This review aims at gathering the most relevant quantum multi-parameter estimation methods that go beyond the direct use of the quantum Fisher information concept. We discuss in detail the Holevo Cramér-Rao bound, the quantum local asymptotic normality approach as well as Bayesian methods. Even though the fundamental concepts in the field have been laid out more than forty years ago, a number of important results have appeared much more recently. Moreover, the field drew increased attention recently thanks to advances in practical quantum metrology proposals and implementations that often involve estimation of multiple parameters simultaneously. Since the topics covered in these review are spread in the literature and often served in a very formal mathematical language, one of the main goals of this review is to provide a largely self-contained work that allows the reader to follow most of the derivations and get an intuitive understanding of the interrelations between different concepts using a set of simple yet representative examples involving qubit and Gaussian shift models.

We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.

We show that for the most general adaptive noiseless estimation protocol, where Uϕ = e iϕΛ is a unitary map describing the elementary evolution of the probe system, the asymptotically saturable bound on the precision of estimating ϕ takes the form ∆ϕ ≥ π n(λ + −λ − ) , where n is the number of applications of Uϕ, while λ+, λ− are the extreme eigenvalues of the generator Λ. This differs by a factor of π from the conventional bound, which is derived directly from the properties of the quantum Fisher information. That is, the conventional bound is never saturable. This result applies both to idealized noiseless situations as well as to cases where noise can be effectively canceled out using variants of quantum-error correcting protocols.

We study the lowest energy states for fixed total momentum, i.e. yrast states, of N bosons moving on a ring. As in the paper of A. Syrwid and K. Sacha [1], we compare mean field solitons with the yrast states, being the many-body Lieb-Liniger eigenstates. We show that even in the limit of vanishing interaction the yrast states possess features typical for solitons, like phase jumps and density notches. These properties are simply effects of the bosonic symmetrization and are encoded in the Dicke states hidden in the yrast states.

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