Shapes of quantum entanglement with persistent homology*

Persistent homology is a modern computational technique that can identify shapes in discrete data. We convert quantum states into barcodes using quantum mutual information as a distance metric [1]. This general approach has three applications. Firstly, it functions as a kind of order parameter, where abrupt changes in the barcode indicate phase transitions. Secondly, it describes the shapes of entanglement present in the quantum state. Thirdly, it is a numerical framework that could prove to be a useful step towards the construction of spacetime. We demonstrate its computational power on two commonly used quantum spin models.