** Andrea Smirne**, from *Ulm University,* will be visiting the Physics Department from Monday, 12 December. He will give a lecture and a talk about quantum parameter estimation.

**Monday 12 December**, *8.45,* *Aula I*

**Lecture** *Classical and quantum limits to the achievable precision in parameter estimation*

**Wednesday 14 December**, *13.30,* *Aula Bonetti*

**Seminar** *Overcoming the classical limits for frequency estimation in the presence of a general class of open-system dynamics*

Here are the abstracts and references for the two talks.

## Classical and quantum limits to the achievable precision in parameter estimation

How precisely can we estimate the value of an unknown parameter? This is the central question of metrology and the answer crucially depends on whether we consider a classical or a quantum framework. In classical experiments involving N sensing particles, i.e. N probes, the best estimation strategies lead to an error (as measured by the variance), which scales at most as 1/N, according to the so-called shot-noise limit (SNL). On the other hand, if we are able to initially prepare the probes in an entangled state, we can get a further factor 1/N of improvement, leading to the Heisenberg limit (HL). This shows in a paradigmatic way that quantum features can be exploited to get a significant advantage compared to any classical strategy [1]. Nevertheless, such a quantum advantage is jeopardized by the interaction of the probes with the surrounding environment, that is, if the probes evolve in a non-unitary way and they have thus to be treated as open quantum systems. In recent years, no-go theorems have been formulated [2,3,4], which state that typical forms of semigroup dynamics constrain the quantum enhancement to a constant factor, bounding the scaling of the error to the SNL.

In the first talk, I will present some basic results about quantum metrology in the presence of a unitary or a semigroup dynamics, relying on the notions of classical and quantum Fisher information, as well as on the corresponding Cramer-Rao bounds.

- V. Giovannetti, S. Lloyd and L. Maccone, Science 306, 1330 (2004)
- S.F. Huelga, C. Macchiavello, T. Pellizzari, A.K. Ekert, M.B. Plenio, and J.I. Cirac, Phys. Rev. Lett. 79, 3865 (1997)
- B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011)
- R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, Nat. Commun. 3, 1063 (2012)

## Overcoming the classical limits for frequency estimation in the presence of a general class of open-system dynamics

As seen in the first talk, quantum metrology protocols allow us to surpass precision limits typical to classical statistics, but the semigroup noise generally constrains the quantum enhancement to a constant factor and then it bounds the error to the standard shot-noise limit (SNL) [1,2]. Nevertheless, the SNL can be truly overcome by taking into account more general and realistic kinds of noise, e.g., by allowing for a time-dependent dephasing rate [3], thus going beyond the semigroup regime. More in general, recently [4] the crucial dynamical features which guarantee to overcome the SNL in the presence of noise have been identified for a wide and significant class of open-system dynamics. Explicitly, it has been shown that for any phase-covariant dynamics, that is, when the action of the noise commutes with the system’s encoding Hamiltonian, one can achieve an asymptotic precision-scaling intermediate between the SNL and the Heisenberg limit if and only if the short-time evolution departs from a semigroup dynamics. Time inhomogeneity at short-time scales is thus the relevant noise feature dictating the asymptotic ultimate precision and allowing to beat the standard limits to precision estimation.

In the second talk, I will discuss this recent result on the maximal achievable precision under phase-covariant dynamics, also illustrating it via an explicit example of an (almost) optimal estimation protocol.

- B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011)
- R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, Nat. Commun. 3, 1063 (2012)
- A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 109 (2012).
- A. Smirne, J. Kolodynski, S.F. Huelga, and R. Demkowicz-Dobrzanski, Phys. Rev. Lett. 116, 120801 (2016)