Quantum Magic Squares

In a new paper in the Journal of Mathematical Physics, ESQ faculty member, quantum physicist Gemma De las Cuevas (Department of Theoretical Physics, Innsbruck University) together with mathematicians Tim Netzer and Tom Drescher (Department of Mathematics, Innsbruck University) introduced the notion of the quantum magic square, and for the first time studied in detail the properties of this quantum version of magic squares.

If a magic square contains real numbers, and every row and column amounts to 1, it is called a doubly stochastic matrix. A matrix that has 0-s everywhere except for one 1 in every column and every row is called a permutation matrix. A famous theorem claims that every doubly stochastic matrix can be shown as a convex combination of permutation matrices. Hence, permutation matrices may be said to ‘contain all the secrets’ of doubly stochastic matrices—more precisely, that the latter can be fully characterised in terms of the former.

The quantum magic square is a magic square, but instead of numbers it contains matrices. This is a non-commutative and thus quantum generalisation of a magic square. The authors show that quantum magic squares cannot be as easily characterised as their “classical” cousins. More precisely, quantum magic squares are not convex combinations of quantum permutation matrices. “They are richer and more complex”, explains Tom Drescher.

Gemma De las Cuevas and Tim Netzer stress that “the work [described in the Journal of Mathematical Physics] lies at the intersection of algebraic geometry and quantum information and showcases the benefits of interdisciplinary collaboration”.


For more information see:

Quantum magic squares: Dilations and their limitations. Gemma De las Cuevas, Tom Drescher, and Tim Netzer. Journal of Mathematical Physics 61, 111704 (2020)

Algebra Group, Department of Mathematics

Mathematical Quantum Physics Group, Department of Theoretical Physics



ESQ office
Austrian Academy of Sciences (ÖAW)
Mag.ª Isabelle Walters
Boltzmanngasse 5
1090 Vienna