Bosonic systems, particularly in quantum optics and atomic physics, are leading platforms for achieving quantum enhanced precision in parameter estimation. By exploiting properties such as mode and particle entanglement, it is possible to attain precisions that surpass the shot noise limit with respect to key resources like probe number or energy. Yet the mechanisms by which these bosonic resources enable quantum enhancement remain unclear. Consequently, the design of optimal probes and evolutions often relies on case by case analyses, where continuous and discrete variable regimes are treated separately and their connection is still unclear. We develop a comprehensive framework for quantum metrology that unifies all known precision enhancement mechanisms based on bosonic systems. Our approach employs a superselection rule compliant representation of the electromagnetic field that explicitly includes the phase reference, thereby enforcing total particle number conservation and bridging the discrete and continuous limits of quantum optics and symmetric massive systems. Within this unified formalism, of which established results emerge as special cases, we identify the distinct roles of mode and particle entanglement for quantum enhanced precision. The framework further provides general measurement optimization strategies for arbitrary multimode entangled probe states and naturally incorporates noise and non-unitary dynamics, ensuring applicability to realistic experimental conditions.
Bosonic systems, particularly in quantum optics and atomic physics, are leading platforms for achieving quantum enhanced precision in parameter estimation. By exploiting properties such as mode and particle entanglement, it is possible to attain precisions that surpass the shot noise limit with respect to key resources like probe number or energy. Yet the mechanisms by which these bosonic resources enable quantum enhancement remain unclear. Consequently, the design of optimal probes and evolutions often relies on case by case analyses, where continuous and discrete variable regimes are treated separately and their connection is still unclear. We develop a comprehensive framework for quantum metrology that unifies all known precision enhancement mechanisms based on bosonic systems. Our approach employs a superselection rule compliant representation of the electromagnetic field that explicitly includes the phase reference, thereby enforcing total particle number conservation and bridging the discrete and continuous limits of quantum optics and symmetric massive systems. Within this unified formalism, of which established results emerge as special cases, we identify the distinct roles of mode and particle entanglement for quantum enhanced precision. The framework further provides general measurement optimization strategies for arbitrary multimode entangled probe states and naturally incorporates noise and non-unitary dynamics, ensuring applicability to realistic experimental conditions.
Since the early days of quantum information science, diverse strategies have been proposed to encode qubits and implement quantum gates across various physical platforms. Modes and bosonic states offer a flexible framework for quantum information processing, from single-photon optics to superconducting circuits, combining massless and massive particles. Universal gate sets have been established for specific encodings—such as those based on Fock states or the field’s quadratures. However, a general framework clarifying how different bosonic resources—such as mode structure, particle-number statistics, Gaussian and non-Gaussian states and operations—combine to enable universality and potential quantum advantage irrespectively of the encoding remains lacking. A main difficulty lies in the coexistence of distinct paradigms that appear to combine loosely in bosonic quantum information: discrete-variable schemes—where information is encoded in qubits or effective finite-dimensional systems—and continuous-variable schemes—which rely on infinite-dimensional Hilbert spaces associated with the field’s quadratures. Despite the success of each approach, they remain seemingly incompatible in many aspects—in particular in terms of physical resource analysis for universality—due to their underlying mathematical differences. In this work, we apply a superselection-rule-compliant framework for bosonic quantum information that unifies these paradigms. This representation explicitly incorporates the phase reference and enforces total particle number conservation, allowing for a consistent treatment of bosonic quantum information protocols and resources across both discrete and continuous regimes. We derive necessary and sufficient conditions expressed in terms of Gaussian and non-Gaussian resources for universality that apply to any encoding, without resorting to specific mappings. Our results show that continuous variable encodings are not fundamentally distinct from discrete ones, but rather emerge as subspaces within a general bosonic structure. Importantly, in such a widely studied framework, non-Gaussian properties of bosonic states and operations can be washed out, providing evidence that the continuous variables regime is not the most suitable to fully assess the quantum properties of bosons.
We present a method to systematically identify and classify quantum optical nonclassical states as classical or nonclassical based on the resources they create on a bosonic quantum computer. This is achieved by converting arbitrary bosonic states into multiple modes, each occupied by a single photon, thereby defining qubits of a bosonic quantum computer. Starting from a bosonic classical-like state in a representation that explicitly respects particle number superselection rules, we apply universal gates to create arbitrary superpositions of states with the same total particle number. The nonclassicality of the corresponding states can then be associated with the operations they induce in the quantum computer. We also provide a correspondence between the adopted representation and the more conventional one in quantum optics, where superpositions of Fock states describe quantum optical states, and we identify how multimode states can lead to quantum advantage. Our work contributes to establish a seamless transition from continuous to discrete properties of quantum optics while laying the grounds for a description of nonclassicality and quantum computational advantage that is applicable to spin systems as well.
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes, or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies that saturate the precision bounds are provided.
As one of the main pillars of quantum technologies, quantum metrology aims to improve measurement precision using techniques from quantum information. The two main strategies to achieve this are the preparation of nonclassical states and the design of optimized measurement observables. We discuss precision limits and optimal strategies in quantum metrology and sensing with a single mode of quantum continuous variables. We focus on the practically most relevant cases of estimating displacements and rotations and provide the sensitivities of the most important classes of states that includes Gaussian states and superpositions of Fock states or coherent states. Fundamental precision limits that are obtained from the quantum Fisher information are compared to the precision of a simple moment-based estimation strategy based on the data obtained from possibly sub-optimal measurement observables, including homodyne, photon number, parity and higher moments. Finally, we summarize some of the main experimental achievements and present emerging platforms for continuous-variable sensing. These results are of particular interest for experiments with quantum light, trapped ions, mechanical oscillators, and microwave resonators.
Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.
Precision measurement plays a vital role in metrology. To achieve the ultimate precision predicted by quantum theory, one can leverage quantum entanglement—correlations between particles of light or matter that cannot be explained in classical physics. The use of quantum entanglement in probing an object enhances measurement precision by a factor of √N beyond what is attainable with classical resources, where precision is typically limited by Poisson noise. Here, N represents the number of entangled particles. To date, most research efforts have focused on understanding and characterizing non-classical correlations between two particles. For instance, in 2022, optical clocks were synchronized using quantum entanglement, achieving frequency uncertainties of √2 lower than what is possible with non-entangled clocks. However, despite rapid advancements in single-particle sensing technologies, the generation and characterization of quantum correlation involving three or more particles remain an ongoing challenge. The absence of rigorous measurement standards for multi-particle entanglement, coupled with the lack of photon-number-resolving (PNR) detection capabilities, can lead to faulty measurements and reduced metrological accuracy and precision. It is therefore essential to establish robust methodologies and standards for the creation and validation of multi-particle entanglement to enable quantum-limited precision measurements. These developments will provide significant impacts in various fields, including optical astronomy, frequency standards, microscopy, and secure communication. The integration of entanglement-assisted sensors with PNR detection will enhance imaging resolution in applications such as astronomy and biological sample analysis. Additionally, standardized metrological frameworks for quantum entanglement and hardware evaluation will enhance the security level of quantum communication by mitigating vulnerabilities associated with inconsistencies arising from the use of dissimilar characterization methodologies. These fundamental research efforts will have far-reaching implications, not only improving the quality of quantum measurements but also guiding national metrology institutes in developing standardized quantum measurement protocols.