Periodic Reporting for period 1 - GRPSTABQIT (Group Stability and Quantum Information Theory)
Período documentado: 2023-07-01 hasta 2025-06-30
The overarching theme of the mathematical content of this proposal is stability for countable discrete groups and its connections with quantum information theory, operator algebras, and ergodic theory. Quantum computing is an upcoming highly active research area with promising implications towards computational applications. However, the mathematical foundations of quantum information are not yet very well understood, and this project aims to advance our understanding of important mathematical concepts underlying quantum information.
In our context, stability refers to the phenomenon of approximate properties being close to actual properties. This is important in a practical context: experimentally, it is nearly always impossible to get exact outcomes, and stability results would guarantee correct solutions if sufficiently good approximations are obtained.
The main goals of this project are threefold:
1. Exploring application of stability in quantum information theory.
2. Establishing stability and approximation results for free product groups and deducing approximation results for factorizable quantum channels.
3. Investigating stability for the class of amenable group.
In our context, stability refers to the phenomenon of approximate properties being close to actual properties. This is important in a practical context: experimentally, it is nearly always impossible to get exact outcomes, and stability results would guarantee correct solutions if sufficiently good approximations are obtained.
The main goals of this project are threefold:
1. Exploring application of stability in quantum information theory.
2. Establishing stability and approximation results for free product groups and deducing approximation results for factorizable quantum channels.
3. Investigating stability for the class of amenable group.
The technical and scientific work for this project consisted of theoretical research, independently by the fellow, as well as collaboratively with researchers from QMath at the University of Copenhagen and international researchers. These research efforts resulted in four publications contributing to the overall objectives of the project, and the obtained results are described in the next section. The international collaborative research included research visits to WWU Münster, the Fields Institute, Université de Paris, SDU Odense, UC San Diego, Chalmers University of Technology, CIRM, KU Leuven, and UC Berkeley.
The work on this project has resulted in four publications contributing to its overall objectives.
1. Laura Mančinska, Pieter Spaas, and Taro Spirig: "Gap-preserving reductions and RE-completeness of Independent Set Games".
In this paper we establish a new stability result for so-called Projective-Valued Measures (PVMs) and use it to unravel the computational complexity of independent set games. In particular, we provide the first example of a family of natural non-local games which is RE-complete, building on the breakthrough work MIP*=RE. As indicated in the paper, this can have far-reaching implications towards the complexity of problems in MIP* (Multiprover Interactive Proof Systems in the quantum setting) and the quantum games PCP conjecture.
2. Adrian Ioana, Pieter Spaas, and Itamar Vigdorovich: "Trace spaces of full free product C*-algebras".
In this paper, we establish novel approximation results in von Neumann algebras to show that the trace simplex of any free product of C*-algebras is the so-called Poulsen simplex (unless an obvious obstruction occurs). We use this to show that any factorizable quantum channel can be approximated by extremal factorizable quantum channels, which are mathematically nicer. Whether this is true had been asked in the literature, and besides providing a complete answer, we also connect this problem with intrinsic properties of the associated operator algebras. We believe that this improved understanding will lead to further advancements in the theory of operator algebras, as well as its applications to quantum information theory.
3. Francesco Fournier-Facio and Maria Gerasimova, and Pieter Spaas: "Local Hilbert–Schmidt stability".
In this paper we introduce a local variant of so-called Hilbert-Schmidt stability, which is one of the main stability notions relevant for quantum information theory. This local version is much more flexible than genuine Hilbert-Schmidt stability, and applies to many more examples. Besides establishing fundamental properties and providing new examples, we investigate local Hilbert-Schmidt stability in detail for amenable groups, connecting it with properties of the character simplex of the group. This work has already led to several follow-up papers by different authors.
4. Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, Forte Shinko, and Pieter Spaas: "Hyperfiniteness for group actions on trees".
This paper concerns group actions on trees and the question of hyperfiniteness for the resulting boundary action. Boundary actions are natural examples of dynamical systems, with connections to many areas in mathematics. Hyperfiniteness and amenability of the corresponding equivalence relation can be viewed as a structural property, allowing for suitable approximations by finite actions. Our main result is the formulation of a natural condition implying hyperfiniteness, providing a simplified approach for many known examples, as well as several new examples.
1. Laura Mančinska, Pieter Spaas, and Taro Spirig: "Gap-preserving reductions and RE-completeness of Independent Set Games".
In this paper we establish a new stability result for so-called Projective-Valued Measures (PVMs) and use it to unravel the computational complexity of independent set games. In particular, we provide the first example of a family of natural non-local games which is RE-complete, building on the breakthrough work MIP*=RE. As indicated in the paper, this can have far-reaching implications towards the complexity of problems in MIP* (Multiprover Interactive Proof Systems in the quantum setting) and the quantum games PCP conjecture.
2. Adrian Ioana, Pieter Spaas, and Itamar Vigdorovich: "Trace spaces of full free product C*-algebras".
In this paper, we establish novel approximation results in von Neumann algebras to show that the trace simplex of any free product of C*-algebras is the so-called Poulsen simplex (unless an obvious obstruction occurs). We use this to show that any factorizable quantum channel can be approximated by extremal factorizable quantum channels, which are mathematically nicer. Whether this is true had been asked in the literature, and besides providing a complete answer, we also connect this problem with intrinsic properties of the associated operator algebras. We believe that this improved understanding will lead to further advancements in the theory of operator algebras, as well as its applications to quantum information theory.
3. Francesco Fournier-Facio and Maria Gerasimova, and Pieter Spaas: "Local Hilbert–Schmidt stability".
In this paper we introduce a local variant of so-called Hilbert-Schmidt stability, which is one of the main stability notions relevant for quantum information theory. This local version is much more flexible than genuine Hilbert-Schmidt stability, and applies to many more examples. Besides establishing fundamental properties and providing new examples, we investigate local Hilbert-Schmidt stability in detail for amenable groups, connecting it with properties of the character simplex of the group. This work has already led to several follow-up papers by different authors.
4. Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, Forte Shinko, and Pieter Spaas: "Hyperfiniteness for group actions on trees".
This paper concerns group actions on trees and the question of hyperfiniteness for the resulting boundary action. Boundary actions are natural examples of dynamical systems, with connections to many areas in mathematics. Hyperfiniteness and amenability of the corresponding equivalence relation can be viewed as a structural property, allowing for suitable approximations by finite actions. Our main result is the formulation of a natural condition implying hyperfiniteness, providing a simplified approach for many known examples, as well as several new examples.