Getting almost all the bits from a quantum random access code

TL;DR

This paper proves that quantum random access codes, which encode multiple bits into fewer qubits, can be measured to recover nearly the entire original data with high probability, even in worst-case scenarios.

Contribution

It demonstrates that all quantum random access codes inherently contain almost complete information about the original data, countering the idea of strong obfuscation.

Findings

Any QRAC allows recovery of the full n-bit string with high probability.

Recovery is possible even for worst-case inputs.

The result applies broadly to all QRAC constructions.

Abstract

A quantum random access code (QRAC) is a map $x\mapsto\rho_x$ that encodes $n$-bit strings $x$ into $m$-qubit quantum states $\rho_x$, in a way that allows us to recover any one bit of $x$ with success probability $\geq p$. The measurement on $\rho_x$ that is used to recover, say, $x_1$ 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 $x$.