Driven-dissipative many-body quantum systems can be exploited in quantum enhanced sensing protocols. Here, non-classical correlations may allow one to reach a measurement precision that surpasses the standard quantum limit. The relevant figure of merit for assessing such quantum advantage is the quantum Fisher information (QFI). In a many-body setting its computation is challenging as it requires knowledge of the full system-environment state. Here we make use of a field-theoretical approach, which is particularly well suited for systems with many degrees of freedom. In particular, we exploit that the QFI can be linked to a Keldysh path-integral. This route leads to a semiclassical expression of the QFI, which is obtained through a controlled expansion in the so-called quantum fields. We benchmark the framework on three systems of increasing complexity: the driven-dissipative harmonic oscillator, the boundary time crystal - which becomes semiclassical at large system sizes - and a two-dimensional array of collectively emitting atoms. We assess in which parameter regimes the semiclassical description holds and gives access to system sizes beyond the reach of exact methods.
Driven-dissipative many-body quantum systems can be exploited in quantum enhanced sensing protocols. Here, non-classical correlations may allow one to reach a measurement precision that surpasses the standard quantum limit. The relevant figure of merit for assessing such quantum advantage is the quantum Fisher information (QFI). In a many-body setting its computation is challenging as it requires knowledge of the full system-environment state. Here we make use of a field-theoretical approach, which is particularly well suited for systems with many degrees of freedom. In particular, we exploit that the QFI can be linked to a Keldysh path-integral. This route leads to a semiclassical expression of the QFI, which is obtained through a controlled expansion in the so-called quantum fields. We benchmark the framework on three systems of increasing complexity: the driven-dissipative harmonic oscillator, the boundary time crystal - which becomes semiclassical at large system sizes - and a two-dimensional array of collectively emitting atoms. We assess in which parameter regimes the semiclassical description holds and gives access to system sizes beyond the reach of exact methods.
Quantum sensors driven into the quantum chaotic regime can have dramatically enhanced sensitivity, which, however, depends intricately on the details of the underlying classical phase space. Here, we develop an accurate semiclassical approach that provides direct and efficient access to the phase-space-resolved quantum Fisher information (QFI), the central quantity that quantifies the ultimate achievable sensitivity. This approximation reveals, in very concrete terms, that the QFI is large whenever a specific dynamical quantity tied to the sensing parameter displays a large variance over the course of the corresponding classical time evolution. Applied to a paradigmatic system of quantum chaos, the kicked top, we show that the semiclassical description is accurate already for modest quantum numbers, i.e., deep in the quantum regime, and it extends seamlessly to very high quantum numbers that are beyond the reach of other methods.
We present a real-time path-integral formulation of the quantum Fisher information for dynamical parameter estimation. For pure states undergoing unitary evolution, we show that the quantum Fisher information can be expressed as a connected symmetrized covariance of a time-integrated action deformation, equivalently as an integrated insertion of ∂λS in the propagator. This reformulation avoids explicit state reconstruction by rewriting the quantum Fisher information in terms of real-time correlators that are natural targets for many-body methods. We further embed the construction into the Schwinger-Keldysh closed-time-path formalism, identifying the quantum Fisher information with the Keldysh component of an appropriate contour-ordered correlator generated by forward and backward propagating sources. Finally, using the Van Vleck-Gutzwiller approximation we re-derive the compact semiclassical quantum Fisher information expression, clarifying how classical trajectory data control leading-order metrological sensitivity.
The continuous monitoring of driven-dissipative systems offers new avenues for quantum advantage in metrology. This approach mixes temporal and spatial correlations in a manner distinct from traditional metrology, leading to ambiguities in how one identifies Heisenberg scalings (e.g.~standard asymptotic metrics like the sensitivity are not bounded by system size). Here, we propose a new metric for continuous sensing, the optimized finite-time environmental quantum Fisher information (QFI), that remedies the above issues by simultaneously treating time and system size as finite resources. In addition to having direct experimental relevance, this quantity is rigorously bounded by both system size and integration time, allowing for a precise formulation of Heisenberg scaling. We also introduce two many-body continuous sensors: the high-temperature superradiant sensor, and the dissipative spin squeezer. Both exhibit Heisenberg scaling of a collective magnetic field for multiple directions. The spin squeezed sensor has a striking advantage over previously studied many-body continuous sensors: the optimal measurement achieving the full QFI does not require the construction of a complex decoder system, but can be achieved using direct photodetection of the cavity output field.
Quantum Fisher information (QFI) has become a fundamental aspect of quantum metrology, connecting the geometry of quantum states to the highest levels of accuracy. In the last ten years, QFI has changed from a purely theoretical concept to a framework that brings together the design of noise-resistant protocols, tests for entanglement in experiments, and benchmarks new quantum platforms. In this Perspective, we examine how QFI is transcending theoretical frameworks to formulate practical solutions for quantum-enhanced sensing in the noisy intermediate-scale quantum (NISQ) era.We emphasize three conceptual borders. First, the interaction between noise and error correction shows that QFI can be both weak and strong, depending on how environmental factors interact with each other. Second, multiparameter estimation and many-body critical phenomena reveal essential trade-offs among measurement incompatibility, entanglement scaling, and decoherence. Third, new experimental platforms, like as photonics, atomic clocks, solid-state qubits, and distributed networks, are increasingly being used with learning-based and random estimation methods that make QFI available on a large scale. We finish by talking about the problems that still need to be solved and the plan for the next ten years. These include making trade-offs between resources in error correction, making sure that critical scaling is strong, and adding QFI to hybrid quantum technologies. We contend that by reconceptualizing QFI as both a theoretical framework and a driver for tangible quantum advantages in sensing, it will be essential in the advancement of quantum technologies.
In this thesis, I focus on the quantum Fisher information (QFI) - a central quantity in parameter estimation theory for quantum systems, which provides a lower bound on the precision one can achieve. In the first part of the thesis, I explore a Lie algebraic approach to evaluate the quantum Fisher information for unitary dynamics. I show how the Lie algebra associated with the unitary evolution, together with the initial state, allows one to compute various quantities of interest, circumventing the computation of the time-evolved state. In particular, I apply this shortcut approach to calculate the QFI and use it for the QFI evaluation in various prototype models, both finiteand infinite-dimensional. In the second part of the thesis, I shift focus to non-unitary dynamics, such as those described by a quantum master equation. In this setting, I theoretically study the description of open system dynamics in the Liouville space of vectorized density matrices and investigate the so-called dissipative quantum Fisher information (DQFI) within that framework. Specifically, I demonstrate how the metric defined in Liouville space differs from its counterpart in Hilbert space. I establish a map between the Hilbert space QFI and the Liouville dissipative quantum Fisher information for qubit and qudit systems, expressed via the quantum state’s purity. Furthermore, I discuss the physical meaning of the DQFI by studying its behavior under completely positive trace-preserving (CPTP) maps. The present results demonstrate that the Lie algebraic approach provides a computationally efficient framework for the QFI evaluation. This approach allows to avoid the explicit solution of quantum dynamics, and remains well-defined in cases when conventional approaches like WeiNorman expansion exhibit singularities.
Quantum Fisher information (QFI) determines the ultimate precision of a quantum sensor after optimization over all allowed measurements, but it does not quantify how difficult the optimal readout is to implement. We introduce a moment Fisher hierarchy FM(k), which measures the fraction of QFI accessible to collective spin moments up to degree k, together with a readout-complexity scale k90 defined by the minimum degree required to recover 90% of the QFI. Applying this framework to a nonlinear collective-spin sensor, we show that many-body dynamics can preserve large QFI while transferring the optimal readout into high-rank SU(2) tensor sectors inaccessible to low-order collective observables. For system sizes up to N=100, first moments access only ∼10−2 of the available QFI, while the degree required to recover 90% of it grows to k90≃28. A rank-resolved analysis of the symmetric logarithmic derivative (SLD) under weak damping shows that the nonlinear protocol pushes the optimal readout toward the highest available tensor ranks while strongly depleting its low-rank weight. These results identify nonlinear tangent-space shear as the mechanism behind a finite-size readout-complexity bottleneck in many-body quantum metrology.