Investigating Pure State Uniqueness in Tomography via Optimization

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.

13 pages, 3 figures

References in corpus (29)

• Scalable multi-particle entanglement of trapped ions
• Continuous-variable optical quantum state tomography
• Quantum state tomography via compressed sensing
• Reconstructing quantum states with generative models
• Reliable Quantum State Tomography
• Cooling-by-measurement and mechanical state tomography via pulsed optomechanics
• Quantum State Tomography via Linear Regression Estimation
• Quantum Tomography under Prior Information
• Permutationally invariant state reconstruction
• Benchmarking a Teleportation Protocol realized in Superconducting Circuits
• Quantum state tomography via reduced density matrices
• Local-measurement-based quantum state tomography via neural networks
• Uniqueness of Quantum States Compatible with Given Measurement Results
• COSMO: A conic operator splitting method for convex conic problems
• Tomography is necessary for universal entanglement detection with single-copy observables
• Strictly-complete measurements for bounded-rank quantum-state tomography
• Informational completeness in bounded-rank quantum-state tomography
• The power of being positive: Robust state estimation made possible by quantum mechanics
• Three-party pure quantum states are determined by two two-party reduced states
• Pure State Tomography with Pauli Measurements
• When is a pure state of three qubits determined by its single-particle reduced density matrices?
• From Ground States to Local Hamiltonians
• Comment on some results of Erdahl and the convex structure of reduced density matrices
• Almost all four-particle pure states are determined by their two-body marginals
• Characterizing ground and thermal states of few-body Hamiltonians
• Unified Quantum State Tomography and Hamiltonian Learning Using Transformer Models: A Language-Translation-Like Approach for Quantum Systems
• Rank Reduction for the Local Consistency Problem
• Pure State Tomography with Fourier Transformation
• A Variational Approach to Unique Determinedness in Pure-state Tomography