**Open Quantum Systems: Prof. Sabrina Maniscalco**

**Introduction to open quantum systems**

In this talk I will give a basic introduction to the dynamics of open quantum systems. After presenting the king and queen of open quantum systems theory, namely the dynamical map and the master equation, I will discuss the projection operator technique to formally derive the equation of motion for the reduced density matrix of the system. Both memory kernel and time-local master equations will be briefly considered and two exemplary solutions will be discussed to illustrate their main properties. Finally, I will shortly present one example of exactly solvable master equation and discussed its main physical phenomena and dynamical regimes.

**Markovian and non-Markovian open quantum dynamics**

I will introduce the concept of divisibility of the dynamical map and review the main properties of quantum systems interacting with their environment in absence and presence of memory effects, i.e., in the Markovian and non-Markovian regimes, respectively. After discussing the recently introduced information theoretical approach to non-Markovianity, I will focus on three of the most used non-Markovianity measures. I will conclude the talk with a few examples illustrating how reservoir engineering allows to exploit memory effects to enhance certain tasks useful for quantum technologies.

**Quantum Entanglement: Prof. Karol Życzkowski**

Geometry of quantum entanglement

Geometry of quantum entanglement

A geometric approach to investigation of quantum entanglement is advocated. We discuss first the geometry of the (N2-1)-dimensional convex body of mixed quantum states acting on an N-dimensional Hilbert space. For composed dimensions, N=K2, one considers the subset of separable states and shows that it has a positive measure. Analyzing its properties contributes to our understanding of quantum entanglement, which in this context can be interpreted as the distance of the given quantum state to the closest separable state.

**Distinguishing generic quantum states**

Properties of random mixed states of dimension N distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large N, due to the concentration of measure phenomenon, the trace distance between two random states tends to a fixed number 1/4+1/π, which yields the Helstrom bound on their distinguishability. To arrive at this result we apply free random calculus and derive the symmetrized Marchenko--Pastur distribution. Asymptotic value for the root fidelity between two random states, √F=3/4, can serve as a universal reference value for further theoretical and experimental studies. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem. Entanglement of a generic mixed state of a bi--partite system is estimated.

**Quantum Metrology: Prof. Susana Huelga**

Foundations of quantum metrology

Foundations of quantum metrology

We will discuss the origin and provide a derivation of fundamental metrological bounds, which characterize the best achievable precision in the local estimation of a phase shift, given certain resources. The problem of phase estimation underlies different forms of metrology, ranging from precision spectroscopy to magnetometry. We will show how quantum metrology protocols allow overcoming precision limits typical to classical statistics, the so-called standard quantum limit (SQL), and achieving Heisenberg resolution using entangled probes.

**Quantum metrology in Open Quantum Systems**

We will discuss in detail the persistence of Heisenberg scaling for open system dynamics and characterize the best achievable precision in the presence of different forms of noise. We will show that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor, and thus bound the error to the SQL. In particular, that is the case of all semigroup (Lindbladian) dynamics that include phase covariant terms, which commute with the system Hamiltonian. Remarkably, the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time-inhomogeneous. We will show that 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, albeit below Heisenberg scaling. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short timescales, while genuine non-Markovianity does not play any specific role.