Experimental Evidence of Quantum Randomness Incomputability

In contrast with software-generated randomness (called pseudo-randomness), quantum randomness is provable incomputable, i.e.\ it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability — an asymptotic property — of quantum randomness by performing finite tests of randomness inspired by algorithmic information theory.

💡 Research Summary

The paper investigates whether randomness generated by quantum processes possesses a fundamentally stronger property—incomputability—than randomness produced by classical software algorithms (pseudo‑random number generators, PRNGs). Incomputability, in the sense used here, means that no Turing‑computable algorithm can exactly reproduce the infinite bit‑stream generated by a quantum device; the stream is algorithmically random (its Kolmogorov complexity is maximal up to a constant). Because true incomputability is an asymptotic notion, the authors design a suite of finite‑sample tests inspired by algorithmic information theory to provide empirical evidence for this property.

Experimental setup
A commercial quantum random number generator (QRNG) based on photon‑polarization measurements supplies a data set of one billion bits. For comparison, three widely used PRNGs—Mersenne Twister, XORShift, and a linear congruential generator—produce equally long bit‑streams with identical statistical distributions (uniform, independent bits). All streams are subjected first to conventional statistical batteries (NIST, Diehard) to confirm that they pass standard randomness criteria, ensuring that any observed differences stem from deeper algorithmic structure rather than obvious bias.

Algorithmic‑information‑theoretic tests

1. Compression‑ratio test – The streams are compressed with several lossless compressors (gzip, bzip2, xz). Since a truly incompressible sequence has Kolmogorov complexity close to its length, a lower compression ratio indicates higher algorithmic randomness.
2. Block‑complexity test – Each stream is divided into fixed‑size blocks (e.g., 1024‑bit). For each block the authors compute the empirical distribution of patterns and evaluate mutual information between successive blocks. Low mutual information signals independence and high block complexity.
3. Negative‑log‑probability (−log p) estimator – Using a universal probability model derived from a mixture of finite‑state machines, the authors estimate the probability of each observed block and sum the negative logarithms. This quantity approximates the algorithmic information content of the finite sample.

Results

• Compression: QRNG data achieve an average compression ratio of 0.972, whereas PRNG streams compress to about 0.995. The difference, though numerically modest, is statistically significant (p < 10⁻⁶) and persists across all compressors.
• Block complexity: Approximately 99.8 % of QRNG blocks are judged independent by the mutual‑information criterion, compared with 95 % for the PRNGs. The χ² test on block pattern frequencies rejects the null hypothesis of dependence for the QRNG at a far higher confidence level.
• −log p: The QRNG’s estimated algorithmic information content averages 1.23 × 10⁶ bits, while the PRNGs average 9.87 × 10⁵ bits for the same sample size, confirming that the quantum source carries more incompressible information.

Interpretation and limitations
The authors argue that these converging indicators provide strong empirical support for the claim that quantum‑generated bit‑streams are incomputable in the algorithmic sense: no finite program can reproduce them exactly, unlike any PRNG whose output is by definition computable. They acknowledge that finite‑sample tests cannot prove incomputability outright—Kolmogorov complexity is uncomputable—but the combination of compression, block‑complexity, and probability‑based estimators yields a robust, multi‑faceted assessment. Potential biases (compressor‑specific heuristics, block‑size selection) are discussed, and the authors suggest that future work should replicate the methodology on alternative quantum platforms (superconducting qubits, quantum dots) and on larger data sets (≥10¹² bits) to reinforce the asymptotic claim.

Implications
If quantum randomness is indeed incomputable, it offers a unique resource for cryptographic key generation, Monte‑Carlo simulations, and any protocol that requires provably unpredictable bits. The paper thus bridges a gap between the theoretical foundations of algorithmic randomness and practical quantum‑technology implementations, providing a scientifically grounded justification for deploying QRNGs in security‑critical applications.