Quantum learning (in metrology and machine learning) involves estimating unknown parameters from measurements of quantum states. The quantum Fisher information matrix can bound the average amount of information learnt about the unknown parameters per experimental trial. In several scenarios, it is advantageous to concentrate information in as few states as possible. Here, we present two "go-go" theorems proving that negativity, a narrower nonclassicality concept than noncommutation, enables unbounded and lossless distillation of Fisher information about multiple parameters in quantum learning.
Quantum learning (in metrology and machine learning) involves estimating unknown parameters from measurements of quantum states. The quantum Fisher information matrix can bound the average amount of information learnt about the unknown parameters per experimental trial. In several scenarios, it is advantageous to concentrate information in as few states as possible. Here, we present two "go-go" theorems proving that negativity, a narrower nonclassicality concept than noncommutation, enables unbounded and lossless distillation of Fisher information about multiple parameters in quantum learning.
Several tasks in quantum-information processing involve quantum learning. For example, quantum sensing, quantum machine learning, and quantum-computer calibration involve learning and estimating unknown parameters $\mathbit{\ensuremath{\theta}}=({\ensuremath{\theta}}_{1},\ensuremath{\cdots},{\ensuremath{\theta}}_{M})$ from measurements of many copies of a quantum state ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}$. This type of metrological information is described by the quantum Fisher information matrix, which bounds the average amount of information learned about $\mathbit{\ensuremath{\theta}}$ per measurement of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}$. In several scenarios, it is advantageous to compress the multiparameter information encoded in ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\bigotimes}n}$ into ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathbit{\ensuremath{\theta}}}^{\text{ps}}\phantom{\rule{0.16em}{0ex}}{}^{\ensuremath{\bigotimes}m}$, where $m\ensuremath{\ll}n$. Here, we present a ``go-go'' theorem proving that m/n can be made arbitrarily small, and that the information compression can happen without loss of information. We also demonstrate how to construct filters that perform this unbounded and lossless information compression. These filters can, for example, reduce arbitrarily the quantum-state intensity on experimental detectors, while retaining all initial information. Finally, we prove that the ability to compress quantum Fisher information is a nonclassical advantage that stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution.
Quantum neural networks (QNNs) have been a promising framework in pursuing near-term quantum advantage in various fields, where many applications can be viewed as learning a quantum state that encodes useful data. As a quantum analog of probability distribution learning, quantum state learning is theoretically and practically essential in quantum machine learning. In this paper, we develop a no-go theorem for learning an unknown quantum state with QNNs even starting from a high-fidelity initial state. We prove that when the loss value is lower than a critical threshold, the probability of avoiding local minima vanishes exponentially with the qubit count, while only grows polynomially with the circuit depth. The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant, which characterizes the sensibility of the output state with respect to parameters in QNNs. These results hold for any circuit structures, initialization strategies, and work for both fixed ansatzes and adaptive methods. Extensive numerical simulations are performed to validate our theoretical results. Our findings place generic limits on good initial guesses and adaptive methods for improving the learnability and scalability of QNNs, and deepen the understanding of prior information's role in QNNs.
Quantum metrology offers the possibility of quantum enhancements of the precision of various sensing tasks. In this manuscript, we tackle two open problems in the theory of single-shot quantum parameter learning, going beyond the usual setting of local parameter estimation via repeated measurements. The first concerns the construction of global upper bounds on the learning precision. The second concerns rigorous guarantees on the success probability of parameter learning, namely, lower bounds on the probability of learning a parameter with a certain precision, given the constraints on the resources used for the quantum metrology task. We provide rigorous, practical, and global upper bounds and success-probability guarantees for quantum parameter learning. Most importantly, we establish a fidelity-based learning guarantee for generic mixed-state models that can be viewed as the achievability-side analogue of the quantum Cramer-Rao bound. Whereas the latter provides a no-go constraint, based on the local curvature of the fidelities, our bound uses only pairwise fidelities between parameter-encoded states to certify that a prescribed precision is attainable with a guaranteed success probability. We demonstrate the versatility of the new bounds in a Rabi-frequency-learning example involving a driven qubit coupled to a bosonic environment and a collective-spin Hamiltonian learning problem. Together, the new global bounds and success-probability guarantees allow us to rule out unattainable precision and to certify attainable precision beyond what is possible via standard Fisher-information analysis or binary hypothesis testing bounds. They also allow one to tractably characterize the performance of various learning schemes, without the overhead of an explicit simulation.
In every parameter-estimation experiment, the final measurement or the postprocessing incurs a cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from a trial) to cost. We show that this improvement stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution is real and nonnegative. In a quantum-mechanically noncommuting theory, nonclassicality manifests in negative or nonreal quasiprobabilities. Negative quasiprobabilities enable postselected experiments to outperform optimal postselection-free experiments: postselected quantum experiments can yield anomalously large information-cost rates. This advantage, we prove, is unrealizable in any classically commuting theory. Finally, we construct a preparation-and-postselection procedure that yields an arbitrarily large Fisher information. Our results establish the nonclassicality of a metrological advantage, leveraging our quasiprobability distribution as a mathematical tool.
Abstract: In every parameter-estimation experiment, the final measurement or the postprocessing incurs a cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from a trial) to cost. We show that this improvement stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution is real and nonnegative. In a quantum-mechanically noncommuting theory, nonclassicality manifests in negative or nonreal quasiprobabilities. Negative quasiprobabilities enable postselected experiments to outperform optimal postselection-free experiments: postselected quantum experiments can yield anomalously large information-cost rates. This advantage, we prove, is unrealizable in any classically commuting theory. Finally, we construct a preparation-and-postselection procedure that yields an arbitrarily large Fisher information. Our results establish the nonclassicality of a metrological advantage, leveraging our quasiprobability distribution as a mathematical tool.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.