Fast Computation of Many-Body Entanglement

Mixed state entanglement measures can act as a versatile probes of many-body systems. However, they are generally hard to compute, often relying on tricky optimizations. One measure that is straightforward to compute is the logarithmic negativity, yet done naively even this is still limited to small system sizes. Here, we introduce a method to compute the logarithmic negativity for arbitrary subsystems of a densely represented state, as well as block subsystems of matrix product states. The method combines lazily evaluated, tensor network representations of the partially transposed density matrix with stochastic Lanczos quadrature, and is easily extendible to other quantities and classes of many-body states. As examples, we compute the entanglement within random pure states for density matrices of up to 30 qubits, explore scrambling in a many-body quench, and match the results of conformal field theory in the ground-state of the Heisenberg model for density matrices of up to 1000 spins. An implementation of the algorithm has been made available in the open-source library \textit{quimb}.