In all chaos there is a cosmos, in all disorder a secret order - Carl Jung
Quantum Chaos in quantum circuits, Hamiltonian systems and Black Holes.
We put forward the notion of entanglement complexity in order to explain the irreversibility of quantum procesess in a many-body system. Irreversibility is the fundamental fact of Nature. As we know, it is very hard to unsplatter an egg. Quantum dynamics is unitary, and fundamentally reversible. In a closed quantum system, evolution is unitary and entropy cannot increase. The old problem of the foundations of statistical mechanics demands again our undivided attention. How is equilibration even possible in quantum many-body systems? Recently, this question has found a renewed interest because of the advance on both theoretical and experimental accessibility of quantum dynamics (e.g., in systems of ultra cold atom gases). From the the experimental point of view, it is now possible to observe unitary dynamics of a quantum many-body away from equilibrium over long times. One observes equilibration and thermalization in the expectation values of the quantities observed, and this has to be explained in the context of unitary dynamics that is reversible and cannot lead to a thermalization of the wave-function. So the wave function cannot become thermal, but the observed quantities behave as if they were in a thermal wave-function. How is that possible? The key to understand this phenomenon is in entanglement. In recent years, it has been understood that typically, a quantum process produces high entangled states that locally look thermal. To this end, it is necessary to produce a large quantity of entanglement. However, this is not the whole story. Integrable systems after a quantum quench do not thermalize but still feature large entanglement. It is clear that how much entanglement there is is not the whole story. The emergence of irreversibility and of the arrow of time is not solely due to the production of a large amount of entanglement. Irreversibility arises when the entanglement produced is complex.