  It was recently observed that in certain thermalizing many-body systems,
measuring the complement of a subsystem that thermalized to infinite
temperature in a suitable orthonormal basis gives rise to approximate quantum
state k-designs as post-measurement states on the subsystem. We prove that this
emergence of approximate k-designs holds true in every large system where some
small subsystem is close to being maximally mixed. On a technical level we show
that any high-dimensional purification of an approximately maximally mixed
state induces an approximate quantum state k-design when measured in a suitable
orthonormal basis. Moreover, we show that this is true with overwhelming
probability for measurement bases chosen uniformly at random from the Haar
measure.
