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arXiv · 2501.00327

Investigating Pure State Uniqueness in Tomography via Optimization

Jiahui Wu·Zheng An·Chao Zhang·Xuanran Zhu·Shilin Huang·Bei Zeng
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Abstract

Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring $\mathcal{O}(d^2)$ measurements for a general $d$-dimensional state. This complexity can be substantially reduced to $\mathcal{O}(d)$ 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 all states (UDA). The UDP determination inherently involves non-convexity 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 Augmented Lagrangian Method (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 parameterizing quantum states and employing ALM to handle the constrained non-convex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems 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.

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Jiahui Wu, Zheng An, Chao Zhang, Xuanran Zhu, Shilin Huang, Bei Zeng. 2024-12-31. Investigating Pure State Uniqueness in Tomography via Optimization. https://doi.org/10.1103/p7kb-k52j

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