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arXiv:2609.19310

Quantum tree networks: complete semidefinite hierarchies and the source-transfer model

Xiangling Xu, Igor Klep, Marc-Olivier Renou

quant-phmath-phmath.OASubmitted Sep 16, 2026• Announced Sep 18, 2026• 0 citations
v1 · Sep 16, 2026v2 · Sep 20, 2026
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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.

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84 pages, 7 figures. Comments welcome!

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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∗C^*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 coRE\mathsf{coRE}coRE-complete.
Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascués et al., New J. Phys. 10, 073013 (2008)], known as the Navascués–Pironio–Acín (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.
It is a fundamental but difficult problem to characterize the set of correlations that can be obtained by performing measurements on quantum mechanical systems. The problem is particularly challenging when the preparation procedure for quantum states is assumed to comply with a given causal structure. Recently, a first completeness result for this quantum causal compatibility problem has been given based on the so-called quantum inflation technique. However, completeness was achieved by imposing additional technical constraints, such as an upper bound on the Schmidt rank of the observables. Here, we show that these complications are unnecessary in the quantum bilocal scenario, a much-studied abstract model of entanglement swapping experiments. We prove that the quantum inflation hierarchy is complete for the bilocal scenario in the commuting observable model of locality. We also give a bilocal version of an observation by Tsirelson, namely, in finite dimensions, the commuting observable model and the tensor product model of locality coincide. These results answer questions recently posed by Renou and Xu [arXiv:2210.09065v2 (2022)]. Finally, we point out that our techniques can be interpreted more generally as giving rise to a semidefinite programming hierarchy that is complete for the problem of optimizing polynomial functions in the states of operator algebras defined by generators and relations. The completeness of this polarization hierarchy follows from a quantum de Finetti theorem for states on maximal C*-tensor products.
We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state ρρρ, our algorithms return a state ∣ψ⟩|ψ\rangle∣ψ⟩ of chosen bond dimension such that ⟨ψ∣ρ∣ψ⟩\langleψ| ρ|ψ\rangle⟨ψ∣ρ∣ψ⟩ is within ε\varepsilonε of the optimum over the model class, without assuming that ρρρ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and 1/ε1/\varepsilon1/ε, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.
Abstract A causal structure is a description of the functional dependencies between random variables. A distribution is compatible with a given causal structure if it can be realized by a process respecting these dependencies. Deciding whether a distribution is compatible with a structure is a practically and fundamentally relevant, yet very difficult problem. Only recently has a general class of algorithms been proposed: These so-called inflation techniques associate to any causal structure a hierarchy of increasingly strict compatibility tests, where each test can be formulated as a computationally efficient convex optimization problem. Remarkably, it has been shown that in the classical case, this hierarchy is complete in the sense that each non-compatible distribution will be detected at some level of the hierarchy. An inflation hierarchy has also been formulated for causal structures that allow for the observed classical random variables to arise from measurements on quantum states—however, no proof of completeness of this quantum inflation hierarchy has been supplied. In this paper, we construct a first version of the quantum inflation hierarchy that is provably convergent. It takes an additional parameter, r , which can be interpreted as an upper bound on the Schmidt rank of the observables involved. For each r , it provides a family of increasingly strict and ultimately complete compatibility tests for correlations that are compatible with a given causal structure under this Schmidt rank constraint. From a technical point of view, convergence proofs are built on de Finetti theorems , which show that certain symmetries (which can be imposed in convex optimization problems) imply independence of random variables (which is not directly a convex constraint). A main technical ingredient to our proof is a Quantum de Finetti Theorem that holds for general tensor products of C∗C^*C∗ C ∗ -algebras, generalizing previous work that was restricted to minimal tensor products.
An important problem in quantum information theory is that of bounding sets of correlations that arise from making local measurements on entangled states of arbitrary dimension. Currently, the best-known method to tackle this problem is the NPA hierarchy; an infinite sequence of semidefinite programs that provides increasingly tighter outer approximations to the desired set of correlations. In this work we consider a more general scenario in which one performs sequences of local measurements on an entangled state of arbitrary dimension. We show that a simple adaptation of the original NPA hierarchy provides an analogous hierarchy for this scenario, with comparable resource requirements and convergence properties. We then use the method to tackle some problems in device-independent quantum information. First, we show how one can robustly certify over 2.3 bits of device-independent local randomness from a two-quibt state using a sequence of measurements, going beyond the theoretical maximum of two bits that can be achieved with non-sequential measurements. Finally, we show tight upper bounds to two previously defined tasks in sequential Bell test scenarios.
Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for all bipartite compiled Bell games via a newly formalized convergent sequential Navascués-Pironio-Acín (NPA) hierarchy: any polynomial-time prover's score is controlled by a finite-level hierarchy value, and finite-level convergence gives a negligible gap to the commuting quantum value, or to the tensor-product quantum value under flat optimality. Second, we develop the sequential NPA hierarchy as a computational tool of independent interest, establishing its convergence and a stopping criterion based on flatness, and identifying its conic dual as a sparse SOS hierarchy.