Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff–von Neumann theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is as follows: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here, we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.
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November 2020
Research Article|
November 16 2020
Quantum magic squares: Dilations and their limitations

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Gemma De las Cuevas
;
Gemma De las Cuevas
1
Institute for Theoretical Physics
, Technikerstr. 21a, A-6020 Innsbruck, Austria
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Tom Drescher
;
Tom Drescher
a)
2
Department of Mathematics
, Technikerstr. 13, A-6020 Innsbruck, Austria
a)Author to whom correspondence should be addressed: [email protected]
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Tim Netzer
Tim Netzer
2
Department of Mathematics
, Technikerstr. 13, A-6020 Innsbruck, Austria
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Gemma De las Cuevas
1
Tom Drescher
2,a)
Tim Netzer
2
1
Institute for Theoretical Physics
, Technikerstr. 21a, A-6020 Innsbruck, Austria
2
Department of Mathematics
, Technikerstr. 13, A-6020 Innsbruck, Austria
a)Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 61, 111704 (2020)
Article history
Received:
July 21 2020
Accepted:
October 08 2020
Connected Content
A companion article has been published:
Matrix-convex set theory reveals non-classical behavior of quantum magic squares
Citation
Gemma De las Cuevas, Tom Drescher, Tim Netzer; Quantum magic squares: Dilations and their limitations. J. Math. Phys. 1 November 2020; 61 (11): 111704. https://doi.org/10.1063/5.0022344
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