Tensoring Positive Maps on Operator Structures

Project Information

TIPTOP

Grant agreement ID: 843414

DOI

10.3030/843414

Project closed

EC signature date 12 April 2019

Start date 1 October 2019

End date 30 September 2021

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 196 707,84

EU contribution

€ 196 707,84

Coordinated by

UNIVERSITE LYON 1 CLAUDE BERNARD
France

Periodic Reporting for period 1 - TIPTOP (Tensoring Positive Maps on Operator Structures)

Reporting period: 2019-10-01 to 2021-09-30

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

[1] We have shown that the tensor square of a qubit map is decomposable if and only if it is positive. Moreover, we have found a general reduction technique to reduce tensorization problems for qubit maps to Pauli-diagonal maps.
This result was published in: A. Müller-Hermes, Decomposable Pauli Diagonal Maps and Tensor Squares of Qubit Maps, Journal of Mathematical Physics (2021), arXiv:2006.14543.
We presented this work in the Operator Algebra Seminar of the University of Kyoto.

[2] We have introduced the problem of entanglement annihilation on proper cones, and we have shown that any entanglement annihilating map on Lorentz cones is entanglement breaking. Using the results from [3] below we have found a tensor product on cones for which non-trivial maps stay positive under tensor powers. Moreover, we have found a general technique producing non-trivial examples of kth order tensorization problems in many different settings.
This result was published in G. Aubrun and A. Müller-Hermes, Annihilating Entanglement Between Cones, arXiv:2110.11825.

[3] We have studied certain regularizations of the operator norm on finite dimensional normed spaces, and we have shown that Euclidean spaces are characterized by the property that the regularizations converge to the nuclear norm.
This result was published in G. Aubrun and A. Müller-Hermes, Asymptotic Tensor Powers of Banach spaces, arXiv:2110.12828.

[4] We have shown that communication rates close to capacity are achievable over noisy quantum channels even if the hardware implementing coding operations is noisy.
This result was published in M. Christandl and A. Müller-Hermes, Fault-tolerant Coding for Quantum Communication, arXiv:2009.07161.
We presented this work at the Conference on Quantum Information Processing (QIP), the workshop for Beyond-iid information theory, and the seminars of the quantum information theory groups in Copenhagen and Lyon.

[5] We have proven a lower bound on the space overhead of fault-tolerant quantum computation.
O. Fawzi, A. Müller-Hermes and A. Shayeghi, A lower bound on the space overhead of fault-tolerant quantum computation, 13th Innovations in Theoretical Computer Science Conference (2022).

Besides the named events, I have also given talks on general topics (Quantum capacities and the PPT squared conjecture) at to workshops organized at the University of Toulouse. Finally, I have written a public outreach article:

[6] A. Müller-Hermes, Cutting cakes and kissing circles, Mathematical Intelligencer (2021), arXiv:2008.11458.

No homepage has been created for the project.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

We have initialized the study of tensorization problems in the mathematical framework of finite-dimensional ordered vector spaces characterized by proper cones. In particular, we have extended results from qubits to the infinite family of Lorentz cones and we have found connections to the theory of Banach spaces and the theory of entanglement distillation. Building on our work, we expect that more connections between quantum information processing tasks, Banach space theory, and tensorization problems on proper cones will be found. The fundamental question of entanglement distillation is still open, but our research makes progress towards its resolution. In particular, we have found the first case of a tensor product where positivity of any tensor power of a non-trivial positive map can be proven. The existence of these non-trivial examples is linked to the structure of the underlying cone, i.e. they exist whenever the base of the cone is not Euclidean ball. Building on this example, it might be possible to find invariants of protocols in quantum information processing, which might lead to a better understanding of limitations of the different kinds of quantum information processing.

Parameters of unital 2-tensor stable positive qubit maps