# Infinite Dimensional Optimisation Problems in Quantum Information — An operator algebra approach to the NPA Hierarchy

### APA

Zeiss, J. (2023). Infinite Dimensional Optimisation Problems in Quantum Information — An operator algebra approach to the NPA Hierarchy. Perimeter Institute. https://pirsa.org/23020049

### MLA

Zeiss, Julius. Infinite Dimensional Optimisation Problems in Quantum Information — An operator algebra approach to the NPA Hierarchy. Perimeter Institute, Feb. 14, 2023, https://pirsa.org/23020049

### BibTex

@misc{ pirsa_23020049, doi = {10.48660/23020049}, url = {https://pirsa.org/23020049}, author = {Zeiss, Julius}, keywords = {Quantum Foundations}, language = {en}, title = {Infinite Dimensional Optimisation Problems in Quantum Information {\textemdash} An operator algebra approach to the NPA Hierarchy}, publisher = {Perimeter Institute}, year = {2023}, month = {feb}, note = {PIRSA:23020049 see, \url{https://pirsa.org}} }

Julius Zeiss Universität zu Köln

## Abstract

The theory of polynomial optimisation considers a polynomial objective function subject to countable many polynomial constraints. In a seminal contribution *Navascués, Pironio and Acín (NPA)* generalised a previous result from Lassere, allowing for its application in quantum information theory by considering its non-commutative variant. Non-commutative variables are represented as bounded operators on potentially infinite dimensional Hilbert spaces. These infinite-dimensional *non-commutative polynomials optimisation (NPO) *problems are recast as a complete hierarchy of *semidefinite programming (SDP)* relaxations by a suitable partitioning of the underlying spaces.

The reformulation into convex optimisation problems allows for numerical analysis. We focus on an operator theoretical approach to the NPA hierarchy and show its equiv-

alence to the original NPA hierarchy. To do so, we introduce the necessary mathematical preliminaries from operator algebra theory and semidefinite programming. We conclude by showing how certain relations on operators translate to SDP relaxations yielding drastically reduced problem sizes.

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