The aim of this note is to present a ``type III'' generalization of a distribution lemma of Connes. We then derive, following Vidick, consequences on infinite-dimensional quantum strategies for non-local games.
Mathematical Applications of Science Fiction We present a rigorous derivation of a non-perturbative isomorphism between the asymptotic growth of quantum circuit complexity and the recoverability of information in scrambled quantum states, formulated within the framework of Type-III von Neumann algebras characteristic of local quantum field theories. By extending the Complexity equals Volume conjecture through Tomita–Takesaki modular theory, we construct a correspondence that links the geodesic length on the manifold of unitaries to the spectral properties of the modular Hamiltonian. We prove that the scrambling time corresponds to a phase transition in the Connes cocycle derivative, demonstrating that perfect information retrieval is dual to the saturation of the Lloyd bound. The work includes ten supporting lemmas and five main theorems, culminating in the Thorne–Reconstruction Theorem, which establishes that modular flow functions as a universal decoding mechanism for Hawking radiation in the infinite-N limit.
We prove that any two-party correlation in the commuting operator model can be approximated using a tracially embeddable strategy, a class of strategies defined on a finite tracial von Neumann algebra, which we define in this paper. Using this characterization, we show that any approximately synchronous correlation can be approximated to the average of a collection of synchronous correlations in the commuting operator model. This generalizes the result from Vidick [JMP 2022] which only applies to finite-dimensional quantum correlations. As a corollary, we show that the quantum tensor code test from Ji et al. [FOCS 2022] follows the soundness property even under the general commuting operator model. Furthermore, we extend the state-dependent norm variant of the Gowers-Hatami theorem to finite von Neumann algebras. Combined with the aforementioned characterization, this enables us to lift many known results about robust self-testing for non-local games to the commuting operator model, including a sample efficient finite-dimensional EPR testing for the commuting operator strategies. We believe that, in addition to the contribution from this paper, this class of strategies can be helpful for further understanding non-local games in the infinite-dimensional setting.
The word ‘stable’ is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and quantum synchronous strategies for non-local games. We observe in particular that simple spectral gap estimates can lead to strong quantitative forms of stability. For example, we prove that the direct product of two (flexibly) Hilbert-Schmidt stable groups is again (flexibly) Hilbert-Schmidt stable, provided that one of them has Kazhdan’s property (T). We also provide a simple form and simple analysis of a non-local game with few questions, with the property that synchronous strategies with large value are close to perfect strategies involving large Pauli matrices. This simplifies one of the steps (the question reduction) in the recent announced resolution of Connes’ embedding problem by Ji, Natarajan, Vidick, Wright and Yuen.