Quantum metrology with full and fast quantum control

Abstract

We establish general limits on how precise a parameter, e.g., frequency or the strength of a magnetic field, can be estimated with the aid of full and fast quantum control. We consider uncorrelated noisy evolutions of N qubits and show that fast control allows to fully restore the Heisenberg scaling (∼ 1/N 2 ) for all rank-one Pauli noise except dephasing. For all other types of noise the asymptotic quantum enhancement is unavoidably limited to a constant-factor improvement over the standard quantum limit (∼ 1/N ) even when allowing for the full power of fast control. The latter holds both in the single-shot and infinitelymany repetitions scenarios. However, even in this case allowing for fast quantum control helps to improve the asymptotic constant factor. Furthermore, for frequency estimation with finite resource we show how a parallel scheme utilizing any fixed number of entangled qubits but no fast quantum control can be outperformed by a simple, easily implementable, sequential scheme which only requires entanglement between one sensing and one auxiliary qubit.

Fundamental limits to frequency estimation: a comprehensive microscopic perspective

“…Let us stress that we did not invoke the Born-Markov approximation [41] in our derivation-the above time-local master equation includes fully general non-Markovian effects and it will provide us with a satisfactory description of the noisy evolution of the probes as long as the interaction with the environment is weak enough (i.e., the higher orders of the TCL expansion can be neglected). In addition, we are taking into account the dependence of the coefficients of the dissipative part of the master equation on the free system frequency ω 0 , see equation (36), i.e., on the parameter to be estimated. This is a natural consequence of the detailed microscopic derivation of the system dynamics [41], in contrast with the phenomenological approaches, where the master equation is postulated on the basis of the noise effects to be described.…”

“…This is a natural consequence of the detailed microscopic derivation of the system dynamics [41], in contrast with the phenomenological approaches, where the master equation is postulated on the basis of the noise effects to be described. Let us emphasize that only in the case of pure dephasing, for which J p = 2 and all dissipative terms in (36) apart from b zz (t) vanish, the dissipative part of the master equation can be assured not to depend on ω 0 . Otherwise, this is not generally the case unless a special choice of J(w) is made (e.g., discussed later in section 5.2).…”

“…where the rate γ(t) and the dissipative operator s are given by It is worth noting that the dissipative part of the master equation is fixed by one single operator s , i.e., we have the most general qubit rank-one Pauli noise, recently proved to be correctable in the semigroup case (γ(t) = const) in quantum metrology by ancilla-assisted error-correction [36], which has been demonstrated experimentally for transversal coupling, ϑ=0, in [93]. In addition, the only noise rate γ(t) is a positive function of time, which guarantees not only the CP of the dynamics, but also that the dynamical maps can be always split into CP terms.…”

Many-body effects in quantum metrology

“…When inspecting the asymptotic behavior of n ã we then note that the dominant scaling of n ã with N will be determined by the coefficients h (and h) which have the highest scaling exponent-the terms corresponding to noise operators with the lowest nonlinearity level. Therefore, in the scaling formula (15), we need to take l l min i * = ({ }). The presence of higher order nonlinearities (i.e.…”

Practical quantum metrology in noisy environments

“…However, the QFI also enters in both the van Trees inequality [32] and the Ziv-Zakai bound [33], which can provide tighter and more versatile bounds on the mean-square error δ 2 ϕ in the relevant case of finite M (including Bayesian settings). Therefore, we shall adopt the QFI as our main figure of merit, in keeping with the quantum metrology bulk literature (see, e.g., [34] for a recent discussion) and in compliance with the spirit of this paper, which focuses on the use of finite resources to retain quantum enhancements in the estimation sensitivity. We will also assume maximization of F over the initial state of the probe unless stated otherwise.…”