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arXiv:1111.1208 · Nature Physics

Experimental estimation of the dimension of classical and quantum systems

Martin Hendrych, Rodrigo Gallego, Michal Mičuda, Nicolas Brunner, Antonio Acín, Juan P. Torres

Published Jun 7, 2012• 138 citations
arXivHTMLPDF
DOI
10.1038/nphys2334

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Estimating the dimension of an Hilbert space is an important component of quantum system identification. In quantum technologies, the dimension of a quantum system (or its corresponding accessible Hilbert space) is an important resource, as larger dimensions determine, e.g., the performance of quantum computation protocols or the sensitivity of quantum sensors. Despite being a critical task in quantum system identification, estimating the Hilbert space dimension is experimentally challenging. While there have been proposals for various dimension witnesses capable of putting a lower bound on the dimension from measuring collective observables that encode correlations, in many practical scenarios, especially for multiqubit systems, the experimental control might not be able to engineer the required initialization, dynamics, and observables. Here we propose a more practical strategy that relies not on directly measuring an unknown multiqubit target system, but on the indirect interaction with a local quantum probe under the experimenter's control. Assuming only that the interaction model is given and the evolution correlates all the qubits with the probe, we combine a graph-theoretical approach and realization theory to demonstrate that the system dimension can be exactly estimated from the model order of the system. We further analyze the robustness in the presence of background noise of the proposed estimation method based on realization theory, finding that despite stringent constrains on the allowed noise level, exact dimension estimation can still be achieved.
Quantum-enhanced fractal dimension estimation presents a novel approach to data analysis, leveraging quantum entanglement to facilitate a more accurate and comprehensive exploration of fractal structures than conventional classical methods. This paper explores the potential of this technique, focusing on its ability to mitigate the limitations of classical algorithms in handling high-dimensional data and accurately characterize complex fractal patterns. We detail the fundamental principles of quantum entanglement and its application to this task, presenting a theoretical framework and preliminary experimental results demonstrating improved fractal dimension estimation accuracy. The core mechanism centers around utilizing entanglement to create a more holistic representation of the data's underlying fractal structure, offering a potential paradigm shift in the field of data analysis.