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Tensoring Positive Maps on Operator Structures

Project description

Uniting mathematical problems in quantum information theory under a common theory

Many important problems in quantum information theory can be formulated in terms of how linear maps between matrix algebras behave under tensor powers. Examples include the distillation problem, local entanglement annihilation and the positive partial transpose squared conjecture. A general theory for solving them has been lacking so far. Funded by the Marie Skłodowska-Curie Actions programme, the TIPTOP project will formulate these problems in terms of abstract operator systems. The aim is to gain a deeper understanding of how the properties of operator system structures affect the properties of linear maps under tensor powers and find settings where only the trivial examples of completely positive and copositive maps stay positive under any tensor power. Ultimately, the project will explore settings where tensorisation problems become easier.

Objective

Many important problems in quantum information theory can be formulated in terms of how linear maps between matrix algebras behave under tensor powers. Examples include the distillation problem (fundamental for quantum communication), the problem of local entanglement annihilation (important for entanglement distribution in quantum networks), and the PPT squared conjecture important for (quantum key repeaters). Despite their importance for quantum communication, these problems are wide open, and no general theory is known for solving them. I realized that these problems can be formulated in the framework of abstract operator systems. Here, they correspond to characterizing which linear maps stay positive under tensor powers with respect to different operator system structures over the matrix algebras at the input and output. Completely positive maps and completely copositive maps (compositions of completely positive maps with a transposition) are always trivial examples, corresponding to known examples in quantum information theory. The question is, whether other examples exist. So far this type of problem has only been studied (indirectly) in the few special cases of operator systems over the matrix algebras corresponding to the above problems. There is a much richer theory of abstract operator systems (even over the matrix algebras) and different tensor products to combine them. In my project, I want to study such tensorization problems for other operator system structures over the matrix algebras and beyond. I want to understand how properties of these structures affect properties of linear maps under tensor powers, and find settings where only the trivial examples of completely positive and completely copositive maps stay positive under any tensor power. Finally, I aim to identify settings where tensorization problems become easier, and where I can construct examples of positive maps with properties we are currently lacking in quantum information theory.

Coordinator

UNIVERSITE LYON 1 CLAUDE BERNARD
Net EU contribution
€ 196 707,84
Address
BOULEVARD DU 11 NOVEMBRE 1918 NUM43
69622 Villeurbanne Cedex
France

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Region
Auvergne-Rhône-Alpes Rhône-Alpes Rhône
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 196 707,84