The paper provides a full understanding on the maximal achievable quantum Fisher information (QFI) in quantum sensing protocols, with continuous time dynamics in the presence of Markovian noise, assuming arbitrary quantum controls. It provides simple algebraic conditions on the character of short time and long time scalings of the QFI (quadratic or linear) as well as characterization of the corresponding time scales.
The paper provides a full understanding on the maximal achievable quantum Fisher information (QFI) in quantum sensing protocols, with continuous time dynamics in the presence of Markovian noise, assuming arbitrary quantum controls. It provides simple algebraic conditions on the character of short time and long time scalings of the QFI (quadratic or linear) as well as characterization of the corresponding time scales.
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4 and T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.
Frequency estimation, a cornerstone of quantum metrology, has been significantly enhanced by advanced quantum sensing strategies. However, most protocols rely either on static or time-independent encoding mechanisms, inherently limiting their achievable precision scaling, or on control strategies requiring changing the Hamiltonian and/or implementing feedback mechanisms. To overcome this, we investigate a simpler dynamical encoding protocol where the quantum oscillator is driven by a general continuous temporal frequency modulation Ω(t)=ω0f(t). We analytically demonstrate that for a given modulation profile f(t) and its corresponding time-integral F(t), the quantum Fisher information (QFI) scales as O(F(t)2). This enhancement stems from the fact that temporal encoding fundamentally alters the mechanism of dynamical phase accumulation. Crucially, when evaluated under the energy and evolution-time constraints, this framework reveals a genuine precision enhancement over the conventional time-independent baseline. By analyzing explicit polynomial and exponential modulations, we establish that arbitrary precision scaling can be deterministically engineered, with ultimate bounds that are asymptotically saturable via optimal homodyne detection. Our framework provides a universal paradigm for exploiting time-dependent quantum control in next-generation sensors.
The problem of quantum metrology in the context of a particular non-Markovian quantum evolution is explored. We study the dynamics of the quantum Fisher information (QFI) of a composite quantum probe coupled to a Lorentzian environment, for a full variety of different classes of parameters. We are able to find the best metrological state, which is not maximally entangled but is the one which evolves the most rapidly. This is shown by demonstrating a connection between QFI and different quantum speed limits. At the same time, by optimizing a control field acting on the probes, we show how the total information flow is actively manipulated by the control so as to enhance the parameter estimation at a given final evolution time. Finally, under this controlled scenario, a sharp interplay between the dynamics of QFI, non-Markovianity, and entanglement is revealed within different control schemes.
The quantum Fisher information (QFI) measures the amount of information that a quantum state carries about an unknown parameter. The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. Both the channel QFI and the optimal input state could be solved via a semidefinite program (SDP). In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N→∞, which we call the asymptotic QFI. It was known that the asymptotic QFI grows either linearly or quadratically with N. Here we obtain a simple criterion that determines whether the scaling is linear or quadratic. In both cases, we found a quantum error correction protocol achieving the asymptotic QFI and an SDP to solve the optimal code. When the asymptotic QFI is quadratic, the Heisenberg limit, a feature once thought unique to unitary quantum channels, is recovered. When the asymptotic QFI is linear, we show it is still in general larger than N times the channel QFI, showing the non-additivity of the channel QFI of general quantum channels.
Understanding how thermal fluctuations modify or suppress quantum-enhanced sensing is a central problem in quantum metrology. Closely related questions arise in quantum thermodynamics, particularly regarding the relation between the work induced by a signal and the information acquired by a sensor. Here, we establish general relations among the sensitivity of thermal quantum states, as quantified by different quantum Fisher information metrics, their temperature dependence, and the work induced by a unitary signal. We further show that, at high temperature, all monotone QFI metrics coincide to leading order with a universal quantity that can be expressed as the variance of a simple observable and itself defines a lesser-known QFI metric. This emergent uniqueness is reminiscent of the uniqueness of classical Fisher information in information geometry. In continuous-variable systems, the infinite-temperature limit is finite and generically non-zero for quadratic Gaussian signals, while it diverges for higher-degree signals. Remarkably, for any purely quadratic signal generator, the infinite-temperature QFI is at least twice the zero-temperature SLD QFI or, equivalently, at least eight times the ground-state variance of the generator. Consequently, the Cramér--Rao lower bound on the estimator variance is reduced by at least a factor of two. We illustrate these results using harmonic chains of bosonic modes relevant to trapped-ion platforms and quantum field-theoretic models.
The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter.The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla.In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N → ∞, which is called the asymptotic QFI.Over the years, researchers found various types of upper bounds of the asymptotic QFI, but they were proven achievable only in several specific situations.It was known that the asymptotic QFI of an arbitrary quantum channel grows either linearly or quadratically with N .Here we show that a simple criterion can determine whether the scaling is linear or quadratic.In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are computable via a semidefinite program.When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered.When the scaling is linear, we show that the asymptotic QFI is still in general larger than N times the single-channel QFI and, furthermore, that sequential estimation strategies provide no advantage over parallel ones.