H2020Индивидуална стипендия2019–2021

TIPTOP · Tensoring Positive Maps on Operator Structures

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2019-10-01 → 2021-09-30
Финансиране от ЕС: 196 708 €
Участници: 1
Схема: MSCA-IF-EF-ST

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Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Tensoring Positive Maps on Operator Structures

Quantum information theory studies how information can be processed under the physical laws of quantum mechanics. Currently, there is a lot of effort to harness quantum phenomena in order to either improve existing technology or to find entirely new solutions to challenging problems. An example of this is quantum entanglement, which is necessary for most quantum technologies and, to give a particular application, enables the secure transmission of private classical information. While such applications have been known for quite some time, fundamental questions about the manipulation of quantum entanglement remained open: It is unknown which forms of quantum entanglement can be transformed into pure state entanglement needed for applications using local operations and classical communication, and it is likewise unknown which quantum processes destroy entanglement in multipartite quantum systems when affecting each local system individually. Solving these problems and problems of a similar kind, would have important consequences for the foundation of quantum information theory and could potentially lead to more efficient quantum cryptographic protocols or to more robust processing of quantum information. While these problems seem different at first, they have a common mathematical description: Mathematically, they ask for the characterization of a particular class of linear maps for which some property related some order structure is preserved under taking tensor powers. The goal of this proposal is to study this problem in its general mathematical form and to find instances (possibly different from the aforementioned ones) where it can be solved. By doing so, I aim to learn more about the underlying mathematical structure of these problems and to devise new methods that could lead to a solution of the problems relevant in quantum information theory.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

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.

Оригинален текст от CORDIS (на английски).

Участници

UNIVERSITE LYON 1 CLAUDE BERNARD · Villeurbanne CedexКоординаторФранция