Science & Technology

Quantum Magic Squares Cannot Be As Easily Characterized As Their “Classical” Cousins

Quantum Magic Squares Cannot Be As Easily Characterized As Their “Classical” Cousins

The magic of arithmetic is especially mirrored in magic squares. Just lately, quantum physicist Gemma De las Cuevas and mathematicians Tim Netzer and Tom Drescher launched the notion of the quantum magic sq., and for the primary time studied intimately the properties of this quantum model of magic squares.

Magic squares belong to the creativeness of humanity for a very long time. The oldest identified magic sq. comes from China and is over 2000 years previous. Some of the well-known magic squares will be present in Albrecht Dürer’s copper engraving Melencolia I. One other one is on the facade of the Sagrada Família in Barcelona. A magic sq. is a sq. of numbers such that each column and each row sums to the identical quantity. For instance, within the magic sq. of the Sagrada Família each row and column sums to 33.

Picture: Element from Melencolia I by Albrecht Dürer. Credit score: Nationwide Library of Spain CC BY-NC-SA 4.0

If the magic sq. can comprise actual numbers, and each row and column sums to 1, then it’s known as a doubly stochastic matrix. One explicit instance could be a matrix that has 0’s in all places apart from one 1 in each column and each row. That is known as a permutation matrix. A well-known theorem says that each doubly stochastic matrix will be obtained as a convex mixture of permutation matrices. In phrases, which means that permutation matrices “comprise all of the secrets and techniques” of doubly stochastic matrices—extra exactly, that the latter will be absolutely characterised when it comes to the previous.

In a brand new paper within the Journal of Mathematical Physics, Tim Netzer and Tom Drescher from the Division of Arithmetic and Gemma De las Cuevas from the Division of Theoretical Physics have launched the notion of the quantum magic sq., which is a magic sq. however as an alternative of numbers, one places in matrices. This can be a non-commutative, and thus quantum, generalization of a magic sq.. The authors present that quantum magic squares can’t be as simply characterised as their “classical” cousins. Extra exactly, quantum magic squares are usually not convex mixtures of quantum permutation matrices. “They’re richer and extra sophisticated to know”, explains Tom Drescher. “That is the final theme when generalizations to the non-commutative case are studied.” 

“The work is on the intersection of algebraic geometry and quantum data and showcases the advantages of interdisciplinary collaboration”, say Gemma De las Cuevas and Tim Netzer.

Reference: “Quantum magic squares: Dilations and their limitations featured” by Gemma De las Cuevas, Tom Drescher and Tim Netzer, 16 November 2020, Journal of Mathematical Physics.
DOI: 10.1063/5.0022344

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