Emergent Space and Entanglement

The holographic correspondence relates a non-gravitational theory to a gravitational theory in a higher number of space-time dimensions. In many instances of holography, the additional space dimensions are emergent: internal degrees of freedom in large matrix valued fields metamorphose into space, and their dynamics becomes re-interpreted as gravitational dynamics. This connection necessitates an understanding of notions of quantum entanglement in target space, rather than in base space. In this talk I will discuss recent progress in defining such notions and how they may lead to a finite entanglement entropy in the resulting gravitational theory. I will also discuss how these notions are of interest to general many-body systems where the quantum states are understood in a first quantized language, but difficult to formulate in second quantization.