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

• URL: http://arxiv.org/abs/2608.24061v1
• Date: Tue, 25 Aug 2026 04:47:02 GMT
• Title: Rank and Range Criteria for Mixed-State Determination from Local Marginals
• Abstract summary: determining whether a mixed quantum state is uniquely determined among all states by its k-body marginals is a fundamental problem in quantum system certification.<n>We develop a range-based approach to this problem by analyzing the structure of the range of the global state.<n>For a channel-based multipartite family, we characterize exactly when a state is $(n-1)$-UDA and show that lower-order marginals never suffice.
• Score: 4.255492892266806
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Determining whether a mixed quantum state is uniquely determined among all states by its k-body marginals (k-UDA) is a fundamental problem in quantum system certification. We develop a range-based approach to this problem by analyzing the structure of the range of the global state. For three-qubit states, we show that states with GHZ-SLOCC-free ranges are 2-UDA at ranks one, three, and four. We derive a necessary and sufficient range criterion for rank-two 2-UDA states and reduce it to a finite quadratic-form test. To cover the remaining range configurations, we formulate an exact range-restricted semidefinite programming criterion and extend it to arbitrary finite-dimensional tripartite states. We also show that every three-qubit state of rank at least five is not 2-UDA, and further extend high-rank obstructions to multipartite systems. For a channel-based multipartite family, we characterize exactly when a state is $(n-1)$-UDA and show that lower-order marginals never suffice. Finally, we apply these results to the certification of genuine multipartite entanglement.

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