For unitary transformations, the quantum Fisher information (QFI) of a pure state is given by the uncertainty of the generator in that state. In post-selected metrology, the QFI is given by a modified expression describing conditional quantum statistics of the generator. Here, we show that the conditional generator uncertainties defined by post-selected QFI correspond to Ozawa-Hall uncertainties known from the theoretical analysis of quantum measurements. The post-selected measurement outcome updates the generator uncertainty according to the quantum statistics of that outcome. Enhancements of QFI beyond the maximal uncertainties of the generator eigenvalues are possible because post-selection tends to concentrate the largest part of the initial generator uncertainty in low probability outcomes of the post-selection measurement. Anomalous conditional uncertainties thus explain the extreme sensitivities that can be achieved in post-selected metrology.
For unitary transformations, the quantum Fisher information (QFI) of a pure state is given by the uncertainty of the generator in that state. In post-selected metrology, the QFI is given by a modified expression describing conditional quantum statistics of the generator. Here, we show that the conditional generator uncertainties defined by post-selected QFI correspond to Ozawa-Hall uncertainties known from the theoretical analysis of quantum measurements. The post-selected measurement outcome updates the generator uncertainty according to the quantum statistics of that outcome. Enhancements of QFI beyond the maximal uncertainties of the generator eigenvalues are possible because post-selection tends to concentrate the largest part of the initial generator uncertainty in low probability outcomes of the post-selection measurement. Anomalous conditional uncertainties thus explain the extreme sensitivities that can be achieved in post-selected metrology.
Quantum metrology uses small changes in the output probabilities of a quantum measurement to estimate the magnitude of a weak interaction with the system. The sensitivity of this procedure depends on the relation between the input state, the measurement results, and the generator observable describing the effect of the weak interaction on the system. This is similar to the situation in weak measurements, where the weak value of an observable exhibits a symmetric dependence on initial and final conditions. In this paper, it is shown that the phase sensitivity of a quantum measurement is in fact given by the variance of the imaginary parts of the weak values of the generator over the different measurement outcomes. It is then possible to include the limitations of a specific quantum measurement in the uncertainty bound for phase estimates by subtracting the variance of the real parts of the weak values from the initial generator uncertainty. This uncertainty relation can be interpreted as the time-symmetric formulation of the uncertainty limit of quantum metrology, where the real parts of the weak values represent the information about the generator observable in the final measurement result.
Statistical paradoxes such as the Hardy paradox and the enhancement of phase estimation via post-selection both draw upon the same non-classical features of quantum statistics described by non-positive quasi-probabilities. In this paper, we introduce a post-selected quantum metrology scenario where the initial state, the dynamics associated with the phase shift, and the post-selection are all inspired by the Hardy paradox. Specifically, we identify an anomalous weak value that is characteristic of both the Hardy paradox and the potential enhancement of sensitivity by the post-selection. We find that the efficiency of the enhancement is reduced when the expectation value associated with the anomalous weak value is different from the inverse of this value. We conclude that the relation between enhanced phase estimation and the Hardy paradox requires a detailed understanding of the relation between weak values and expectation values.
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.
Abstract We demonstrate that the contextuality of measurement selection can enhance the precision of quantum metrology with a simple linear optical experiment. Contextuality is a nonclassical property known as a resource for various quantum information processing tasks. Recent studies show that contextuality by anomalous weak values can be utilized to enhance metrological precision, unraveling the role of contextuality in quantum metrology. Our contextual quantum metrology (coQM) scheme can elevate the precision inferred from the observed Fisher information as much as six times the precision limit given by the quantum Fisher information (QFI), and, when the Godambe information is used, the precision inferred becomes about two times lower than the precision limit. We achieve the contextuality-enabled precision with two mutually complementary measurements, whereas, in the conventional method, some optimal measurements to achieve the precision limit are either theoretically challenging to find or experimentally infeasible to realize. These results highlight that the contextuality of measurement selection is applicable in practice for quantum metrology.
Abstract Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with a thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal quantum Fisher information (QFI) after the postselection for an accepted event (conversely, the postselection for a rejected event) can attain a level of thermal fluctuation, without surpassing the total QFI, and further, that QFI which increases with increasing temperature can constantly improve measurement precision. These results are much better than those of weak measurement with a pure state (i.e. Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction systems with weak measurement, and by using a thermal state pointer, we obtain in the phase space the accepted postselection and the postselected measurement both achieving the Heisenberg limit of quantum metrology, and showing that weak measurement with thermal states only obtain classical Fisher information (CFI) which increases with increasing temperature and achieves classical-enhanced scaling of N 2 . Moreover, weak measurement with thermal states has an advantage over that with coherent states or mixed states of light because generating these states with larger uncertainty is limited under current technology, but thermal states with larger uncertainly are very easy to achieve with increasing temperature in nature, such as for optical systems and optomechanical systems.
Entanglement measures such as concurrence, negativity and REE are well-known tools for quantifying non-classical correlations in quantum systems. However, these measures can lead to different state orderings for non-maximally entangled states. On the other hand, Quantum Fisher Information (QFI), while not an entanglement measure, provides a framework for analyzing a state's metrological potential. In this study, we numerically analyze the relationship between these entanglement measures and QFI for a large ensemble of random two-qubit states. We specifically focus on the maximized QFI (MQFI) obtained through local unitary rotations. Our findings demonstrate a strong correlation between entanglement and a state's metrological capacity, confirming that entanglement is a valuable resource. We show that while a state's QFI with a fixed generator can vary widely for a given entanglement value, local optimization to find the MQFI leads to a tighter, more predictable relationship. Our results also reinforce the principle that the metrological performance of all mixed states is fundamentally bounded by the theoretical limit of pure states. The polynomial fit equations for the upper and lower bounds of our data provide a quantitative description of these complex relationships.