Our research project is centered around the notion of randomness in quantum theory. There are three

main ways in which probabilistic phenomena appear in quantum mechanics:

- as models for typical quantum states and operations, usually described by random matrices
- intrinsically, as the results of quantum measurements and their effects on quantum systems
- as noise, providing stochastic models for environments interacting with quantum systems.

These three manifestations of stochastic processes in quantum theory are strongly interconnected. Our proposal is grounded on synergies between the members of our consortium having different areas of expertise and approaches to studying these problems. More precisely, we shall focus on three main topics, each pertaining to one of the categories above: random quantum states and channels, quantum trajectories and feedback, and many-body stochastic quantum dynamics. The theoretical insights gained on stochastic quantum evolutions will be key to current challenges in quantum physics, such as observable distributions in random quantum circuits and channels, obtaining information from indirect measurements with and without feedback, controlling or correcting noisy quantum systems, taming information or entanglement propagation and transport in many-body systems, etc.

## Work packages

**A. Random quantum states and channels**

WP1: Spectral statistics of quantum channels*Task 1.1: Rigorous results for the spectral gap of random quantum channelsTask 1.2: Correlation length of random matrix product states*

WP2: Various capacities of quantum channels

*Task 2.1: Additive bounds for entropic quantities*

Task 2.2: Characterization of the output set of tensor products generic quantum channels

Task 2.2: Characterization of the output set of tensor products generic quantum channels

WP3: Entanglement of multipartite quantum states

*Task 3.1: Multi-partite entanglement via tensor norms*

**B. Quantum trajectories and feedback**

WP4: Statistical aspects of quantum trajectories*Task 4.1: Estimation of the measurement efficiency parameterTask 4.2: Estimation of the interaction parameters*

WP5: Convergence and robustness under measurement-based feedback

Task 5.1: Simple output feedback scheme stabilizing a target subspace

Task 5.2: Quantum-state feedback scheme stabilizing a target subspace

**C. Many-body stochastic quantum dynamics**

WP6: Random quantum circuits*Task 6.1: Rigorous results for random quantum circuitsTask 6.2: Perfect tensors and dual unitary quantum circuits*

WP7: Noisy Hamiltonian systems

*Task 7.1: Exact solution to Quantum Symmetric Simple Exclusion Process and applications*

Task 7.2: Exact solution to Quantum Asymmetric Simple Exclusion Process and applications

Task 7.2: Exact solution to Quantum Asymmetric Simple Exclusion Process and applications