Quantum metrology promises high-precision measurements of classical parameters with far reaching implications for science and technology. So far, research has concentrated almost exclusively on quantum-enhancements in integrable systems, such as precessing spins or harmonic oscillators prepared in non-classical states. Here we show that large benefits can be drawn from rendering integrable quantum sensors chaotic, both in terms of achievable sensitivity as well as robustness to noise, while avoiding the challenge of preparing and protecting large-scale entanglement. We apply the method to spin-precession magnetometry and show in particular that the sensitivity of state-of-the-art magnetometers can be further enhanced by subjecting the spin-precession to non-linear kicks that renders the dynamics chaotic.

Recently proposed quantum-chaotic sensors achieve quantum enhancements in measurement precision by applying nonlinear control pulses to the dynamics of the quantum sensor while using classical initial states that are easy to prepare. Here, we use the cross-entropy method of reinforcement learning (RL) to optimize the strength and position of control pulses. Compared to the quantumchaotic sensors with periodic control pulses in the presence of superradiant damping, we find that decoherence can be fought even better and measurement precision can be enhanced further by optimizing the control. In some examples, we find enhancements in sensitivity by more than an order of magnitude. By visualizing the evolution of the quantum state, the mechanism exploited by the RL method is identified as a kind of spin-squeezing strategy that is adapted to the superradiant damping.

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We study quantum metrology for unitary dynamics. Analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision. This represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available. In particular, we provide a generalization to mixed states of an upper bound on measurement precision for time-dependent Hamiltonians that can be saturated with optimal Hamiltonian control. These results make precise and reveal the full potential of mixed states for quantum metrology.The standard paradigm of quantum metrology involves the preparation of an initial state, a parameterdependent dynamics, and a consecutive quantum measurement of the evolved state. From the measurement outcomes the parameter can be estimated [1][2][3]. Naturally, it is the goal to estimate the parameter as precisely as possible, i.e., to reduce the uncertainty ∆α = Var(α) 1 2 of the estimatorα of the parameter α that we want to estimate. Quantum coherence and nonclassical correlations in quantum sensors help to reduce this uncertainty compared to what is possible with comparable classical resources [4,5]. The ultimate precision limit is given by the quantum Cramér-Rao bound ∆α ≥ (M I α ) −1 2 which depends on the number of measurements M and the quantum Fisher information (QFI) I α which is a function of the state [6,7]. When the number of measurements is fixed, as they correspond to a limited resource, precision is optimal when the QFI is maximal which involves an optimization with respect to the state.In this Letter, we consider a freely available state ρ, unitary freedom to prepare an initial state from ρ, and unitary parameter-dependent dynamics of the quantum system (see Fig. 1). The parameter-dependent dynamics will be called sensor dynamics in the following in order to distinguish it from the state preparation dynamics. For instance, in a spin system the unitary freedom can be used to squeeze the spin before it is subjected to the sensor dynamics, as it is the case in many quantum-enhanced measurements [8][9][10][11]. In the worst case scenario, only the maximally mixed state is available, which does not change under unitary state preparation or unitary sensor dynamics and, thus, no information about the parameter can be gained. In the best-case scenario the available state is pure, when the maximal QFI as well as the optimal state to be prepared are well-known [12,13].The appeal and advantage of the theoretical study of unitary sensor dynamics lies in the analytic solutions that can be found that allow fundamental insights in the limits of quantum metrology and the role of resources such as measurement time and system size. The QFI maximized available state unitary state preparation initial state readout unitary sensor dynamics available state Figure 1. Schematic representation of the metrology protocol.with respect to initial stat...