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After an introduction to the notion of Leonard pairs, I explain their different uses. Then, I provide the definition of the associated Heun operator and how it allows us to simplify the computation of the quantum entanglement entropy. Finally, I show, in the simplest example, how the Bethe ansatz can be used to diagonalize the Heun operator.

Quantum biology explores how phenomena like superposition, tunneling, and entanglement influence biological systems, with growing experimental evidence supporting their role in processes such as photosynthesis, enzymatic reactions, DNA mutations, and animal navigation. These discoveries have profound implications for medicine, renewable energy, and our understanding of life. Quantum neurobiology applies this perspective to the brain, exploring how brain cells function beyond traditional models of electrical activity and chemical neurotransmission. Moving beyond classical models of chemical and electrical neuronal signaling, quantum approaches could revolutionize diagnostics and treatments for neurological disorders while offering deeper insights into cognition, behavior, and consciousness. Here will be presented an overview of quantum biology, examining the current state of quantum neurobiology, both theory and experiment, and exploring its potential applications in diagnosing and treating neuroinflammatory conditions such as Alzheimer’s and Parkinson’s disease.

In this talk, I will introduce our recent studies on the effect of decoherence on fractional quantum Hall fluids and their robustness as topological quantum memories. Specifically, we examine the behavior of Laughlin states and Moore-Read states under dephasing noise using the plasma analogy as well as the field theory description. Notably, we identify a critical filling (which is 1/4 for the Renyi-2 case and between 1/4 and 1/8 in the replica limit) separating two regimes of mixed-state phase transition. Below the critical filling, a Berezinskii-Kosterlitz-Thousless (BKT) transition occurs at finite noise strength, separating the topologically ordered phase from a critical phase. This transition is characterized by quantum-information quantities that are non-linear functions of the density matrix. On a torus, the BKT transition marks the critical decoherence above which the quantum memory encoded in the ground state manifold is degraded. In contrast, above the critical filling, the quantum memory is extremely resilient against dephasing noise, with the transition occurring only at infinite noise strength. For the Moore-Read state, we further show that the transition also characterizes whether different fusion outcomes of non-Abelian anyons remain distinguishable under decoherence.