[Paper Review] Numerical Estimation Schemes for Quantum Tomography

Yong Siah Teo|arXiv (Cornell University)|Feb 14, 2013

Quantum Information and Cryptography9 references3 citations

TL;DR

This thesis proposes novel numerical estimation schemes for quantum tomography, introducing a conjugate-gradient algorithm for efficient maximum-likelihood quantum state estimation and a joint maximum-likelihood and maximum-entropy (MLME) method to resolve ambiguities in informationally incomplete data. It further develops an adaptive framework for entanglement detection using witness bases and extends the MLME approach to quantum process estimation, demonstrating improved robustness and accuracy in simulations and experimental data.

ABSTRACT

This is a PhD dissertation on the latest numerical quantum estimation schemes as of 2012, submitted to the National University of Singapore. The main content of the thesis focuses on accessing quantum information with informationally incomplete measurements to reconstruct quantum states of large quantum systems, as well as to reduce the amount of resources to reconstruct quantum channels.

Motivation & Objective

• . The paper aims to improve the efficiency and reliability of quantum state estimation under informationally incomplete measurement data.
• . It addresses the fundamental challenge of non-uniqueness in quantum state reconstruction when data is incomplete.
• . The objective is to develop a systematic method that balances likelihood and entropy to produce the least-biased, most-likely estimator.
• . The work extends to quantum process estimation and entanglement detection using adaptive witness measurements.
• . It compares the MLME approach with the hedged maximum-likelihood (HML) method, evaluating their performance under incomplete data.

Proposed method

• . Proposes a conjugate-gradient algorithm as a more efficient alternative to the steepest-ascent method for maximizing the likelihood functional in quantum state estimation.
• . Introduces the MLME estimator by jointly maximizing the likelihood and von Neumann entropy functionals to resolve ambiguity in incomplete data.
• . Derives an iterative HML algorithm to maximize a hedged likelihood functional, incorporating prior knowledge and uncertainty.
• . Applies the MLME method to both simulated and experimental data, including two-qubit systems and entanglement detection via witness bases.
• . Develops adaptive strategies for measuring witness bases, optimizing input states and measurement settings to enhance detection efficiency.
• . Extends the MLME principle to quantum process estimation, yielding a unique estimator from incomplete process tomography data.

Experimental results

Research questions

• RQ1 . How can quantum state estimation be made more efficient and robust when measurement data is informationally incomplete?
• RQ2 . What is the optimal trade-off between likelihood and entropy in selecting a quantum state estimator under incomplete data?
• RQ3 . How does the MLME estimator compare to the hedged maximum-likelihood (HML) estimator in terms of bias and accuracy?
• RQ4 . Can adaptive witness basis measurements improve the efficiency of entanglement detection compared to conventional methods?
• RQ5 . How can the principles of maximum-likelihood and maximum-entropy be extended to quantum process estimation with incomplete data?

Key findings

• . The conjugate-gradient algorithm significantly outperforms the steepest-ascent method in convergence speed and computational efficiency for quantum state estimation.
• . The MLME estimator consistently produces the most likely, least-biased state estimate under incomplete data, as validated by Monte Carlo simulations.
• . The HML estimator exhibits improved robustness to noise and model misspecification compared to standard maximum-likelihood estimation.
• . Adaptive witness basis measurements, combined with state estimation, increase the detection rate of entanglement by fully utilizing measurement data, outperforming expectation-value-only strategies.
• . The MLME approach for quantum process estimation yields a unique, physically meaningful estimator even from incomplete process tomography data.
• . Numerical simulations confirm that the MLME method maintains high fidelity and stability across various noise levels and data sparsity regimes.