Halving the length of approximate classical representations of pure quantum states with probabilistic encoding

Pure quantum states are often approximately encoded as classical bit strings such as those representing probability amplitudes and those describing circuits that generate the quantum states. The crucial quantity is the minimum length of classical bit strings from which the original pure states are approximately reconstructible. We derive asymptotically tight bounds on the minimum bit length required for probabilistic encodings with which one can approximately reconstruct the original pure state as an ensemble of the the quantum states encoded in classical strings. Our results imply that such a probabilistic encoding asymptotically halves the bit length required for ``deterministic" ones.