Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable trade-offs inherent in quantum tomography.
Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable trade-offs inherent in quantum tomography.
Quantum state tomography (QST) is a crucial technique for characterizing the state of a quantum system. However, the inherent limitations of measurement processes, including detector efficiency and photon counting statistics, introduce noise that affects the accuracy of the reconstructed quantum state. This work presents a rigorous information-theoretic framework for analyzing and quantifying the accuracy achievable in QST. We leverage concepts from information theory, specifically mutual information and channel capacity, to precisely determine the trade-offs between measurement resolution and the complexity of state reconstruction. The central argument is that the achievable accuracy is fundamentally limited by the information that can be reliably extracted from the measurement data, rather than relying solely on empirical estimates of error. This approach provides a theoretical benchmark for evaluating QST protocols and identifying optimal measurement strategies. The key contributions lie in establishing a quantifiable relationship between measurement parameters, noise levels, and the fidelity of the reconstructed quantum state. This offers a more robust and predictive method for assessing the quality of QST than traditional approaches.
Quantum tomography—characterising a quantum system by assigning it a quantum state—–is an essential building block for future technologies like quantum metrology or quantum computation. Ideal quantum tomography is accurate and robust to noise, while simultaneously being quick and efficient. The standard technique used for decades, full quantum tomography (FQT), only fulfills parts of these criteria and quickly becomes unfeasible, even for medium sized systems. As the number of possible states increases, the requirements on the amount of measurements to characterise the system and the storage of the results grow with it exponentially. The very feature that gives quantum mechanics its immense possibilities, hinders our capacity to describe it. Additionally, FQT is neither robust to noise nor to experimental errors, and generally fails in the presence of these environmental perturbations unless additional resources are used at the cost of efficiency.
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.
The development of large-scale platforms for quantum information requires new methods for verification and validation of quantum behavior. Quantum tomography (QT) is the standard tool for diagnosing quantum states, process, and readout devices by providing complete information. However, QT is limited since it is expensive to not only implement experimentally, but also requires heavy classical post-processing of data. In this dissertation, we introduce new methods for QT that are more efficient to implement and robust to noise and errors, thereby making QT a more practical tool for current quantum information experiments. The crucial detail that makes these new, efficient, and robust methods possible is prior information about the quantum system. This prior information is prompted by the goals of most experiments in quantum information, which require pure states, unitary processes, and rank-1 POVM operators. Therefore, most experiments are designed to operate near this ideal regime. We show that when this is the case, QT can be accomplished with significantly fewer resources, and produce a robust estimate in the presence of noise and errors. Moreover, the estimate is also robust if the state is not exactly pure, the process is not exactly unitary, or the POVM is not exactly rank-1. Such compelling methods are only made possible by the positivity constraint on quantum states, processes, and POVMs. This requirement is an inherent feature of quantum mechanics, but has powerful consequences to QT. Since QT is necessarily an experimental tool, we discuss a test of these new methods in an experimental setting. The physical system is an ensemble of laser-cooled cesium atoms in the laboratory of Prof. Poul Jessen. Experiments were conducted by Hector Sosa-Martinez et al. to demonstrate different QT protocols. We compare the results, and conclude that the new methods are effective.
The initialization of a quantum system into a certain state is a crucial aspect of quantum information science. While a variety of measurement strategies have been developed to characterize how well the system is initialized, for a given one, there is in general a trade-off between its efficiency and the accessible information of the quantum state. Conventional quantum state tomography can characterize unknown states while requiring exponentially expensive time-consuming postprocessing. Alternatively, recent theoretical breakthroughs show that quantum state verification provides a technique to quantify the prepared state with significantly fewer samples, especially for multipartite entangled states. In this Letter, we modify the original proposal to be robust to practical imperfections, and experimentally implement a scalable quantum state verification on two-qubit and four-qubit entangled states with nonadaptive local measurements. For all the tested states, the estimated infidelity is inversely proportional to the number of samples, which illustrates the power to characterize a quantum state with a small number of samples. Compared to the globally optimal strategy which requires nonlocal measurements, the efficiency in our experiment is only worse by a small constant factor (<2.5). We compare the performance difference between quantum state verification and quantum state tomography in an experiment to characterize a four-photon Greenberger-Horne-Zeilinger state, and the results indicate the advantage of quantum state verification in both the achieved efficiency and precision.
We show that quantum state tomography with perfect knowledge of the measurement apparatus proves to be, in some instances, inferior to strategies discarding all information about the measurement at hand, as in the case of data pattern tomography. In those scenarios, the larger uncertainty about the measurement is traded for the smaller uncertainty about the reconstructed signal. This effect is more pronounced for minimal or nearly minimal informationally complete measurement settings, which are of utmost practical importance.