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• W3100713059 abstract "We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>n</mml:mi></mml:math> of all QMDS codes with local dimension <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>D</mml:mi></mml:math> and distance <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>d</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:math> is bounded by <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math>. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>r</mml:mi></mml:math>-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement." @default.
• W3100713059 created "2020-11-23" @default.
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• W3100713059 date "2020-06-18" @default.
• W3100713059 modified "2023-10-10" @default.
• W3100713059 title "Quantum Codes of Maximal Distance and Highly Entangled Subspaces" @default.
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• W3100713059 doi "https://doi.org/10.22331/q-2020-06-18-284" @default.
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