[Paper Review] Certifiable Quantum Dice - Or, testable exponential randomness expansion

Umesh Vazirani, Thomas Vidick|arXiv (Cornell University)|Nov 25, 2011

Quantum Computing Algorithms and Architecture14 references19 citations

TL;DR

This paper presents a certifiable quantum randomness expansion protocol that generates $ n $ bits of true randomness from $ O(\log n) $ uniformly random seed bits using only the no-signaling principle as a security assumption. The protocol enables testable, exponential randomness expansion with security against quantum adversaries, achieving $ \Omega(n) $ bits of certified randomness from a small seed, even when the devices are entangled with an eavesdropper.

ABSTRACT

We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits of true randomness from a seed of O(log n) uniform bits. The bits generated are certifiably random based only on a simple statistical test that can be performed by the user, and on the assumption that the devices obey the no-signaling principle. No other assumptions are placed on the devices' inner workings. A modified protocol uses a seed of O(log^3 n) uniformly random bits to generate $n$ bits of true randomness even conditioned on the state of a quantum adversary who may have had prior access to the devices, and may be entangled with them.

Motivation & Objective

• To develop a protocol that generates certified random bits from minimal randomness, ensuring the output is truly random even if the devices are adversarially controlled.
• To achieve exponential randomness expansion—producing $ n $ bits of randomness from a seed of only $ O(\log n) $ bits—while maintaining testability.
• To provide security against a quantum adversary who may have prior entanglement with the devices, ensuring the output remains random even when conditioned on the adversary's quantum state.
• To design a protocol that relies solely on a simple statistical test (CHSH game violation) and the no-signaling principle, avoiding assumptions about device internals.

Proposed method

• The protocol uses a sequence of CHSH game rounds where most rounds use fixed inputs (0,0), while a small fraction are 'Bell blocks' with random inputs to test non-classical correlations.
• It applies a statistical test on the output: the CHSH condition must be satisfied in at least 84% of rounds in each block to accept the output.
• The protocol uses a small random seed to select Bell blocks and ensures that any deviation from quantum behavior is detectable via statistical deviation from the expected success rate.
• A reduction to a guessing game is used to prove security: if an adversary could predict outputs with high accuracy, it would imply signaling, violating the no-signaling principle.
• The protocol groups Bell blocks into chunks and uses a chain rule argument to show that with high probability, there exists at least one block where the adversary’s prediction is correct and the CHSH condition is satisfied.
• The final output is extracted from the CHSH game outcomes in the Bell blocks, with a final decision rule based on agreement with predicted outputs to certify randomness.

Experimental results

Research questions

• RQ1 Can a small random seed be used to generate a large number of certifiably random bits using only the no-signaling principle?
• RQ2 Can the protocol remain secure against a quantum adversary who may be entangled with the devices?
• RQ3 Is it possible to achieve exponential randomness expansion (i.e., $ n $ bits from $ O(\log n) $ seed) with only a statistical test and no device trust?
• RQ4 Can the protocol be made testable in practice, relying only on observable statistics and not on device details?

Key findings

• The protocol generates $ \Omega(n) $ bits of certifiably random output from a seed of $ O(\log n) $ uniformly random bits, achieving exponential randomness expansion.
• The security of the output is guaranteed with probability $ 1 - \exp(-\Omega(\Delta)) $, where $ \Delta $ controls the error rate and security level.
• A modified protocol using $ O(\log^3 n) $ seed bits generates $ n $ bits of randomness that are secure even when conditioned on a quantum adversary’s prior entanglement with the devices.
• The protocol ensures that any attempt by an adversary to predict outputs with high accuracy would imply signaling, which violates the no-signaling principle and thus cannot occur.
• The existence of at least one Bell block where both the CHSH condition is satisfied and the adversary’s prediction is correct with high probability is proven via a chain rule argument.
• The final output is extracted using a decision rule based on agreement with predicted outputs, ensuring that the final bits are indistinguishable from uniform to any adversary with bounded prediction success.