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arXiv:2512.00432

The Connes-Kirchberg Problem and infinite-dimensional phenomena in quantum information theory

Magdalena Musat

Published Nov 29, 2025• 0 citations
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Abstract

We give an overview of results tying together a circle of problems connected to the Connes Embedding Problem, Kirchberg's reformulations thereof, Tsirelson's conjecture and its relation to quantum information theory, and a class of quantum channels, called factorizable, introduced by Anantharaman-Delaroche. While parts of the article are more expository, there are new results, including obstructions for channels to being $k$-noisy (admitting a factorization through a full matrix algebra).

DOI
10.48550/arxiv.2512.00432

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We give an overview of results tying together a circle of problems connected to the Connes Embedding Problem, Kirchberg's reformulations thereof, Tsirelson's conjecture and its relation to quantum information theory, and a class of quantum channels, called factorizable, introduced by Anantharaman-Delaroche. While parts of the article are more expository, there are new results, including obstructions for channels to being kkk-noisy (admitting a factorization through a full matrix algebra).
More precisely, we give a simple and very short proof of "the Connes embedding problem implies the synchronous Tsirelson conjecture" that relies on only two elementary ingredients: 1) the well-known description of synchronous correlations as traces on the algebra per player C∗(player)=C∗(inputs∣outputs)C^*(\mathrm{player})=C^*(\mathrm{inputs}|\mathrm{outputs})C∗(player)=C∗(inputs∣outputs) and 2) an elementary lifting result by Kim, Paulsen and Schafhauser. Moreover, this bypasses every of the deep results by Kirchberg as well as any other implicit reformulation as the microstates conjecture and thelike. Meanwhile, we also give a different construction of Connes' algebra Rω\mathcal{R}^ωRω appearing in the Connes embedding problem, which is more suitable for the purposes of quantum information theory and much easier to comprehend for the reader without any prior knowledge in operator algebras. Most importantly, however, we present this proof for the following reason: Since the recent refutation of the synchronous Tsirelson conjecture by MIP*=RE, there exists a nonlocal game which violates the synchronous Tsirelson conjecture, and by the proof of MIP*=RE even a synchronous such game. The approach however is based on contradiction with the undecidability of the Halting problem, and so remains implicit. As such the quest now has started to give an explicit example of a synchronous game violating the synchronous Tsirelson conjecture together with a direct argument for the failure, and the current article serves as a direct translation to the corresponding operator algebra and its tracial state violating the Connes embedding problem.
Motivated by a recent progress concerning quantum commuting and quantum tensor models of composed systems we investigate a notion of (generalized) unitary induced quantum channel. Using properties of Brown algebras we provide an equivalent characterization of discussed families in both tensor and commuting paradigms. In particular, we provide an equivalent formulation of Tsirelson's conjecture (Connes' embedding problem) in terms of considered paradigms based on protocols which do not require measurements performed on infinite-dimensional subsystems. As a result we show that there is a difference between quantum commuting and quantum tensor models for generalized unitary induced channels.
Connes’ Embedding Problem is a deep question on approximability of certain tracial von Neumann algebras by finite-dimensional matrix algebras. We survey the connections between operator algebras, quantum information and theoretical computer science that enabled the recent resolution of this problem. The resolution goes through an equivalent formulation, known as Tsirelson’s problem, in terms of separating convex sets whose definition is motivated by the study of nonlocality in quantum mechanics. We construct an explicit separating hyperplane using the theory of two-player games from complexity theory.
Abstract Information theory deals with the efficient representation of information sources as well as providing fundamental limits to the amount of information communicated reliably over channels. These sources and channels are generally classical, i.e., represented by standard probability distributions. Quantum information theory takes it to the next level where we allow for the sources as well as channels to be quantum. From the representation of quantum states to the communication over quantum channels, the theory not only essentially encapsulates classical information theoretic methods but also accounts for quantum effects such as superposition, entanglement, interference, etc. In this article, we will review and focus on the information theoretic analysis of quantum channels with infinite dimensions. Infinite dimensionality is needed to model quantum optical channels which are ubiquitous in today’s practical networks, distributed quantum communication and quantum internet. The infinite dimensionality introduces some unique problems when compared with finite-dimensional channels and has not been deeply explored in literature from the quantum information theoretic perspective. For these channels, we provide the essential concepts and state-of-the-art channel capacity results. To make this paper self-contained, we also recall the finite dimensional results.
Tsirelson once claimed that the set of quantum correlations, defined by strategies of non-local two-player games, does not depend on which of two possible models is chosen: the tensor product model or the commuting operator model. He later came back from this claim, and the resulting conjecture is now known as Tsirelson’s problem. The problem has since been proven equivalent to notoriously hard problems in operator theory, such as the Connes’ Embedding Problem and the QWEP conjecture. In this master thesis, we look at the finite dimensional case of Tsirelson’s problem, working out all the details of an existing proof and giving a new, shorter proof which also extends to the nuclear case. Moreover, we give an overview of the equivalence of Tsirelson’s problem and two of Kirchberg’s conjectures, including the QWEP conjecture. Finally, we give some results and considerations for the three-player case of Tsirelson’s problem. The appendix contains proofs of many related results used throughout the thesis, and also a beginner’s introduction to quantum mechanics.