⚛️ quantum physics

Maximal global device-independent randomness from projective measurements in every dimension

This paper demonstrates that projective measurements on bipartite quantum systems of local dimension $d$ can certify the theoretically maximal $2\log(d)$ bits of device-independent randomness for any $d \ge 2$ using explicit protocols based on mutually unbiased bases, while also proving their robustness against experimental noise.

Original authors: Máté Farkas, Piotr Mironowicz, Remigiusz Augusiak

Published 2026-06-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Máté Farkas, Piotr Mironowicz, Remigiusz Augusiak

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 generate a truly random number, like a secret code that no one else can guess. In the quantum world, we can do this by measuring particles. But there's a catch: usually, you have to trust your measuring equipment perfectly. If your machine is slightly broken or biased, your "random" numbers might actually be predictable.

Device-Independent Randomness is the "gold standard" of security. It's like a magic trick where you don't need to trust the magician or the props. You only look at the results of the trick. If the results show a specific, impossible-to-fake pattern (called a "Bell violation"), you know for a fact that the numbers are truly random, even if the equipment is a black box you know nothing about.

The Problem: How Much Randomness Can We Get?

Think of a quantum system as a die.

A standard coin (2 sides) gives you 1 bit of randomness.
A 6-sided die gives you about 2.58 bits.
A die with $d$ sides gives you $\log(d)$ bits.

The paper tackles a big question: If two people (Alice and Bob) share a pair of these "quantum dice" (entangled particles), how much total random information can they extract together?

Previous research showed that if you use very complex, hard-to-build measurements, you could get the theoretical maximum: $2 \log(d)$ bits. This is the absolute limit of what's mathematically possible for a system of that size. However, those complex measurements are like trying to build a house with a hammer made of glass—they are theoretically possible but practically very difficult to do in a lab.

The Breakthrough: Simple Tools, Maximum Results

The authors of this paper asked: Can we get that same maximum amount of randomness using only "projective measurements"?

In everyday language, projective measurements are the "standard" way physicists measure quantum systems. They are the workhorses of the lab—easy to build and very reliable.

The Answer: Yes.
The paper proves that for any size of quantum system (any dimension $d$ ), Alice and Bob can use these simple, standard measurements to extract the maximum possible amount of randomness ( $2 \log(d)$ bits).

The Analogy:
Imagine you have a locked safe (the quantum system).

Old Method: To get the most treasure (randomness), you needed a master key that was incredibly fragile and hard to forge (complex measurements).
This Paper's Method: The authors found a way to use a simple, sturdy screwdriver (projective measurements) to open the safe and get the exact same amount of treasure. They didn't need the fragile key; the simple tool was enough to reach the theoretical limit.

How Did They Do It?

They designed a specific "game" (a Bell inequality protocol) that Alice and Bob play.

The Setup: Alice and Bob share a pair of entangled quantum dice.
The Game: They roll the dice using specific settings. Alice has two types of rolls, and Bob has many.
The Check: They compare their results. If the results match a specific, highly correlated pattern (the "maximal violation"), it proves two things:
- Their dice are perfectly entangled (like a pair of magical twins).
- Their measurements are perfectly aligned with the "secret" of the dice.
The Result: Because the game proves the system is perfect, they can mathematically guarantee that the outcomes of their final rolls are completely unpredictable to anyone else. They get the full $2 \log(d)$ bits of randomness.

Is It Robust? (The "Real World" Test)

In a perfect lab, everything works perfectly. But in the real world, there is noise (static, interference, imperfect machines).

The authors ran computer simulations to see what happens when the experiment isn't perfect.

The Finding: Even with a little bit of noise, their method still produces a huge amount of randomness.
The Comparison: They compared their method (using larger dice, $d=3$ and $d=4$ ) against the old standard method (using a coin, $d=2$ ).
The Result: Their method is more robust. Even when the noise is high, the larger systems ( $d=3, 4$ ) still manage to certify more than 2 bits of randomness, whereas the simple coin system ( $d=2$ ) starts to fail.

Summary

This paper shows that you don't need exotic, impossible-to-build technology to get the maximum possible security from quantum randomness. By using standard, reliable measurement tools (projective measurements) on quantum systems of any size, you can certify the absolute maximum amount of private, unpredictable randomness allowed by the laws of physics. Furthermore, this method holds up well even when the experiment isn't perfect, making it a very practical step forward for secure communication.

Technical Summary: Maximal Global Device-Independent Randomness from Projective Measurements in Every Dimension

Problem Statement
Device-independent quantum random number generation (DIQRNG) allows for the certification of private randomness based solely on observed measurement statistics, without requiring trust in the internal workings of the devices. While it is theoretically established that a single party can certify a maximum of $2 \log(d)$ bits of randomness in a local dimension $d$ using generalized (non-projective) measurements, the practical implementation of such measurements is challenging. Projective measurements are the most experimentally feasible option. However, a critical open question remained: can the theoretical maximum of $2 \log(d)$ bits of global randomness (certified for both parties jointly) be achieved using only projective measurements for all dimensions $d \ge 2$ ? Previous results confirming this were limited to the qubit case ( $d=2$ ).

Methodology
The authors address this by constructing explicit DIQRNG protocols based on a family of Bell inequalities originally introduced in Ref. [15], adapted to include a specific measurement setting for randomness generation.

Bell Inequality Construction: The protocol utilizes a Bell inequality $\hat{W}_d(p)$ parametrized by an integer $d \ge 2$ . In this scenario:
- Alice has two settings ( $x \in \{1, 2\}$ ), each with $d$ outcomes.
- Bob has $d^2 + 1$ settings. The first $d^2$ settings are used to maximize the violation of the base inequality from Ref. [15], while the final setting, labeled 'r', is specifically designed to generate randomness.
- The inequality combines terms from the original Ref. [15] with a new term involving Bob's 'r' setting and Alice's first setting.
Self-Testing and Certification: The authors employ a sum-of-squares (SOS) decomposition of the Bell operator to derive the maximal quantum value, which is $\sqrt{d(d-1)} + 1$ . By analyzing the conditions required to reach this maximal violation, they derive algebraic constraints on the state and measurements:
- The underlying state must be locally isometric to a maximally entangled state $|\phi^+_d\rangle$ in dimension $d$ .
- Alice's measurements must satisfy specific algebraic relations characteristic of mutually unbiased bases (MUBs).
- Crucially, the constraints force Bob's 'r' measurement operators to be projective and perfectly correlated with Alice's first measurement operators (via transposition in the computational basis).
Entropy Calculation: Using these self-testing constraints, the authors calculate the classical-quantum state $\sigma_{ABE}$ corresponding to the outcomes of Alice's second setting and Bob's 'r' setting. They demonstrate that for any state and measurements compatible with the maximal violation, the conditional von Neumann entropy $H(A, B|E)$ —which quantifies the device-independent randomness—equals $\log(d^2) = 2 \log(d)$ .
Robustness Analysis: Recognizing that experimental imperfections prevent exact maximal violation, the authors numerically compute lower bounds on the certified randomness (min-entropy) for noisy scenarios. They utilize the Navascués-Pironio-Acín (NPA) hierarchy at level $1+AB$ to bound the guessing probability for dimensions $d=3$ and $d=4$ , comparing these results against the standard modCHSH inequality ( $d=2$ ).

Key Contributions and Results

Theoretical Maximum Achievement: The paper proves that for every dimension $d \ge 2$ , it is possible to certify the theoretical maximum of $2 \log(d)$ bits of global device-independent randomness using only projective measurements. This resolves the open question regarding the global limit for projective measurements, extending previous knowledge which was restricted to $d=2$ .
Explicit Protocols: The authors provide explicit protocols based on mutually unbiased bases that achieve this bound. The protocols rely on a specific Bell inequality where the maximal violation certifies that the system behaves as a maximally entangled state measured in mutually unbiased bases.
Noise Robustness: Numerical simulations demonstrate that the protocols are robust to experimental noise. For dimensions $d=3$ and $d=4$ , the certified randomness exceeds 2 bits (the maximum for $d=2$ ) at noise levels $\eta \approx 0.997$ and $\eta \approx 0.998$ , respectively. These noise levels are comparable to those reported in current qubit experiments, suggesting that the advantages of higher-dimensional systems are within reach of current technology.

Significance
The work establishes that the theoretical limit of device-independent randomness generation is achievable in any dimension using the most experimentally accessible type of measurement (projective). By showing that global randomness scales as $2 \log(d)$ , the paper motivates the use of higher-dimensional quantum systems to maximize the rate of secure random number generation without requiring complex non-projective measurements. The authors note that while analytic techniques for robust self-testing would further strengthen these protocols, the current numerical evidence confirms the practical viability of these schemes. The paper also acknowledges that similar results regarding global randomness certification were independently developed in Ref. [30].

The Problem: How Much Randomness Can We Get?

The Breakthrough: Simple Tools, Maximum Results

How Did They Do It?

Is It Robust? (The "Real World" Test)

Summary

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