[ad_1]
Magic squares have been in the imagination of humanity for a long time. The oldest known magic square comes from China and is over 2000 years old. One of the most famous magic squares is found in the copper engraving by Albrecht Dürer Melencolia I. Another is found on the facade of the Sagrada Familia in Barcelona. A magic square is a square of numbers such that each column and row add up the same number. For example, in the magic square of the Sagrada Família each row and column is equal to 33.
If the magic square can contain real numbers and each row and column adds up to 1, then it is called a doubly stochastic matrix. A particular example would be a matrix that has 0 everywhere except a 1 in every column and every row. This is called a permutation matrix. A famous theorem says that any doubly stochastic matrix can be obtained as a convex combination of permutation matrices. In words, this means that permutation matrices “contain all the secrets” of doubly stochastic matrices – more precisely, that the latter can be fully characterized in terms of the former.
In a new article in Journal of Mathematical Physics, Tim Netzer and Tom Drescher of the Department of Mathematics and Gemma De las Cuevas of the Department of Theoretical Physics introduced the notion of a quantum magic square, which is a magic square but instead of numbers it is put into matrices. This is a non-commutative, and hence quantum, generalization of a magic square. The authors show that quantum magic squares cannot be easily characterized as their “classical” cousins. More precisely, quantum magic squares are not convex combinations of quantum permutation matrices. “They are richer and more complicated to understand,” explains Tom Drescher. “This is the general theme when studying generalizations to the non-commutative case.”
###
“The work is at the crossroads between algebraic geometry and quantum information and shows the benefits of interdisciplinary collaboration,” say Gemma De las Cuevas and Tim Netzer.
Publication: Quantum magic squares: delays and their limitations. Gemma De las Cuevas, Tom Drescher and Tim Netzer. Journal of Mathematical Physics 61, 111704 (2020) [arXiv:1912.07332]
Disclaimer: AAAS and EurekAlert! are not responsible for the accuracy of press releases published on EurekAlert! by contributing institutions or for the use of any information through the EurekAlert system.
.
[ad_2]
Source link
