Researchers from Keio University, Lund University, and the Technical University of Denmark have demonstrated a new method for certifying quantum randomness that requires fewer tests than previously thought. The team’s scheme certifies randomness from an untrusted measurement device using only a limited number of trusted “probe” states, simplifying implementation with existing technology and reducing costs.

This approach bypasses the need to assume specific noise characteristics in the measurement, and the work shows randomness can be certified even with imperfections. The researchers report the protocol applies to measurements with any number of outcomes, experimentally realizing it with intensity modulation of a telecom-wavelength pilot laser and homodyne detection.

Untrusted Measurement & Few States Enable Quantum Randomness Certification

Homodyne detection of vacuum states, combined with analog-to-digital conversion, now requires fewer probe states to certify quantum randomness than previously understood. The demonstrated scheme achieves certification using a limited set of trusted input states, coherent states with varying amplitude, simplifying the implementation of quantum random number generators and potentially reducing their cost. This contrasts with earlier methods that demanded a larger number of precisely calibrated states for reliable randomness verification.

The certification process does not rely on assumptions about the characteristics of noise or imperfections within the measurement device itself. Instead, the method focuses on lower-bounding the entropy of the output data based on observed outcome distributions, allowing the use of potentially flawed or commercially available detectors.

The researchers modeled both additive Gaussian noise and non-Gaussian noise originating from the analog-to-digital converter to assess the amount of randomness achievable. The experimental setup involved probing an untrusted detector with the vacuum state and a series of coherent states, then performing homodyne detection and discretizing the signal with an analog-to-digital converter. By analyzing the resulting distributions, the team could establish a lower bound on the min-entropy of the data, effectively certifying the randomness of the generated numbers.

This approach is applicable to measurements with any finite number of outcomes, offering flexibility in experimental design and implementation. “No assumptions are made about the particular form of the noise or the measurement,” the researchers write, emphasizing the robustness of their technique.

Many real-world detectors exhibit deviations from ideal Gaussian behavior, and this scheme demonstrates resilience to such non-ideal conditions. This is important for practical applications where perfect noise characteristics are difficult or impossible to achieve. The work demonstrates a method for randomness certification from a trusted source in prepare-and-measure setups, where the measurement is partially characterized using a small set of known input states.

Makoto Ishihara of Keio University, Carles Roch i Carceller of Lund University and ICFO and Kieran Neil Wilkinson, Casper Ahl Breum, Tobias Gehring, and Jonatan Bohr Brask of the Technical University of Denmark contributed equally to this work. The team showed significant randomness can be produced using just a few probe states.

The implications of this research extend to a range of applications requiring secure and unpredictable random numbers, including numerical simulation, statistical sampling, gaming and cryptographic key generation. In cryptography, the inability of an adversary to predict random numbers is paramount for maintaining security, and this certification scheme provides a rigorous method for establishing those bounds. The demonstrated technique offers a pathway toward more accessible and reliable quantum random number generators, potentially enhancing security in various technological domains.

Prepare-and-Measure Protocol for Bounding Output Entropy

The ability to rigorously bound output entropy in quantum randomness certification relies on a mathematical technique using Gauss-Radau quadrature approximation to the relative entropy, as detailed in recent work. This allows researchers to establish a lower bound on the conditional von Neumann entropy of the output, even when the measurement device itself is untrusted, by minimizing a semidefinite program constrained by observed data.

The resulting dual semidefinite program uses weights and nodes derived from the quadrature approximation, with the minimization process restricted to positive semidefinite matrices possessing a specific block structure. The protocol’s resilience to both Gaussian additive noise and non-Gaussian imperfections is a key advantage.

The researchers modeled both types of noise, including that originating from the analogue-to-digital converter, to assess expected randomness extraction. The peak-to-peak voltage driving the modulator directly influences the amplitudes of these states, which are the trusted inputs for the measurement device. Results indicate that even with substantial additive noise, the protocol maintains performance, with entropy beginning to saturate at reasonable signal-to-noise ratios.

The team observed that even when half of the measurement bins are effectively removed, the entropy degrades only slightly in the optimal regime. Further analysis accounted for finite-size effects on the output entropy using the generalized entropy accumulation theorem, revealing that a considerable amount of randomness can be generated with a relatively small number of probe states.

While the finite-size correction increases with the number of rounds, the minimum entropy remains above 0. 4 even with 10% estimation errors in the probe state amplitudes. “We observe that, in the case of low estimation errors and ignoring finite-size effects, the minimum entropy increases with R,” the researchers report. “However, the finite-size correction also increases with R, eliminating the advantage in most cases.” The team notes that single-round min-entropy bounds, while simpler to calculate, significantly underestimate the certifiable randomness, even when assuming independent and identically distributed data.

This highlights the importance of incorporating more sophisticated techniques to accurately assess the security of the generated random numbers. In summary, the demonstrated protocol offers a robust and secure method for semi-device-independent randomness certification, simplifying implementation while maintaining high performance.

Homodyne Detection of Vacuum with Coherent State Probes

Homodyne detection offers a pathway to certifying quantum randomness using a minimal set of trusted states, as demonstrated through experiments utilizing telecom-wavelength pilot lasers. This streamlined approach relies on intensity modulation of the pilot laser followed by standard homodyne detection and digitization via an analog-to-digital converter, simplifying implementation with existing technology. The use of homodyne detection, a well-established technique, further contributes to the feasibility of this method, as it allows for the measurement of vacuum states probed by coherent states.

The team achieved a total detection efficiency of at least 90%, with any inefficiency stemming from propagation loss, imperfect visibility and the quantum efficiency of the homodyne detector diodes. The protocol’s resilience extends to the non-Gaussian noise introduced by the analog-to-digital converter, a common source of error in practical quantum random number generators.

By discretizing continuous outcomes into a finite number of intervals, the ADC introduces imperfections that could compromise randomness certification, but the demonstrated scheme maintains performance even in the presence of these distortions. The researchers also accounted for finite-size effects in their analysis, using a generalized entropy accumulator to refine the lower bound on extractable randomness. The observed bit value distributions, recorded by the ADC, were optimized by tuning the local oscillator power and ADC range to ensure the shot noise trace spanned the entire digitization range.

This optimization process maximized the certifiable randomness, demonstrating the sensitivity of the protocol to experimental parameters. “Finally, from we see that the protocol is very robust against the non-Gaussian ADC noise,” the researchers conclude, emphasizing the method’s adaptability to real-world conditions.

Conditional Min-Entropy & Entropy Accumulation Theorem

This efficiency stems from using the generalized entropy accumulation theorem, a mathematical tool allowing researchers to lower-bound total conditional min-entropy without assuming independent and identically distributed device behavior in each measurement round. This approach bypasses a common restriction in earlier methods, offering a more practical pathway to secure random number generation. Calculating the lower bound on certifiable randomness relies on a semidefinite program, a computational technique optimized to account for potential adversarial strategies.

This mathematical formulation allows for a rigorous quantification of randomness, even in the presence of noise and imperfections, and is more powerful than simpler approaches that rely on the single-round conditional min-entropy. Experimental validation involved a straightforward optical setup employing up to four probe states, demonstrating the protocol’s feasibility with existing technology.

Results show that even with 10% uncertainty in estimating the probe state amplitudes, the entropy per round remains above 0. 4, indicating a substantial level of certifiable randomness. This careful accounting for real-world limitations, such as the finite number of measurements and the inherent imperfections in detectors, is important for translating theoretical security guarantees into practical applications. The study’s findings demonstrate that the protocol maintains performance even with additive Gaussian noise, and importantly, remains robust against non-Gaussian imperfections introduced by the analog-to-digital converter.

The researchers found that the protocol’s resilience extends to these non-Gaussian effects, a significant advantage over methods sensitive to specific noise characteristics. This robustness is critical for deploying quantum random number generators in real-world scenarios where perfect noise conditions are rarely, if ever, met. The team’s work provides a pathway toward building secure and reliable random number generators that are less reliant on idealised assumptions about measurement devices.

Experimental Demonstration with Up to Four Probe States

The implementation used homodyne detection of the vacuum state alongside coherent states possessing varying amplitudes, a technique that simplifies the experimental demands compared to approaches requiring extensive state preparation or measurement device characterization. This method relies on discretizing the measurement outcomes using an analog-to-digital converter, a common component in telecommunications and signal processing, further highlighting the potential for practical application. Analysis of the conditional min-entropy revealed that the amount of extractable randomness increases with the number of detector outputs, though this gain plateaus beyond four outputs, suggesting an optimal balance between complexity and performance.

Detailed examination of the experimental data focused on quantifying the certified randomness under varying conditions, including different levels of Gaussian noise and the number of probe states used. The results show that the protocol remains effective even when estimating probe state amplitudes with 10% uncertainty, indicating its robustness against real-world imperfections.

The team’s implementation included a digital signal processing chain to extract the X-quadrature distributions of the probe states, enabling accurate amplitude estimation and downsampling of the raw data. The researchers write, emphasizing the practical implications of their work. The observed saturation of certifiable randomness beyond four outputs suggests a trade-off between increasing the number of measurement outcomes and the diminishing returns in entropy gain, a factor key for optimizing the design of future quantum random number generators.

👉 More information
🗞 Quantum randomness certification with untrusted measurements and few probe states
✍️ Makoto Ishihara, Carles Roch i Carceller, Kieran Neil Wilkinson, Casper Ahl Breum, Tobias Gehring and Jonatan Bohr Brask
🧠 DOI: http://link.aps.org/doi/10.1103/d7pg-hjgs

More like this