Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.04896

Neural networks learn to reconstruct multipartite entanglement from quantum marginals

Matreyee Kandpal, Arvind, Kavita Dorai

quant-phSubmitted Sep 4, 2026• Announced Sep 7, 2026• 0 citations
arXivHTMLPDF

Abstract

Different sets of local correlations are not equivalent: some fragments of reduced data uniquely determine a global quantum state, while others leave it ambiguous. The quantum marginal problem asks whether a collection of reduced density matrices uniquely determines a compatible global quantum state. Although generic quantum states are uniquely specified by suitable sets of marginals, different collections of marginals are not equally informative: some uniquely determine the global state, whereas others leave it ambiguous. Identifying when uniqueness holds, and reconstructing the global state from partial information, remains computationally demanding and experimentally challenging. We show that information about the multipartite entanglement class and reconstructability in four-qubit systems is compactly encoded in small sets of two- and three-qubit marginals. Using semidefinite programming, we chart the reconstructability landscape across 49 inequivalent SLOCC entanglement classes and show that uniqueness strongly depends on both entanglement structure and marginal order. Neural networks trained only on reduced density matrices learn this structure directly. They accurately classify marginal reconstructability and, when uniqueness holds, reconstruct the full four-qubit density matrix with high fidelity from two- and three-qubit marginals. We benchmark the approach on a four-qubit nuclear magnetic resonance quantum processor and demonstrate that reconstructions from experimentally measured marginals remain faithful despite phase damping and control imperfections. Our results show that neural networks can learn when local correlations uniquely specify a global quantum state, and reveal how global quantum structure is encoded in reduced data.

Comments
13 pages, 11 figures

Related papers

Almost all tripartite pure states are uniquely determined by two overlapping marginals

HyunHo Cha

In many-body quantum physics, the global state encodes all correlations, but the information most naturally available is often local. This makes it important to understand when a small collection of reduced density matrices already fixes the whole state. Such uniqueness questions are especially important for tomography and for the study of entanglement, because agreement with local data from a pure state does not by itself exclude a different mixed global state. We prove a generic uniqueness theorem for finite-dimensional tripartite systems. Previous results were restricted either to uniqueness against pure competitors or to uniqueness against all competitors under stronger dimension assumptions or balanced subsystem dimensions. Our result removes both restrictions.

•Jun 24, 2026•0 citations

Reconstructing the whole from its parts

Daniel Uzcátegui Contreras, Dardo Goyeneche

The quantum marginal problem consists in deciding whether a given set of marginal reductions is compatible with the existence of a global quantum state or not. In this work, we formulate the problem from the perspective of dynamical systems theory and study its advantages with respect to the standard approach. The introduced formalism allows us to analytically determine global quantum states from a wide class of self-consistent marginal reductions in any multipartite scenario. In particular, we show that any self-consistent set of multipartite marginal reductions is compatible with the existence of a global quantum state, after passing through a depolarizing channel. This result reveals that the complexity associated to the marginal problem can be drastically reduced when restricting the attention to sufficiently mixed marginals. We also formulate the marginal problem in a compressed way, in the sense that the total number of scalar constraints is smaller than the one required by the standard approach. This fact suggests an exponential speedup in runtime when considering semi-definite programming techniques to solve it, in both classical and quantum algorithms. Finally, we reconstruct $n$-qubit quantum states from all the $\binom{n}{k}$ marginal reductions to $k$ parties, generated from randomly chosen mixed states. Numerical simulations reveal that the fraction of cases where we can find a global state equals 1 when $5\leq n\leq12$ and $\lfloor(n-1)/\sqrt{2}\rfloor\leq k\leq n-1$, where $\lfloor\cdot\rfloor$ denotes the floor function.

arXiv:2209.14154•Sep 28, 2022•0 citations

Machine learning approach to reconstruct density matrices from quantum marginals

Daniel Uzcátegui Contreras, Antonio Guerra, Sebastián Niklitschek +1 more

Abstract In this work, we propose a machine learning (ML)-based approach to address a specific aspect of the Quantum Marginal Problem: reconstructing a global density matrix compatible with a given set of quantum marginals. Our method integrates a quantum marginal imposition technique with convolutional denoising autoencoders. The loss function is carefully designed to enforce essential physical constraints, including Hermiticity, positivity, and normalization. Through extensive numerical simulations, we demonstrate the effectiveness of our approach, achieving high success rates and accuracy. Furthermore, we show that, in many cases, our model offers a faster alternative to state-of-the-art semidefinite programming solvers without compromising solution quality. These results highlight the potential of ML techniques for solving complex problems in quantum mechanics.

arXiv:2410.11145•Machine Learning Science and Technology•Jun 13, 2025•1 citation

Local-measurement-based quantum state tomography via neural networks

Tao Xin, Sirui Lu, Ningping Cao +5 more

Abstract Quantum state tomography is a daunting challenge of experimental quantum computing, even in moderate system size. One way to boost the efficiency of state tomography is via local measurements on reduced density matrices, but the reconstruction of the full state thereafter is hard. Here, we present a machine-learning method to recover the ground states of $$k$$ k -local Hamiltonians from just the local information, where a fully connected neural network is built to fulfill the task with up to seven qubits. In particular, we test the neural network model with a practical dataset, that in a 4-qubit nuclear magnetic resonance system our method yields global states via the 2-local information with high accuracy. Our work paves the way towards scalable state tomography in large quantum systems.

arXiv:1807.07445•npj Quantum Information•Nov 29, 2019•79 citations

Distinguishing graph states by the properties of their marginals

Lina Vandré, Jarn de Jong, Frederik Hahn +3 more

Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.

arXiv:2406.09956•Physical Review A•May 28, 2025•5 citations

Distinguishing graph states by the properties of their marginals

Lina Vandré, Jarn de Jong, Frederik Hahn +3 more

Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.

•May 28, 2025•0 citations
Different sets of local correlations are not equivalent: some fragments of reduced data uniquely determine a global quantum state, while others leave it ambiguous. The quantum marginal problem asks whether a collection of reduced density matrices uniquely determines a compatible global quantum state. Although generic quantum states are uniquely specified by suitable sets of marginals, different collections of marginals are not equally informative: some uniquely determine the global state, whereas others leave it ambiguous. Identifying when uniqueness holds, and reconstructing the global state from partial information, remains computationally demanding and experimentally challenging.
We show that information about the multipartite entanglement class and reconstructability in four-qubit systems is compactly encoded in small sets of two- and three-qubit marginals. Using semidefinite programming, we chart the reconstructability landscape across 49 inequivalent SLOCC entanglement classes and show that uniqueness strongly depends on both entanglement structure and marginal order. Neural networks trained only on reduced density matrices learn this structure directly. They accurately classify marginal reconstructability and, when uniqueness holds, reconstruct the full four-qubit density matrix with high fidelity from two- and three-qubit marginals.
We benchmark the approach on a four-qubit nuclear magnetic resonance quantum processor and demonstrate that reconstructions from experimentally measured marginals remain faithful despite phase damping and control imperfections. Our results show that neural networks can learn when local correlations uniquely specify a global quantum state, and reveal how global quantum structure is encoded in reduced data.
In many-body quantum physics, the global state encodes all correlations, but the information most naturally available is often local. This makes it important to understand when a small collection of reduced density matrices already fixes the whole state. Such uniqueness questions are especially important for tomography and for the study of entanglement, because agreement with local data from a pure state does not by itself exclude a different mixed global state. We prove a generic uniqueness theorem for finite-dimensional tripartite systems. Previous results were restricted either to uniqueness against pure competitors or to uniqueness against all competitors under stronger dimension assumptions or balanced subsystem dimensions. Our result removes both restrictions.
The quantum marginal problem consists in deciding whether a given set of marginal reductions is compatible with the existence of a global quantum state or not. In this work, we formulate the problem from the perspective of dynamical systems theory and study its advantages with respect to the standard approach. The introduced formalism allows us to analytically determine global quantum states from a wide class of self-consistent marginal reductions in any multipartite scenario. In particular, we show that any self-consistent set of multipartite marginal reductions is compatible with the existence of a global quantum state, after passing through a depolarizing channel. This result reveals that the complexity associated to the marginal problem can be drastically reduced when restricting the attention to sufficiently mixed marginals. We also formulate the marginal problem in a compressed way, in the sense that the total number of scalar constraints is smaller than the one required by the standard approach. This fact suggests an exponential speedup in runtime when considering semi-definite programming techniques to solve it, in both classical and quantum algorithms. Finally, we reconstruct nnn-qubit quantum states from all the (nk)\binom{n}{k}(kn​) marginal reductions to kkk parties, generated from randomly chosen mixed states. Numerical simulations reveal that the fraction of cases where we can find a global state equals 1 when 5≤n≤125\leq n\leq125≤n≤12 and ⌊(n−1)/2⌋≤k≤n−1\lfloor(n-1)/\sqrt{2}\rfloor\leq k\leq n-1⌊(n−1)/2​⌋≤k≤n−1, where ⌊⋅⌋\lfloor\cdot\rfloor⌊⋅⌋ denotes the floor function.
Abstract In this work, we propose a machine learning (ML)-based approach to address a specific aspect of the Quantum Marginal Problem: reconstructing a global density matrix compatible with a given set of quantum marginals. Our method integrates a quantum marginal imposition technique with convolutional denoising autoencoders. The loss function is carefully designed to enforce essential physical constraints, including Hermiticity, positivity, and normalization. Through extensive numerical simulations, we demonstrate the effectiveness of our approach, achieving high success rates and accuracy. Furthermore, we show that, in many cases, our model offers a faster alternative to state-of-the-art semidefinite programming solvers without compromising solution quality. These results highlight the potential of ML techniques for solving complex problems in quantum mechanics.
Abstract Quantum state tomography is a daunting challenge of experimental quantum computing, even in moderate system size. One way to boost the efficiency of state tomography is via local measurements on reduced density matrices, but the reconstruction of the full state thereafter is hard. Here, we present a machine-learning method to recover the ground states of kkk k -local Hamiltonians from just the local information, where a fully connected neural network is built to fulfill the task with up to seven qubits. In particular, we test the neural network model with a practical dataset, that in a 4-qubit nuclear magnetic resonance system our method yields global states via the 2-local information with high accuracy. Our work paves the way towards scalable state tomography in large quantum systems.
Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.
Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.