aiqbench analysis
This research resolves the bounds on correlations for general quantum tree networks by introducing a novel source-transfer model equivalent to a mixed quantum framework. The authors establish two convergent semidefinite program hierarchies, proving that the associated game-value problem is coRE-complete and extending noncommutative real algebraic geometry to multi-state polynomials. These theoretical advances provide rigorous tools for analyzing the fundamental limits of distributed quantum information processing without dimension restrictions.
Quantum tree networks: complete semidefinite hierarchies and the source-transfer model
arXiv:2609.19310v1 Announce Type: new Abstract: Understanding the capabilities and limits of quantum systems is central to quantum information processing. This includes determining which correlations can arise in networks of quantum systems distributed by independent sources. Yet for general quantum networks, no systematic, dimension-free method is known to bound these correlations arbitrarily well. We resolve this problem for all quantum tree networks through a new source-transfer model, which we prove equivalent to a mixed quantum model. The mixed model places no restriction on dimension and reduces to the standard tensor-product model in finite dimensions; the source-transfer model instead describes these correlations through states on $C^*$-algebras and completely positive maps composed recursively along the tree. We prove the equivalence by reconstructing local measurements as Radon-Nikodym derivatives in the spatial tensor products prescribed by the network. Building on this characterization, we establish two convergent outer hierarchies of semidefinite programs (SDPs): a novel source-transfer hierarchy and the inflation-NPA hierarchy. For the source-transfer hierarchy, we extend noncommutative real algebraic geometry to multi-state polynomials with completely positive maps, prove a reconstruction theorem establishing its convergence, and give a sufficient stopping criterion for extracting finite-dimensional realizations. For the inflation-NPA hierarchy, we reconstruct source-transfer realizations from weak-operator limits of averages over source copies. Consequently, the game-value promise problem for quantum tree networks in the mixed model is $\mathsf{coRE}$-complete.
Abstract:Understanding the capabilities and limits of quantum systems is central to quantum information processing. This includes determining which correlations can arise in networks of quantum systems distributed by independent sources. Yet for general quantum networks, no systematic, dimension-free method is known to bound these correlations arbitrarily well. We resolve this problem for all quantum tree networks through a new source-transfer model, which we prove equivalent to a mixed quantum model. The mixed model places no restriction on dimension and reduces to the standard tensor-product model in finite dimensions; the source-transfer model instead describes these correlations through states on $C^*$-algebras and completely positive maps composed recursively along the tree. We prove the equivalence by reconstructing local measurements as Radon-Nikodym derivatives in the spatial tensor products prescribed by the network. Building on this characterization, we establish two convergent outer hierarchies of semidefinite programs (SDPs): a novel source-transfer hierarchy and the inflation-NPA hierarchy. For the source-transfer hierarchy, we extend noncommutative real algebraic geometry to multi-state polynomials with completely positive maps, prove a reconstruction theorem establishing its convergence, and give a sufficient stopping criterion for extracting finite-dimensional realizations. For the inflation-NPA hierarchy, we reconstruct source-transfer realizations from weak-operator limits of averages over source copies. Consequently, the game-value promise problem for quantum tree networks in the mixed model is $\mathsf{coRE}$-complete.
84 pages, 7 figures. Comments welcome!
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Operator Algebras (math.OA)
arXiv:2609.19310 [quant-ph]
(or arXiv:2609.19310v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.19310 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 321 words
