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arXiv:2512.20133 · IEEE Transactions on Information Theory

On the mixed UDA states and additivity

Xinyu Qiu, Lin Chen, Genwei Li, Delin Chu

quant-phSubmitted Dec 23, 2025• Announced Dec 24, 2025• 0 citations• Revised Jan 21, 2026 (v2)
v1 · Dec 23, 2025v2 · Jan 21, 2026v3 · Jun 6, 2026
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Abstract

Mixed states that are uniquely determined among all (UDA) states are vital in efficient quantum tomography. We show the necessary and sufficient conditions by which some multipartite mixed states are UDA by their $k$-partite reduced density matrices. The case for $k=2$ is mostly studied, which requires minimal local information and shows practical benefits. Based on that, we establish a systematic method for determining UDA states and provide a complete characterization of the additivity of UDA bipartite and three-qubit product states. We show the application of mixed UDA states and their characterization from the perspectives of tomography and other tasks.

DOI
10.1109/TIT.2026.3699321

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Mixed states that are uniquely determined among all (UDA) states are vital in efficient quantum tomography. We show the necessary and sufficient conditions by which some multipartite mixed states are UDA by their kkk-partite reduced density matrices. The case for k=2k=2k=2 is mostly studied, which requires minimal local information and shows practical benefits. Based on that, we establish a systematic method for determining UDA states and provide a complete characterization of the additivity of UDA bipartite and three-qubit product states. We show the application of mixed UDA states and their characterization from the perspectives of tomography and other tasks.
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)(n-1)(n−1)-UDA and show that lower-order marginals never suffice. Finally, we apply these results to the certification of genuine multipartite entanglement.
The pure states that can be uniquely determined among all (UDA) states by their marginals are essential to efficient quantum state tomography. We generalize the UDA states from the context of pure states to that of arbitrary (whether pure or mixed) states, motivated by the efficient state tomography of low-rank states. The concept of additivity of kkk-UDA states for three different composite types of tensor product applies if the composite state of two kkk-UDA states is still uniquely determined by the kkk-partite marginals for the corresponding type of tensor product. We show that the additivity holds if one of the two initial states is pure and present the conditions under which the additivity holds for two mixed UDA states. One of the three composite types of tensor product is also adopted to construct genuinely multipartite entangled (GME) states. Therefore, it is effective to construct multipartite kkk-UDA states with genuine entanglement by uniting the additivity of kkk-UDA states and the construction of GME states.
We qualify the entanglement of arbitrary mixed states of bipartite quantum systems by comparing global and marginal mixednesses quantified by different entropic measures. For systems of two qubits we discriminate the class of maximally entangled states with fixed marginal mixednesses, and determine an analytical upper bound relating the entanglement of formation to the marginal linear entropies. This result partially generalizes to mixed states the quantification of entanglement with marginal mixednesses holding for pure states. We identify a class of entangled states that, for fixed marginals, are globally more mixed than product states when measured by the linear entropy. Such states cannot be discriminated by the majorization criterion.
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