⚛️ quantum physics

Deterministic randomness extraction for quantum random number generation with partial trust

This paper extends deterministic randomness extraction from device-independent to partial-trust and semi-device-independent prepare-and-measure scenarios, demonstrating that specific functions serve as effective extractors for memoryless quantum devices and achieving positive key rates in simulations with as few as 7,000 rounds.

Original authors: Pablo Tikas Pueyo, Tomás Fernández Martos, Gabriel Senno

Published 2026-02-27

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CC BY 4.0

Original authors: Pablo Tikas Pueyo, Tomás Fernández Martos, Gabriel Senno

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake the perfect loaf of bread (perfect randomness) for a very important recipe (encryption). You have a bag of flour (your data source), but there's a problem: the flour isn't 100% pure. It might have a few tiny pebbles or bits of dirt in it.

In the world of classical computing, if you just sift this flour through a standard sieve (a deterministic procedure), you can't guarantee you'll get a perfect loaf if the dirt is arranged in a tricky, unpredictable way. You usually need a "magic ingredient" (a random seed) to help you clean it up.

This paper is about a new, clever way to bake that bread without the magic ingredient, even when you aren't 100% sure about your kitchen equipment.

Here is the breakdown of their breakthrough using simple analogies:

1. The Problem: The "Slightly Suspicious" Kitchen

In Quantum Random Number Generation (QRNG), we use the weirdness of quantum physics (like a spinning coin that is both heads and tails) to generate raw data.

The Ideal Scenario: You trust your oven and your flour completely.
The Real World: Sometimes, you don't fully trust the machine making the flour (the state preparation), or you don't fully trust the sieve you're using (the measurement). Maybe a hacker (Eve) is peeking into your kitchen or tampering with the machine.

Previously, if you didn't trust everything, you had to assume the worst-case scenario, which made it very hard to get good randomness. Or, you had to use a "seed" (a tiny bit of pre-existing randomness) to clean up the data. The authors wanted to know: Can we clean the flour perfectly without any magic seeds, even if we only trust part of the kitchen?

2. The Solution: The "Dual-Check" Filter

The authors took a technique that was previously only used for "Device-Independent" scenarios (where you trust nothing and just look for a specific "Bell violation" like a magic handshake between particles) and adapted it for "Partially Trusted" scenarios.

Think of it like this:

The Old Way (Device-Independent): You have to watch a magic show where two magicians perform a trick that is impossible to fake. If they succeed, you know the randomness is real. But this is very hard to do in a real kitchen.
The New Way (Partial Trust): You trust the flour but not the sieve, OR you trust the sieve but not the flour.
- The authors realized that instead of looking for a "magic trick" (Bell inequality), you can look for a mathematical balance scale.
- They created a special "Dual-Check" filter. This filter looks at the relationship between the input (what you put in) and the output (what comes out). If the relationship passes a specific mathematical test (derived from a "guessing probability" game), it proves that the output must be random, even if a hacker is watching.

3. The "Spot-Checking" Protocol

How do you use this in practice? You can't check every single grain of flour, or you'd run out of bread.

The Strategy: Imagine you are baking 1,000 loaves.
- For 90% of them, you just bake them and sell them (these are your Random Numbers).
- For 10% of them, you stop and perform a rigorous "Spot Check." You measure the flour's properties and the sieve's performance very carefully.
- If the Spot Check passes the "Dual-Check" test, you know with mathematical certainty that the other 90% (the ones you didn't check) are also pure and random.
- If the Spot Check fails, you throw the whole batch away (abort the protocol).

4. The Results: Fast and Efficient

The authors simulated this process on a computer using a realistic model of how quantum lasers work.

The Speed: They found that you don't need millions of rounds of testing to get a good result. You can get a positive, secure result with as few as 7,000 rounds. That's like baking a small batch of cookies and knowing they are safe to eat immediately.
The Noise Tolerance: Even if the kitchen is a bit messy (noisy equipment), this method still works better than many other quantum security methods. It's like having a sieve that still catches the dirt even if the flour is a bit damp.

Summary

In plain English:
This paper proves that you can turn "messy" quantum data into "perfect" random numbers without needing a pre-existing random seed, even if you don't fully trust your equipment. They did this by creating a new mathematical "spot-check" system. If the system passes a specific test on a small sample of data, it guarantees the rest of the data is secure. This makes quantum random number generators faster, cheaper, and more practical for real-world use, like securing your bank transactions or encrypting secret messages.

The Metaphor:
They found a way to prove your bread is pure by tasting just one crumb, even if you aren't sure if your oven is broken or if your flour was stolen, as long as you trust either the oven or the flour, but not necessarily both.

1. Problem Statement

The paper addresses a fundamental limitation in Quantum Random Number Generation (QRNG): deterministic randomness extraction.

The Impossibility: In classical information theory, it is proven that no deterministic procedure can extract near-ideal randomness from an arbitrary entropy source with a known min-entropy lower bound (Santha-Vazirani result).
The Quantum Context: While Quantum Random Number Generators (QRNGs) exploit inherent quantum unpredictability, standard protocols often rely on seeded extraction (requiring a short, perfect random seed) or multi-source extraction.
The Gap: Recent work by Foreman and Masanes (2025) demonstrated that deterministic extraction is possible in a fully Device-Independent (DI) setting (where Bell inequality violations replace entropy bounds) for memoryless devices. However, DI protocols are experimentally demanding.
The Challenge: Can deterministic extractors be constructed for Prepare-and-Measure (PM) scenarios with partial trust? These scenarios are more experimentally feasible but involve different trust assumptions (e.g., trusting the state preparation but not the measurement, or vice versa).

2. Methodology

The authors adapt the techniques from the DI setting to three specific partial-trust PM scenarios:

Setting (S1) - Untrusted Preparation: The state preparation is untrusted (potentially correlated with an eavesdropper, Eve), but the measurement is trusted and characterized.
Setting (S2) - Untrusted Measurement: The state preparation is trusted and characterized, but the measurement is untrusted (though characterized).
Setting (S3) - Semi-Device-Independent (Semi-DI): Two trusted preparations with a known lower bound on their fidelity (overlap $\delta$ ) are used, while the measurement is untrusted.

Key Technical Steps:

Dual SDP Formulation: The authors formulate the eavesdropper's (Eve's) guessing probability as a Semidefinite Program (SDP). They derive the dual of this SDP.
Operator Inequalities: Instead of using Bell functionals (as in DI), they use solutions to the dual SDP to construct operator inequalities. These inequalities bound the predictability of the outcomes based on the observed statistics.
- Crucially, by using the Naimark dilation picture, these operators act on the Hilbert space of the measurement device's ancillary system (where Eve's side information is correlated).
Deterministic Extractors: They prove that specific functions, previously shown to be extractors in the DI setting, remain valid extractors in these PM settings.
- Single-bit: The XOR function ( $\bigoplus a_i$ ).
- Multi-bit: A family of functions satisfying specific Fourier coefficient bounds (similar to those in Foreman & Masanes).
Protocol Design: They design a spot-checking protocol where a fraction of rounds ( $p_e$ ) are used to estimate the relevant expectation values (to verify the operator inequality holds), and the remaining rounds generate the raw key.

3. Key Contributions

A. Theoretical Extension

The paper proves that the deterministic extractors from the DI setting can be ported to Prepare-and-Measure scenarios with partial trust. The central theorem (Theorem 1) establishes that for any behavior $p$ in the considered settings, if a dual solution exists, the following operator inequality holds:
$\pm \text{Tr}_S [(\rho_S \otimes \mathbb{I}_M)(\Pi^0_{SM} - \Pi^1_{SM})] \preceq G_M$
where $G_M$ is constructed from the dual variables. This inequality bounds the predictability of the output, enabling deterministic extraction without a seed.

B. Security Proofs

Theorem 2 & 3: Prove that the XOR function and the multi-bit family of functions are valid $(m, \epsilon)$ -deterministic randomness extractors for the resulting classical-quantum (cq) states in settings (S1)-(S3).
Theorem 4: Establishes the security of the spot-checking protocol, showing that the trace distance between the generated key and an ideal uniform key is bounded by the security parameter $\epsilon$ .

C. Numerical Simulations

The authors simulate the protocol using a novel family of behaviors based on Vertical-Cavity Surface-Emitting Lasers (VCSELs) with polarization switching.

Finite-Size Regime: They demonstrate that positive key rates are achievable with as few as $7 \times 10^3$ rounds.
Asymptotic Regime: They show that successful extraction is possible even when the fraction of estimation rounds ( $p_e$ ) vanishes, provided the behavior is not completely deterministic.
Noise Tolerance: The protocol exhibits robust noise tolerance. While rates decrease with noise, the degradation is less steep compared to other quantum cryptographic primitives (like B92 QKD).

4. Results

Feasibility: Deterministic extraction is possible in semi-DI and partially trusted PM settings without requiring a random seed.
Efficiency:
- In the asymptotic limit, the best rates are an order of magnitude higher than those reported for fully Device-Independent protocols (due to weaker assumptions).
- However, the rates are still significantly lower than what seeded extraction could provide for the same setting (since seeded extraction utilizes the full min-entropy).
Practicality: The protocol requires a relatively small number of rounds ($7,000$) to generate a positive key, making it suitable for near-term experimental implementation.
XOR vs. Multi-bit: For the XOR extractor, unlike the DI case where $p_e > 0.5$ was required for extraction near the local bound, the authors find that for any $p_e > 0$ , extraction is possible as long as the overlap $\delta \notin \{0, 1\}$ (i.e., the state is not perfectly predictable).

5. Significance and Impact

Bridging Theory and Experiment: This work moves deterministic randomness extraction from the highly demanding Device-Independent regime to more practical Semi-Device-Independent and partially trusted regimes.
Seedless QRNGs: It provides a rigorous framework for seedless QRNGs, which are desirable for reducing hardware complexity and avoiding the need for a pre-shared random seed.
New Experimental Targets: By identifying a specific family of behaviors realizable with VCSELs, the paper offers a concrete path for experimentalists to implement these protocols.
Limitations and Future Work: The authors acknowledge that while the rates are promising, they are currently lower than seeded alternatives. Future work aims to extend these constructions to general device-dependent settings with zero trust and improve the efficiency of the protocols for industrial applications.

In summary, the paper successfully generalizes the concept of deterministic randomness extraction to a broader class of quantum entropy sources, proving that structural knowledge (like state overlap or trusted components) can replace the need for random seeds in generating certified randomness.