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arXiv:1708.09361 · Physical Review A

Heisenberg scaling with classical long-range correlations

Samuel Fernández-Lorenzo, Jacob Dunningham, Diego Porras

Published Feb 27, 2018• 11 citations
arXivHTMLPDF

Abstract

The Heisenberg scaling is typically associated with nonclassicality and entanglement. In this work, however, we discuss how classical long-range correlations between lattice sites in many-body systems may lead to a $1/N$ scaling in precision with the number of probes in the context of quantum optical dissipative systems. In particular, we show that networks of coupled single qubit lasers can be mapped onto a classical $XY$ model, and a Heisenberg scaling with the number of sites appears when estimating the amplitude and phase of a weak periodic driving field.

DOI
10.1103/physreva.97.023843

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The Heisenberg scaling is typically associated with nonclassicality and entanglement. In this work, however, we discuss how classical long-range correlations between lattice sites in many-body systems may lead to a 1/N1/N1/N scaling in precision with the number of probes in the context of quantum optical dissipative systems. In particular, we show that networks of coupled single qubit lasers can be mapped onto a classical XYXYXY model, and a Heisenberg scaling with the number of sites appears when estimating the amplitude and phase of a weak periodic driving field.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
Quantum metrology, such as quantum phase estimation, can surpass classical sensing limits, reaching the Heisenberg‐scaling precision. So far, this kind of metrology has been thought to be only implementable with the quantum systems, which, however, are fragile to environmental noise and hardly contribute to the practical detections. Herein, it is demonstrated both theoretically and experimentally that the parameter encoded by the optical phase can also be estimated at the Heisenberg scaling in classical optics. Inspired by the quantum‐entanglement‐enhanced sensing scheme, the estimation is performed by using classically correlated beams as probes, and obtaining the probes readout after their interaction with the target system. Because the correlated beams considered are spatially separable, a distributed phase estimation scheme is given, which can sense the linear combinations of the phase shifts induced by distinct systems. The results of our experiments show an error reduction up to 3.89 dB below the classical limit when the correlated beam number for probing is 6, approaching the Heisenberg limit. Compared with quantum strategies, the proposal shows a better robustness against the environmental disturbance and keeps their performances even when the correlated beam number is relatively large. Hence, it indicates promising practical applications in the future.
Theoretical proposals to reach the Heisenberg scaling of the measurement precision typically require carefully engineered interactions or initial entanglement. In studying optical magnetometry, we show that the continuous collective measurement process itself can generate the necessary many-body quantum correlations to achieve the elusive Heisenberg scaling of the quantum Fisher information in a dissipative, steady-state system without direct inter-atomic interactions. By contrasting a correlation-neglecting but otherwise consistent semiclassical model, which can violate the quantum Cramér-Rao bound (QCRB) by several orders of magnitude, with a collective quantum model, we isolate measurement-induced correlations as the essential mechanism. The violation of the QCRB serves thereby as a fundamental sanity check for semiclassical spectroscopic theories. This work reveals measurement-induced correlations as a widely unexplored quantum resource for quantum-enhanced sensing, establishes a new paradigm for achieving Heisenberg scaling in open quantum systems, and provides a direct path to test the foundations of quantum mechanics using macroscopic atom ensembles.
Abstract We consider the estimation of an arbitrary parameter φ, such as the temperature or a magnetic field, affecting in a distributed manner the components of an arbitrary linear optical passive network, such as an integrated chip. We demonstrate that Heisenberg scaling precision (i.e. of the order of 1/N, where N is the number of probe photons) can be achieved without any iterative adaptation of the interferometer hardware and by using only a simple, single, squeezed light source and well-established homodyne measurements techniques. Furthermore, no constraint on the possible values of the parameter is needed but only a preliminary shot-noise estimation (i.e. with a precision of N ) easily achievable without any quantum resources. Indeed, such a classical knowledge of the parameter is enough to prepare a single, suitable optical stage either at the input or the output of the network to monitor with Heisenberg-limited precision any variation of the parameter to the order of 1 / N without the need to iteratively modify such a stage.
Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time TTT, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, RRR: when the stochastic signal exhibits high-rank correlations across the RRR channels and the probe is entangled, the average variance (per parameter) scales no better than Ω(1/(TR2))Ω(1/(TR^2))Ω(1/(TR2)). For collective kkk-body dissipation, R=Θ(Nk)R=Θ(N^k)R=Θ(Nk), signifying super-Heisenberg scaling with the system size NNN. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.
We address the issue of precisely estimating small parameters encoded in a general linear transformation of the modes of a bosonic quantum field. Such Bogoliubov transformations frequently appear in the context of quantum optics. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires nonclassical but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to monitor only finitely many modes, for which we identify a lower bound.