Ultimate Precision Limits for Noisy Frequency Estimation

Abstract

Quantum metrology protocols allow to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor, and thus bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time-inhomogeneous. In this case, the ultimate precision is determined by the system short-time behaviour, which when exhibiting the natural Zeno regime leads to a non-standard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short timescales, while non-Markovianity does not play any specific role.Introduction.-Parameter estimation, ranging from the precise determination of atomic transition frequencies to external magnetic field strengths, is a central task in modern physics [1][2][3][4][5][6][7]. Quantum probes made up of N entangled particles can attain the so-called Heisenberg limit (HL), where the estimation mean squared error (MSE) scales as ∼ 1/N 2 , as compared with the standard quantum limit (SQL) ∼ 1/N of classical statistics [8].Heisenberg resolution relies on the unitarity of the time evolution. In realistic situations, however, quantum probes decohere as a result of the unavoidable interaction with the surrounding environment [9]. Such interactions can have a dramatic effect on estimation precision-even infinitesimally small uncorrelated dephasing noise, modelled as a semigroup (time-homogeneous-Lindbladian) evolution [10], forces the MSE to eventually follow the SQL [11]. This result was proven to be an instance of the quantum Cramér-Rao bound (QCRB) [12] for generic Lindbladian dephasing and thus holds even when using optimized entangled states and measurements [13][14][15][16]. The question then arises of what is the ultimate precision limit when the noisy time evolution is not governed by a dephasing dynamical semigroup [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. The SQL has been shown to be surpassable in the presence of time-inhomogeneous (non-semigroup) dephasing noise [24], noise with a particular geometry [25] and correlated timehomogeneous dephasing [27], or when the noise geometry allows for error correction techniques [28].Here, we derive the ultimate lower bounds on the MSE for the noisy frequency estimation scenario depicted in Fig. 1 where probe systems are independently affected by the decoherence. In particular, we focus on uncorrelated phase-covariant noise, that is, noise-types commuting with the parameter-encoding Hamiltonian, as these underpin the asymptotic SQL-like precision in the semigroup case [16,25]. Yet, most importantl...

Noisy metrology: a saturable lower bound on quantum Fisher information

“…For such measurements F ↓ c = tr(ρ ′2 ω ) and the saturability of Eq. (16) in the main context is guaranteed.…”

“…In this case each qubit is locally affected by a special noise type named as phase-covariant noise, that is, a noise type commuting with the parameter encoding Hamiltonian, σ z , [16,20]. In the case of semigroup dynamics, the noise type is one of the most destructive noise due to constraining the quantum enhancement to a constant factor [12,14,21].…”

“…In the case of semigroup dynamics, the noise type is one of the most destructive noise due to constraining the quantum enhancement to a constant factor [12,14,21]. However, this is not the case for a non-semigroup dynamics [16,[22][23][24][25]. Regarding U ω as the encoding unitary map and J as the parameter-independent noise map, the state of N probes at any instant is described as…”

“…Considering the most general form of the phase-covariant noise, the matrix form of Φ ω in the computational basis is obtained as [16] …”

Quantum force estimation in arbitrary non-Markovian Gaussian baths

“…Most of the publications dealing with non-Markovian noise concern two-level system quantum metrology [16][17][18].…”