Noisy Metrology beyond the Standard Quantum Limit

Abstract

Parameter estimation is of fundamental importance in areas from atomic spectroscopy and atomic clocks to gravitational wave-detection. Entangled probes provide a significant precision gain over classical strategies in the absence of noise. However, recent results seem to indicate that any small amount of realistic noise restricts the advantage of quantum strategies to an improvement by at most a multiplicative constant. Here we identify a relevant scenario in which one can overcome this restriction and attain super-classical precision scaling even in the presence of uncorrelated noise. We show that precision can be significantly enhanced when the noise is concentrated along some spatial direction, while the Hamiltonian governing the evolution which depends on the parameter to be estimated can be engineered to point along a different direction. In the case of perpendicular orientation, we find super-classical scaling and identify a state which achieves the optimum.Estimation of an unknown parameter is essential across disciplines from atomic spectroscopy and clocks [1][2][3] to gravitational wave-detection [4]. It is typically achieved by letting a probe, e.g. light, interact with the system under investigation, picking up information about the desired parameter. As seen in Fig. 1, a metrology protocol can be understood in four main steps [5,6]: i) preparation of the probe, ii) interaction with the system, iii) readout of the probe, and iv) construction of an estimate of the unknown parameter from the results. Steps (i)-(iii) may be repeated many times before the final construction of the estimate. FIG. 1.General metrology protocol where a known probe state evolves according to a physical evolution depending on an unknown parameter ω. After sufficient amount of data is collected an estimate for the parameter is constructed.The estimate uncertainty will depend on the available resources, here the probe size N and the total time T available for the experiment (other choices are possible [7]). By the central limit theorem, for N uncorrelated particles, the best uncertainty scales as 1/ √ νN , where ν = T /t is the number of evolve-and-measure rounds. This bound is known as the shot-noise or standard quantum limit (SQL). By making use of quantum phenomena, a metrology protocol may surpass the SQL, reaching instead the limits imposed by the quantum uncertainty relations. For probes of non-interacting particles, the best possible scaling compatible with these relations is 1/( √ νN ), known as the Heisenberg limit. Without noise, the Heisenberg limit can be attained using entangled input states, e.g. Greenberger-HorneZeilinger (GHZ) states for atomic spectroscopy [8]. In the presence of noise however, the picture is much less clear, as the optimal strategy depends strongly on the model of decoherence considered. Nevertheless, the SQL has been significantly surpassed in experiments of optical magnetometry [9,10], which proved that some sources of noise can be effectively counterbalanced [11,12]. However, unless one can k...

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