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,{{Cite journal |last1=Goyeneche |first1=Dardo |last2=Alsina |first2=Daniel |last3=Latorre |first3=José I. |last4=Riera |first4=Arnau |last5=Życzkowski |first5=Karol |date=2015-09-15 |title=Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices |url=https://link.aps.org/doi/10.1103/PhysRevA.92.032316 |journal=Physical Review A |volume=92 |issue=3 |pages=032316 |doi=10.1103/PhysRevA.92.032316|arxiv=1506.08857 |bibcode=2015PhRvA..92c2316G |hdl=1721.1/98529 |s2cid=13948915 |hdl-access=free }} discrete AdS/CFT correspondence,{{Cite journal |last1=Pastawski |first1=Fernando |last2=Yoshida |first2=Beni |last3=Harlow |first3=Daniel |last4=Preskill |first4=John |date=2015-06-23 |title=Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence |url=https://doi.org/10.1007/JHEP06(2015)149 |journal=Journal of High Energy Physics |language=en |volume=2015 |issue=6 |pages=149 |doi=10.1007/JHEP06(2015)149 |arxiv=1503.06237 |bibcode=2015JHEP...06..149P |s2cid=256004738 |issn=1029-8479}} AdS/CMT correspondenc e, 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.{{Cite web |last1=Huber |first1=F. |last2=Wyderka |first2=N. |title=Table of AME states |url=https://www.tp.nt.uni-siegen.de/+fhuber/ame.html}}{{Cite journal |last1=Huber |first1=Felix |last2=Eltschka |first2=Christopher |last3=Siewert |first3=Jens |last4=Gühne |first4=Otfried |date=2018-04-27 |title=Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity |url=https://iopscience.iop.org/article/10.1088/1751-8121/aaade5 |journal=Journal of Physics A: Mathematical and Theoretical |volume=51 |issue=17 |pages=175301 |doi=10.1088/1751-8121/aaade5 |issn=1751-8113|arxiv=1708.06298 |bibcode=2018JPhA...51q5301H |s2cid=12071276 }}

The existence of the AME state can be transformed into the existence of the solution for a specific quantum marginal problem.{{Cite journal |last1=Yu |first1=Xiao-Dong |last2=Simnacher |first2=Timo |last3=Wyderka |first3=Nikolai |last4=Nguyen |first4=H. Chau |last5=Gühne |first5=Otfried |date=2021-02-12 |title=A complete hierarchy for the pure state marginal problem in quantum mechanics |journal=Nature Communications |language=en |volume=12 |issue=1 |pages=1012 |doi=10.1038/s41467-020-20799-5 |pmid=33579935 |issn=2041-1723|pmc=7881147 |arxiv=2008.02124 |bibcode=2021NatCo..12.1012Y }}

The AME can also be used to build a kind of quantum error-correcting code called holographic error-correcting code.{{Cite book |title="Holographic code", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. |chapter=Holographic code |date=2022 |chapter-url=https://errorcorrectionzoo.org/c/holographic}}{{Cite journal |last1=Pastawski |first1=Fernando |last2=Preskill |first2=John |date=2017-05-15 |title=Code Properties from Holographic Geometries |url=https://link.aps.org/doi/10.1103/PhysRevX.7.021022 |journal=Physical Review X |volume=7 |issue=2 |pages=021022 |doi=10.1103/PhysRevX.7.021022|arxiv=1612.00017 |bibcode=2017PhRvX...7b1022P |s2cid=44236798 }}

==References==

Category:Quantum information science

Category:Quantum states