The Markov Marginal Problem for Density Operators
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(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 Tr(T(ℛ))=1 is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to T(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 gI(𝒢)ρ associated with a chordal graph 𝒢 and show that it is a relative-entropy discrepancy from ρ 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.
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BibTeX
@misc{lauritzen2026markov,
title = {{The Markov Marginal Problem for Density Operators}},
author = {Steffen Lauritzen and Piotr Zwiernik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.19453},
eprint = {2605.19453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.19453},
}
RIS
TY - UNPB
TI - The Markov Marginal Problem for Density Operators
AU - Steffen Lauritzen
AU - Piotr Zwiernik
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.19453
UR - https://arxiv.org/abs/2605.19453
AB - 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(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 Tr(T(ℛ))=1 is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to T(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 gI(𝒢)_ρ associated with a chordal graph 𝒢 and show that it is a relative-entropy discrepancy from ρ 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.
ER -