Getting almost all the bits from a quantum random access code

A quantum random access code (QRAC) is a map xmapstoρx that encodes n-bit strings x into m-qubit quantum states ρx, in a way that allows us to recover any one bit of x with success probability ≥ p. The measurement on ρx that is used to recover, say, x1 may destroy all the information about the other bits; this is in fact what happens in the well-known QRAC that encodes n=2 bits into m=1 qubits. Does this generalize to large n, i.e., could there exist QRACs that are so "obfuscated" that one cannot get much more than one bit out of them? Here we show that this is not the case: for every QRAC there exists a measurement that (with high probability) recovers the full n-bit string x up to small Hamming distance, even for the worst-case x.