 10.1103/PhysRevA.92.032316. 1506.08857 . 2015PhRvA..92c2316G . 1721.1/98529 . 13948915 . free .
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 . en . 2015 . 6 . 149 . 10.1007/JHEP06(2015)149 . 1503.06237 . 2015JHEP...06..149P . 256004738 . 1029-8479.
3. Web site: 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 . 10.1088/1751-8121/aaade5 . 1751-8113. 1708.06298 . 2018JPhA...51q5301H . 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 . en . 12 . 1 . 1012 . 10.1038/s41467-020-20799-5 . 33579935 . 2041-1723. 7881147 . 2008.02124 . 2021NatCo..12.1012Y .
6. Book: "Holographic code", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. . Holographic code . 2022 . https://errorcorrectionzoo.org/c/holographic.
7. Pastawski . Fernando . Preskill . John . 2017-05-15 . Code Properties from Holographic Geometries . Physical Review X . 7 . 2 . 021022 . 10.1103/PhysRevX.7.021022. 1612.00017 . 2017PhRvX...7b1022P . 44236798 .