⚛️ quantum physics

Unambiguous randomness from a quantum state

This paper introduces the concept of unambiguous randomness, where an eavesdropper may return inconclusive outcomes instead of errors, and demonstrates that the maximal unambiguous randomness of a quantum state is proportional to its smallest eigenvalue while revealing that joint correlations between a noisy state and measurement can completely eliminate private randomness beyond a critical error threshold.

Original authors: Fionnuala Curran

Published 2026-02-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fionnuala Curran

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 playing a high-stakes game of "Guess the Secret" with a friend, Alice. Alice has a special coin that isn't just heads or tails; it's a quantum coin. When she flips it, the result is truly random. But there's a catch: a sneaky eavesdropper, Eve, is watching.

In the world of quantum physics, this randomness is the foundation of unbreakable security. But what happens when the coin isn't perfect? What if it's a bit "noisy" or "wobbly"? That's the question Fionnuala Curran answers in this paper.

Here is the breakdown of the research using simple analogies:

1. The Setup: The Perfect vs. The Noisy Coin

Imagine Alice has a perfect quantum coin. If she flips it, the result is 100% unpredictable. Eve, the spy, knows nothing about the outcome.

But in the real world, devices aren't perfect. Alice's coin might be slightly bent, or the table she flips it on is shaking. This is called a "noisy state."

The Problem: When a coin is noisy, it's like a mixture of a perfect coin and a broken one. Eve might know which version of the coin Alice is using, even if Alice doesn't. If Eve knows the coin is "broken," she can guess the outcome better.
The Goal: We want to know: How much true randomness is left? And more importantly, how much can Eve guess?

2. The New Rule: "I Give Up" (Inconclusive Outcomes)

Traditionally, we assume Eve tries to guess the result every single time. She might be wrong, but she always guesses.

Curran introduces a new, smarter way for Eve to play. Imagine Eve has a rule: "I will never guess wrong, but sometimes I will say 'I don't know'."

The Analogy: Think of a detective solving a crime.
- Old Way: The detective guesses who the killer is every time. Sometimes they are right, sometimes wrong.
- New Way (Unambiguous Randomness): The detective says, "I am 100% sure it was the Butler," or "I am 100% sure it was the Maid." But if the evidence is too blurry, she says, "I can't tell." She never makes a mistake, but she admits defeat sometimes.

The paper asks: If Eve is allowed to say "I don't know" sometimes, how often can she still be 100% right?

3. The Big Discovery: The "Smallest Piece" Matters

The paper finds a beautiful, simple rule for the "perfect" scenario (where Eve never guesses wrong):

The Rule: The amount of randomness you can guarantee is directly tied to the "smallest piece" of the noise in the system.
The Metaphor: Imagine the quantum state is a pizza. The "noise" is like toppings scattered unevenly. The "smallest eigenvalue" is the size of the tiniest slice of cheese. If that tiny slice is big, Eve can guess a lot. If it's tiny, there is still a lot of true randomness left.
The Result: The more "noisy" the system is, the less randomness you have. But surprisingly, even with noise, there is a specific mathematical limit to how much Eve can know if she refuses to make mistakes.

4. The "Double Trouble" Scenario (Shared Noise)

This is the most surprising part of the paper.

Imagine Alice has a noisy coin, and she also uses a noisy table (a noisy measurement device) to flip it.

Scenario A (Single Noise): Eve only spies on the coin.
Scenario B (Joint Noise): Eve spies on both the coin AND the table.

The Finding: Eve is much better at guessing when she spies on both.

The Analogy: Imagine trying to guess the result of a dice roll.
- If you only know the dice is slightly weighted (noisy coin), you have a slight advantage.
- But if you also know the table is tilted and bumpy (noisy measurement), you can predict the roll with almost perfect accuracy.
The Critical Point: The paper calculates a "tipping point." If the noise (the tilt of the table and the weight of the dice) gets too high, Eve can guess 100% of the time. At that point, the "randomness" is completely gone, even though the system looks random to Alice.

5. Why This Matters

This research is like a safety manual for the future of Quantum Random Number Generators (which are used to create unbreakable encryption keys for the internet).

The Warning: If you build a device that generates random numbers, you can't just assume the noise is harmless. If you don't model the noise correctly (specifically, if you ignore that the measurement device itself might be noisy), you might think you have a secure random number, but Eve might actually know what it is.
The Takeaway: To be truly secure, you must account for the fact that the "ruler" you use to measure the randomness might be broken, too. If the ruler and the object are both broken, the spy wins.

Summary in One Sentence

This paper proves that if you want to generate truly secret random numbers using quantum physics, you must be careful: if your equipment is noisy, a clever spy who watches both the equipment and the noise can figure out your "random" numbers, unless you strictly limit how much noise is allowed.

1. Problem Statement

The paper addresses the quantification of intrinsic quantum randomness in adversarial scenarios, specifically focusing on how noise affects the security of Quantum Random Number Generators (QRNGs).

Context: In standard QRNGs, randomness is generated by measuring a quantum state in a basis where it is not diagonal. However, real-world devices produce mixed (noisy) states.
The Adversarial Model: An eavesdropper (Eve) is assumed to possess side information about the state preparation. In standard models, Eve attempts to guess the measurement outcome on every round, inevitably making some errors.
The Gap: The paper introduces two new adversarial capabilities inspired by Quantum State Discrimination (QSD):
1. Unambiguous Randomness: Eve is allowed to output an "inconclusive" result but must never be wrong (zero error rate).
2. Fixed Rate of Inconclusive Outcomes (FRIO): Eve aims to maximize her guessing probability subject to a fixed probability $Q$ of returning an inconclusive outcome.
Specific Challenge: The paper investigates how noise in both the state and the measurement device (joint noise) impacts the eavesdropper's ability to guess outcomes compared to scenarios where only the state is noisy.

2. Methodology

The author employs a rigorous mathematical framework combining Quantum Information Theory and Semidefinite Programming (SDP).

Theoretical Framework:
- Utilizes the Schrödinger-Gisin-HJW theorem to map the randomness generation scenario (where Eve holds a purification of the state) to a Quantum State Discrimination scenario. This allows the use of established QSD bounds.
- Formulates the eavesdropper's strategy as an optimization problem: Eve chooses a decomposition of the mixed state $\rho = \sum p_\mu \rho_\mu$ to maximize her guessing probability.
Optimization Problems:
- Unambiguous Case: Maximize $\sum p_j$ subject to $\rho \succeq \sum p_j |m_j\rangle\langle m_j|$ . This ensures that if Eve guesses outcome $j$ , the state was definitely $|m_j\rangle$ .
- FRIO Case: Maximize guessing probability subject to a fixed probability $Q$ of an inconclusive outcome ( $p_0 = Q$ ).
Analytical Solutions:
- Derives closed-form solutions for arbitrary dimensions and specific cases (qubits, isotropic noise).
- Uses Bloch vector representations for qubit analysis to provide geometric intuition.
- Constructs specific joint decompositions for the "shared noise" scenario where Eve correlates with both the noisy state and the noisy measurement.

3. Key Contributions and Results

A. Maximal Unambiguous Randomness

General Solution: The paper proves that the maximal unambiguous randomness of any quantum state $\rho$ (optimized over all rank-one projective measurements) is proportional to the smallest eigenvalue of the state:
$P^*_{UD}(\rho) = d \cdot \lambda_{\min}(\rho)$
where $d$ is the dimension.
Optimal Measurement: This bound is saturated when the measurement basis is unbiased with respect to the eigenvector corresponding to the minimum eigenvalue.
Qubit Case: For a qubit state with Bloch vector length $r$ , the maximal unambiguous randomness is $1-r$ .

B. FRIO Randomness (Fixed Rate of Inconclusive Outcomes)

Qubits: The author derives exact analytic expressions for the guessing probability $P_Q$ for any qubit state and measurement. The solution reveals a geometric transition in the eavesdropper's optimal decomposition strategy depending on whether the state lies within the convex hull of the measurement vectors and the inconclusive vector.
Noisy States (Isotropic Noise): For a noisy state $\rho_\varepsilon = (1-\varepsilon)|\phi\rangle\langle\phi| + \frac{\varepsilon}{d}\mathbb{I}$ measured in an unbiased basis, the FRIO randomness is solved for any dimension $d$ . The guessing probability decreases monotonically with the inconclusive rate $Q$ .

C. The "Joint Noise" Adversary (Critical Finding)

The paper introduces a scenario where the eavesdropper has correlations with both the noisy state and the noisy measurement device (modeled as a depolarizing channel $\Delta_\varepsilon$ ).

Comparison: The author compares a "single noise" Eve (correlated only to the state, with total noise $\delta$ ) against a "joint noise" Eve (correlated to both state and measurement, each with noise $\varepsilon$ , such that the total Born-rule statistics are preserved).
Result: The joint noise Eve always outperforms the single noise Eve for any fixed error bound.
Critical Threshold: There exists a critical noise parameter $\varepsilon_{crit} = \frac{d}{2(d+\sqrt{d})}$ $ε_{cr i t} = \frac{d}{2 ( d + d )}$ (which is $< 0.5$ $< 0.5$ for all $d$ $d$ ).
- If the noise $\varepsilon \ge \varepsilon_{crit}$ , the joint noise Eve achieves perfect guessing probability (probability of error = 0).
- This implies that beyond this critical noise level, no private randomness can be generated, even if the device statistics appear statistically random to the user.

4. Significance and Implications

Redefining Security Bounds: The work challenges the standard assumption that "statistical randomness" guarantees "private randomness." It demonstrates that noise can be exploited by an adversary who has joint access to the state preparation and measurement devices to eliminate all private randomness.
Device-Dependent QRNGs: The results are crucial for device-dependent QRNGs. They show that accurate modeling of noise is essential; underestimating the correlation between state and measurement noise leads to a severe overestimation of the extractable secure random bits.
Connection to QSD: By bridging the gap between randomness generation and quantum state discrimination, the paper provides a unified theoretical toolkit. The derived bounds on unambiguous discrimination for ensembles of linearly independent states are of independent interest in quantum information theory.
Practical Warning: The identification of the critical error parameter serves as a warning for experimentalists: even in a "wholly device-dependent" setting with perfect statistical randomness, noise can allow an eavesdropper to perfectly predict outcomes, rendering the system insecure.

5. Conclusion

Fionnuala Curran's paper provides a comprehensive analytical framework for quantifying randomness under relaxed adversarial constraints (unambiguous and FRIO). The most significant finding is the demonstration that joint noise correlations between the source and the measurement device can completely destroy private randomness at surprisingly low noise levels, necessitating stricter noise modeling in quantum cryptographic protocols.