Rank and Range Criteria for Mixed-State Determination from Local Marginals

Determining whether a global quantum state is uniquely determined among all possible states by its local marginals (the UDA problem) is a fundamental challenge in quantum certification. While pure states have been studied extensively, mixed states remain less understood. This paper investigates the 2-UDA problem for three-qubit mixed states by analyzing the geometric structure of the state's range and extends these findings to multipartite systems and genuine multipartite entanglement (GME) certification.

The authors employ a range-based approach, using the rank of the density matrix and the geometric structure of its range as primary invariants. They utilize projective geometry and the Cayley hyperdeterminant to classify GHZ-SLOCC-free subspaces. For rank-two states, they derive a necessary and sufficient criterion reduced to a finite quadratic-form test. For higher-rank states, they formulate an exact range-restricted semidefinite programming (SDP) criterion. Finally, they use characteristic classes and spin-flip refinements to establish universal high-rank obstructions for multipartite systems.

• Three-Qubit Classification: Three-qubit states with GHZ-SLOCC-free ranges are 2-UDA at ranks one, three, and four. Rank-two states are 2-UDA if they satisfy a specific range-dependent quadratic-form test. No three-qubit state of rank five or higher is 2-UDA.
• Multipartite Extensions: The authors derive general high-rank obstructions for $k$-UDA states in arbitrary finite dimensions. For $n$-qubit systems, they show that rank at least $2^n - n$ rules out $k$-UDA for $k \leq n-2$.
• Channel-Based Family: They characterize a multipartite family where the $(n-1)$-UDA property is determined by the purity and pairwise nonorthogonality of channel outputs, showing that lower-order marginals are insufficient.
• GME Certification: The derived criteria allow for the exact determination of GME detection lengths for various nonsymmetric families and demonstrate that separable 2-marginals can jointly certify GME in certain $n$-qubit families, even when they fail to uniquely determine the global state.