Investigating Pure State Uniqueness in Tomography via Optimization

TL;DR

The paper addresses the problem of determining when a pure quantum state is uniquely specified by a subset of measurements in quantum state tomography, distinguishing UDP and UDA. It develops a unified ALM-based optimization framework and proves a low-rank existence theorem, showing that equivalent density matrices can be found with rank $<\sqrt{m+2}$ (and $<\sqrt{m+3}$ under extension), thereby reducing variable complexity. Through extensive experiments on qutrits and four-qubit symmetric states, the authors validate the framework, classify states into UDP, UDA, or neither, and demonstrate close agreement with SDP where applicable. These results enhance practical state reconstruction in high-dimensional systems and clarify how measurement selection and degeneracies influence uniqueness.

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.

Figures (3)

• Figure 1: Distribution of UDA and UDP cases under (a) $\mathbf{A_7}$ and (b) $\mathbf{A_6}$ on the surface of a unit ball with $a_1$ on the vertical axis and $a_0$, $a_2$ on the horizontal axes
• Figure 2: The minimal fidelity of UDP optimization problems for four-qubit generalized GHZ states, which are represented as $|\psi_\text{GHZ} \rangle = \sin{\Theta}|\omega_0\rangle + \cos{\Theta}|\omega_4\rangle$.
• Figure 3: (a) Parameter space of the symmetric state $\ket{\psi}$ in the form (\ref{['num:eq:psi']}), showing the colored curves corresponding to four types of 2-RDM eigenvalue degeneracies: (i) $\lambda_1 = \lambda_2$, (ii) $\lambda_3 = \lambda_4$, (iii) $\lambda_1 = \lambda_3$ or $\lambda_1 = \lambda_4$, and (iv) $\lambda_2 = \lambda_3$ or $\lambda_2 = \lambda_4$. (b) Classification of UDA and UDP cases across the parameter space of $\ket{\psi}$ in the form (\ref{['num:eq:psi']}), categorized into three groups: (A) UDA, (B) UDP but not UDA, and (C) neither UDP nor UDA.

Theorems & Definitions (19)

• Definition 1: Measurement Vector
• Definition 2: UDP
• Definition 3: UDA
• Theorem 1: Existence of Low-Rank Solution in QST
• proof
• Corollary 1: Existence of Equivalent Low-Rank Density Matrix in UDA Problem
• proof
• Theorem 2: UDP Property of Qutrit States
• proof
• Theorem 3: Optimal Solution for UDP Problem of Generalized GHZ States