aiqbench analysis
This research addresses the fundamental challenge of reconstructing global quantum states from local marginals when noncommutativity prevents standard classical junction-tree methods from working. The authors establish precise trace conditions that determine feasibility and introduce a corrected logarithmic construction to handle these obstructions within chordal graph structures. These findings are critical for advancing quantum information theory, particularly in understanding conditional independence and maximum entropy principles in complex quantum systems.
The Markov Marginal Problem for Density Operators
arXiv:2605.19453v3 Announce Type: replace Abstract: We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a canonical logarithmic construction $T(\mathcal R)$, the noncommutative analogue of the junction-tree formula for decomposable graphical models. Unlike in the classical case, this formal construction may fail: noncommutativity can prevent it from being a normalized state with the prescribed marginals. We prove that this obstruction is captured exactly by a trace condition. For two overlapping marginals, and for clique marginals on a chordal graph, the condition ${\rm Tr}(T(\mathcal R))=1$ is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to $T(\mathcal R)$, and selected by the maximum entropy principle. In the two-clique case, we also give an equivalent conditional reconstruction characterization: the two natural one-sided sandwich reconstructions agree if and only if the trace condition holds. We introduce the global quantum information $g{\rm I}(\mathcal{G})_\rho$ associated with a chordal graph $\mathcal{G}$ and show that it is a relative-entropy discrepancy from $\rho$ to the logarithmic candidate, with a trace correction when the candidate is not normalized. We also prove an intersection property for strictly positive quantum conditional independence. Three-qubit Pauli examples illustrate how the quantum obstructions are real: local consistency, feasibility, Markov feasibility, and maximum entropy can all separate.
Abstract:We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a canonical logarithmic construction $T(\mathcal R)$, the noncommutative analogue of the junction-tree formula for decomposable graphical models. Unlike in the classical case, this formal construction may fail: noncommutativity can prevent it from being a normalized state with the prescribed marginals. We prove that this obstruction is captured exactly by a trace condition. For two overlapping marginals, and for clique marginals on a chordal graph, the condition ${\rm Tr}(T(\mathcal R))=1$ is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to $T(\mathcal R)$, and selected by the maximum entropy principle. In the two-clique case, we also give an equivalent conditional reconstruction characterization: the two natural one-sided sandwich reconstructions agree if and only if the trace condition holds. We introduce the global quantum information $g{\rm I}(\mathcal{G})_\rho$ associated with a chordal graph $\mathcal{G}$ and show that it is a relative-entropy discrepancy from $\rho$ to the logarithmic candidate, with a trace correction when the candidate is not normalized. We also prove an intersection property for strictly positive quantum conditional independence. Three-qubit Pauli examples illustrate how the quantum obstructions are real: local consistency, feasibility, Markov feasibility, and maximum entropy can all separate.
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Probability (math.PR)
arXiv:2605.19453 [quant-ph]
(or arXiv:2605.19453v3 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2605.19453 Focus to learn more arXiv-issued DOI via DataCite
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