The Markov Marginal Problem for Density Operators

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.