Entanglement and Private Information in Many-Body Thermal States

We use concepts from quantum cryptography to relate the entanglement in many-body mixed states to standard correlation functions. If a system can be used as a resource for distilling private keys-random classical bits that are shared by spatially separated observers but hidden from an eavesdropper having access to the environment-we can infer that the state of the system is entangled. For thermal states, we derive a simple relation between the information accessible to the eavesdropper and the linear response of the system. This relation allows us to determine which spatial correlations can be used to detect entanglement across wide varieties of physical systems, and provides a new experimental probe of entanglement. We also show that strong symmetries of a density matrix imply the existence of correlations that are always hidden from the environment. This result implies that, although grand canonical ensembles are separable above a finite temperature, canonical ensembles are generically entangled at all finite temperatures.