Measurements with randomly chosen settings determine many important properties of quantum states without the need for a shared reference frame or calibration. They naturally emerge in the context of quantum communication and quantum computing when dealing with noisy environments, and allow the estimation of properties of complex quantum systems in an easy and efficient manner. In this review, we present the advancements made in utilising randomised measurements in various scenarios of quantum information science . We describe how to detect and characterise different forms of entanglement, including genuine multipartite entanglement and bound entanglement. Bell inequalities are discussed to be typically violated even with randomised measurements, especially for a growing number of particles and settings. Furthermore, we also present an overview on the estimation of non-linear functions of quantum states and shadow tomography from randomised measurements. Throughout the review, we complement the description of theoretical ideas by explaining key experiments.
Measurements with randomly chosen settings determine many important properties of quantum states without the need for a shared reference frame or calibration. They naturally emerge in the context of quantum communication and quantum computing when dealing with noisy environments, and allow the estimation of properties of complex quantum systems in an easy and efficient manner. In this review, we present the advancements made in utilising randomised measurements in various scenarios of quantum information science . We describe how to detect and characterise different forms of entanglement, including genuine multipartite entanglement and bound entanglement. Bell inequalities are discussed to be typically violated even with randomised measurements, especially for a growing number of particles and settings. Furthermore, we also present an overview on the estimation of non-linear functions of quantum states and shadow tomography from randomised measurements. Throughout the review, we complement the description of theoretical ideas by explaining key experiments.
Quantum entanglement is one of the core features of quantum theory. While it is typically revealed by measurements along carefully chosen directions, here we review different methods based on so-called random or randomized measurements. Although this approach might seem inefficient at first, sampling correlations in various random directions is a powerful tool to study properties which are invariant under local-unitary transformations. Based on random measurements, entanglement can be detected and characterized without a shared reference frame between the observers or even if local reference frames cannot be defined. This overview article discusses different methods using random measurements to detect genuine multipartite entanglement and to distinguish SLOCC classes. Furthermore, it reviews how measurement directions can efficiently be obtained based on spherical designs.
The concept of randomized measurements on individual particles has proven to be useful for analyzing quantum systems and is central for methods like shadow tomography of quantum states. We introduce collective randomized measurements as a tool in quantum information processing. Our idea is to perform measurements of collective angular momentum on a quantum system and actively rotate the directions using simultaneous multilateral unitaries. Based on the moments of the resulting probability distribution, we propose systematic approaches to characterize quantum entanglement in a collective-reference-frame-independent manner. First, we show that existing spin-squeezing inequalities can be accessible in this scenario. Next, we present an entanglement criterion based on three-body correlations, going beyond spin-squeezing inequalities with two-body correlations. Finally, we apply our method to characterize entanglement between spatially separated two ensembles.
Quantum states and measurements are two central physical objects in the theory of quantum mechanics. The preparation, control and measurement of quantum states lies at the heart of modern quantum information processing protocols ranging from secure communication to quantum computation and precision sensing. And yet, the reliable operation of these advanced protocols ultimately rests on a far more essential prerequisite - our ability to calibrate, certify and benchmark the quantum states we prepare and the quantum measurements we perform. Quantum information theory provides a natural framework for these tasks, meaningfully connecting the resources we expend for characterisation with the accuracy and precision we can expect in return. While decades of advances in quantum information have refined our understanding of quantum characterisation and sharpened our practical ability to calibrate with enhanced precision, important questions about the fundamental limits of quantum characterisation remain unanswered. Although incorporating entanglement at the measurement stage is known to enhance the estimation of unknown quantum states, the extent of this improvement remains poorly understood. This leaves open a fundamental question on the utility of complex entangling measurement strategies, impeding progress towards bridging the significant gap between practically achievable precision and the ultimate theoretical quantum limit. Complementary to quantum state estimation, which is relatively well-established in theory and practice, the characterisation of quantum measurements lacks a rigorous information-theoretic foundation, leading to a dearth of principled benchmarks and provably optimal strategies despite growing experimental interest. This thesis develops and applies information-theoretic tools to answer questions in quantum characterisation, with an eye to enabling practical improvements beyond current techniques in the field. First, we introduce a computable measure of entanglement enhancement for quantum state estimation and study its variation across general multi-parameter estimation problems for higher-dimensional quantum states. We show that the entanglement enhancement is maximised for full qudit state tomography but grows only linearly with system dimension even when allowing an asymptotically large number of entangling operations. Our proposed tight upper bound to entanglement enhancement therefore constitutes a strong limitation on the utility of such entangling measurements. Next, we formulate the dual problem of quantum detector estimation, developing a comprehensive framework for maximally precise calibration centred around a newly introduced quantity - the detector quantum Fisher information. This information-theoretic measure stipulates the ultimate precision attainable for measurement calibration and identifies the optimal input states that achieve this precision. Further, our framework enables the first provably optimal detector estimation experiment and reveals the scope for entanglement-based enhancements, thereby placing measurement characterisation on equal footing with state characterisation. Considered together, these results suggest two contrasting conclusions about quantum characterisation. On the one hand, they show that for quantum state estimation, increasingly powerful entangling measurements offer only diminishing returns at the ultimate precision limit. On the other hand, they highlight practical quantum advantages available for detector estimation by providing a framework that can improve calibration performance and guide future experimental design. More broadly, these results complete and advance the triad of state, process, and detector characterisation, positioning quantum information measures as not merely abstract bounds, but practical guides for deciding when complex quantum resources are genuinely worth deploying and where near-term quantum advantages are most likely to be realised.
Symmetric quantum states are fascinating objects. They correspond to multipartite systems that remain invariant under particle permutations. This symmetry is reflected in their compact mathematical characterisation but also in their unique physical properties: they exhibit genuine multipartite entanglement and notable robustness against noise and perturbations. These features make such states particularly well-suited for a wide range of quantum information tasks. Here, we provide a pedagogic analysis of the mathematical structure and relevant physical properties of this class of states. Beyond the theoretical framework, robust tools for certifying and verifying the properties of symmetric states in experimental settings are essential. In this regard, we explore how standard techniques-such as quantum state tomography, Bell tests, and entanglement witnesses-can be specifically adapted for symmetric systems. Next, we provide an up-to-date overview of the most relevant applications in which these states outperform other classes of states in specific tasks. Specifically, we address their central role in quantum metrology, highlight their use in quantum error correction codes, and examine their contribution in computation and communication tasks. Finally, we present the current state-of-the-art in their experimental generation, ranging from systems of cold atoms to implementations via quantum algorithms. We also review the most significant results obtained in the different experimental realizations. Despite the notable progress made in recent years with regard to the characterisation and application of symmetric quantum states, several intriguing questions remain unsolved. We conclude this review by discussing some of these open problems and outlining promising directions for future research.
The fact that quantum mechanics predicts stronger correlations than classical physics is an essential cornerstone of quantum information processing. Indeed, these quantum correlations are a valuable resource for various tasks, such as quantum key distribution or quantum teleportation, but characterizing these correlations in an experimental setting is a formidable task, especially in scenarios where no shared reference frames are available. By definition, quantum correlations are reference-frame independent, i.e., invariant under local transformations; this physically motivated invariance implies, however, a dedicated mathematical structure and, therefore, constitutes a roadblock for an efficient analysis of these correlations in experiments. Here we provide a method to directly measure any locally invariant property of quantum states using locally randomized measurements, and we present a detailed toolbox to analyze these correlations for two quantum bits. We implement these methods experimentally using pairs of entangled photons, characterizing their usefulness for quantum teleportation and their potential to display quantum nonlocality in its simplest form. Our results can be applied to various quantum computing platforms, allowing simple analysis of correlations between arbitrary distant qubits in the architecture.
The experimental detection of multipartite entanglement usually requires a number of appropriately chosen local quantum measurements that are aligned with respect to a previously shared common reference frame. The latter, however, can be a challenging prerequisite, e.g., for satellite-based photonic quantum communication, making the development of alternative detection strategies desirable. One possibility for avoiding the distribution of classical reference frames is to perform a number of local measurements with settings distributed uniformly at random. In this Letter, we follow such a treatment and show that an improved detection and characterization of multipartite entanglement is possible by combining statistical moments of different order. To do so, we make use of designs that are pseudorandom processes allowing us to link the present entanglement criteria to ordinary reference frame independent ones. The strengths of our methods are illustrated in various cases, starting with two qubits and followed by more involved multipartite scenarios.