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arXiv:2308.11089 · Physical Review A

Additivity of states uniquely determined by marginals

Yi Shen, Lin Chen

Published Dec 18, 2023• 1 citation
arXivHTMLPDF

Abstract

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 $k$-UDA states for three different composite types of tensor product applies if the composite state of two $k$-UDA states is still uniquely determined by the $k$-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 $k$-UDA states with genuine entanglement by uniting the additivity of $k$-UDA states and the construction of GME states.

DOI
10.1103/physreva.108.062418

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Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring O(d2)\mathcal{O}({d}^{2})O(d2) measurements for a general ddd-dimensional state. This complexity can be substantially reduced to O(d)\mathcal{O}(d)O(d) in pure state tomography, indicating that full measurements are unnecessary for pure states. In this paper, we investigate the conditions under which a given pure state can be uniquely determined by a subset of full measurements, focusing on the concepts of uniquely determined among pure states (UDP) and uniquely determined among aall states (UDA). The UDP determination inherently involves nonconvexity challenges, while the UDA determination, though convex, becomes computationally intensive for high-dimensional systems. To address these issues, we develop a unified framework based on the mugmented Lagrangian ethod (ALM). Specifically, our theorem on the existence of low-rank solutions in QST allows us to reformulate the UDA problem with low-rank constraints, thereby reducing the number of variables involved. Our approach entails parametrizing quantum states and employing ALM to handle the constrained nonconvex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems, generalized Greenberger-Horne-Zeilinger states and four-qubit symmetric states not only validate theoretical findings but also reveal the complete distribution of quantum states across three uniqueness categories: (a) UDA, (b) UDP but not UDA, and (c) neither UDP nor UDA. This work provides a practical approach for determining state uniqueness, advancing our understanding of quantum state reconstruction.