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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We exploit a few-to many-body approach to study strongly interacting dipolar bosons in the quasi-one-dimensional system. The dipoles attract each other while the short range interactions are repulsive. Solving numerically exactly the multi-atom Schrödinger equation, we discover that such systems can exhibit not only the well known bright soliton solutions but also novel quantum droplets for a strongly coupled case. For larger systems, basing on microscopic properties of the found few-body solution, we propose a new generalization of the Gross-Pitaevskii equation (GPE) that incorporates the Lieb-Liniger energy in a local density approximation. Not only does such a framework provide an alternative mechanism of the droplet stability, but it also introduces means to further analyze this previously unexplored quantum phase. In the limiting strong repulsion case, yet another simple multi-atom model is proposed. We stress that the celebrated Lee-Huang-Yang term in the GPE is not applicable in this case.

Two identical dipolar atoms moving in a harmonic trap without an external magnetic field are investigated. Using the algebra of angular momentum a semi -analytical solutions are found. We show that the internal spin -spin interactions between the atoms couple to the orbital angular momentum causing an analogue of Einstein -de Haas effect. We show a possibility of adiabatically pumping our system from the s-wave to the d-wave relative motion. The effective spin-orbit coupling occurs at anti-crossings of the energy levels.PACS numbers: 03.75.-b, 67.85.-d

We study how dark solitons, i.e., solutions of one-dimensional, single-particle, nonlinear, time-dependent Schrödinger equation, emerge from eigenstates of a linear many-body model of contact-interacting bosons moving on a ring, the Lieb-Liniger model. This long-standing problem has been addressed by various groups, which presented different, seemingly unrelated, procedures to reveal the solitonic waves directly from the many-body model. Here, we propose a unification of these results using a simple ansatz for the many-body eigenstate of the Lieb-Liniger model, which gives us access to systems of hundreds of atoms. In this approach, mean-field solitons emerge in a single-particle density through repeated measurements of particle positions in the ansatz state. The postmeasurement state turns out to be a wave packet of yrast states of the reduced system.