Getting almost all the bits from a quantum random access code

2025-06-02

AI Summary

This work investigates the information extractability of quantum random access codes (QRACs): can a QRAC be designed to be sufficiently “fuzzy” such that, beyond the target bit, essentially no information about the original n-bit string is recoverable? We refute this conjecture by proving that, for any n and m, there exists a universal POVM measurement enabling reconstruction of the entire n-bit input string x from an m-qubit QRAC state with high probability and Hamming error o(n). Our result holds without assumptions on the input distribution, transcending the conventional single-bit recovery paradigm. It establishes, for the first time, the intrinsic information richness of QRACs: even when m ≪ n, the m-qubit encoded state implicitly contains nearly complete classical information about the n-bit input. This reveals a fundamental limitation on the privacy or hiding capability of QRACs and provides new insight into the classical information capacity embedded in compact quantum encodings.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

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

Problem

• Investigates quantum random access codes (QRACs) for multi-bit recovery
• Challenges the limitation of recovering only one bit from QRACs
• Demonstrates full n-bit string recovery from QRACs with high probability

Innovation

• Quantum random access code for n-bit encoding
• High-probability full bit string recovery
• Measurement overcoming QRAC obfuscation limits