Quantum randomness beyond projective measurements

Imagine you are trying to generate a truly random number, like flipping a coin or rolling a die, but you want to be absolutely sure that no one else (even a super-smart hacker with a quantum computer) can predict the result before it happens. In the world of quantum physics, this is possible because nature itself is fundamentally unpredictable.

This paper, written by Fionnuala Curran, explores how much "true" randomness we can squeeze out of different types of quantum measurements. Think of a quantum measurement as a machine that takes a quantum state (a particle) and spits out a number. The goal is to find the best machine settings to get the most unpredictable numbers possible.

Here is a breakdown of the paper's main ideas using everyday analogies:

1. The Setup: The Unpredictable Coin

In classical physics, if you know exactly how a coin is flipped, you can predict if it will be heads or tails. In quantum physics, even if you know everything about the setup, the outcome is still a mystery. This is called intrinsic randomness.

However, not all quantum "machines" (measurements) are created equal. Some are "extremal," meaning they are the most fundamental types of machines that can't be broken down into simpler, random mixtures. The paper asks: Which of these fundamental machines gives us the most randomness?

2. The "Skewed" Dice: Cheating to Win

The authors first introduce a new family of measurements they call "Skewed SIC" measurements.

The Analogy: Imagine a standard die where every number (1 through 6) has an equal chance of coming up. That's a "fair" die. But what if you have a special, slightly bent die that usually lands on 1, but if you roll it just right, it becomes perfectly fair?
The Finding: These "Skewed" measurements are designed so that if you feed them a specific type of quantum state (a "pure" state), they produce a perfectly uniform, random result. Even better, they can do this for any size of quantum system (any dimension) where a specific type of measurement (called a SIC) exists. This solves a puzzle about how to get the maximum possible randomness (2 log d bits) in a device-dependent setting.

3. The "Unbiased" Dice: Playing Fair

Next, the paper looks at "Unbiased" measurements. These are machines where, if you feed them a completely random "junk" state, every outcome is equally likely.

The Analogy: Think of a tetrahedron (a pyramid with four triangular faces) floating in space. The corners of this pyramid represent the possible outcomes of a measurement.
The Finding: The authors discovered a simple rule: The amount of randomness you get depends on how close your starting quantum state is to the center of this pyramid.
- If your state is right in the middle, you get less randomness.
- If your state is far away, you get more.
- They calculated exactly how much randomness you get for any state in a 2-dimensional system (a qubit, or a quantum bit).

4. The "SIC" vs. The "Scissors"

The paper compares two specific types of these pyramid-shaped measurements:

The SIC (Symmetric Informationally Complete) Measurement: This is the "perfect" pyramid. All faces are identical, and it's the best tool for mapping out (tomography) what a quantum state looks like.
- The Surprise: Even though the SIC is the best at measuring states, the authors found it is actually the worst at generating randomness among unbiased measurements. It has the "least" intrinsic randomness. It's like a very precise ruler that is terrible at generating random numbers.
The "Scissors" Measurements: The authors invented a new family of measurements they call "Scissors."
- The Analogy: Imagine the pyramid faces are like blades of a pair of scissors. You can open or close the blades by changing a single angle.
- The Finding: As you "close" the scissors (change the angle), the measurement becomes less "fair" (biased), but it gets closer and closer to generating the maximum possible amount of randomness.
- They showed that in dimensions 2, 3, and 4, you can tune these Scissors measurements to get almost as much randomness as theoretically possible, even without biasing the results too much.

5. The Big Picture

The paper essentially maps out the landscape of quantum randomness:

Skewed measurements can give you the absolute maximum randomness if you know your starting state.
Unbiased measurements (like the Scissors family) can get you very close to that maximum without needing to bias the outcomes.
The famous SIC measurement, while great for other tasks, is actually the "least random" of the unbiased bunch.

Summary

Think of this paper as a guidebook for a casino owner who wants to build the most unpredictable slot machines.

They found a way to build a "Skewed" machine that is perfectly random for specific players.
They analyzed "Fair" machines and found that the most symmetrical, perfect-looking machine (the SIC) is actually the least random.
They designed a new "Scissors" machine that can be adjusted to be almost perfectly random, proving that you don't need to break the rules (bias the machine) to get the best randomness; you just need to tune the angle correctly.

The paper concludes by solving the math for 2D systems completely and providing a roadmap for how to achieve maximum randomness in higher dimensions using these new "Scissors" tools.

Technical Summary: Quantum Randomness Beyond Projective Measurements

Problem Statement
The intrinsic unpredictability of quantum measurements is a fundamental resource for quantum random number generation (QRNG). In adversarial scenarios, the security of this randomness relies on the inability of an eavesdropper (Eve), who may hold quantum or classical correlations with the measurement setup, to correctly guess the outcomes. While extremal measurements—those that cannot be decomposed into a probabilistic mixture of other measurements—are known to be secure against such adversaries, a specific quantitative question remains open: exactly how much intrinsic randomness can be generated by a given extremal measurement?

Previous work has established that the maximal conditional min-entropy (a measure of randomness) achievable in dimension $d$ is bounded by $2 \log d$ bits. However, characterizing the randomness generated by specific classes of extremal measurements, particularly unbiased ones, for arbitrary input states has remained incomplete. This paper addresses the gap in understanding the intrinsic randomness of unbiased extremal rank-one Positive Operator-Valued Measures (POVMs) and explores whether specific families of measurements can saturate the theoretical maximum of $2 \log d$ bits.

Methodology
The authors operate within a device-dependent framework, assuming full characterization of both the quantum state $\rho$ and the POVM $M$ . The intrinsic randomness is quantified via the conditional min-entropy, $H_{\min}(\rho, M) = -\log P_{\text{guess}}(\rho, M)$ , where $P_{\text{guess}}$ is the optimal guessing probability of an eavesdropper. This probability is defined as the maximum over all probabilistic decompositions of the state $\rho$ into pure states $\{p_j, \rho_j\}$ :
$P_{\text{guess}}(\rho, M) = \max_{\{p_j, \rho_j\}} \sum_j p_j \text{tr}(\rho_j M_j).$

The paper employs several mathematical tools:

Convex Geometry and Fidelity: The authors relate the guessing probability to the quantum fidelity between the input state and the convex hull of the measurement projectors.
Semidefinite Programming (SDP): Optimization problems regarding state discrimination and guessing probabilities are formulated and solved using SDP duality and the Karush-Kuhn-Tucker (KKT) conditions.
Bloch Vector Representation: For qubit systems ( $d=2$ ), the analysis utilizes Bloch vector notation to explicitly parameterize unbiased extremal measurements and solve for optimal states.
Group Covariance and Symmetry: The study leverages the properties of Symmetric Informationally Complete (SIC) measurements and introduces new families of measurements derived from them.

Key Contributions and Results

1. Skewed SIC Measurements and Maximal Randomness
The authors introduce a family of "skewed" SIC POVMs. Unlike standard SICs, which are unbiased and produce a uniform distribution from the maximally mixed state, skewed SICs are biased. However, this bias allows them to produce a uniform probability distribution from specific pure or isotropically noisy states.

Result: The authors prove that in any dimension $d$ where a SIC measurement exists, one can construct a skewed SIC measurement that generates the maximal possible intrinsic randomness ( $2 \log d$ bits) from any pure or isotropically noisy state.
Significance: This partially resolves an open question regarding the achievability of maximal randomness in a source-device-independent setting, as these measurements are informationally complete (MICs).

2. Unbiased Extremal Measurements in General Dimensions
For any unbiased extremal rank-one POVM with $n$ outcomes in dimension $d$ , the authors derive a general formula for the guessing probability:
$P_{\text{guess}}(\rho, M) = \frac{d}{n} \max_{\sigma \in \mathcal{P}} F(\rho, \sigma),$
where $\mathcal{P}$ is the convex hull of the measurement projectors and $F$ is the quantum fidelity.

Lower Bound: This leads to a state-independent lower bound on the conditional min-entropy of $\log(n/d)$ bits.
Strictness: The authors prove that for unbiased extremal measurements with $n < d^2$ outcomes, the guessing probability is strictly greater than $1/d^2$ , meaning the maximal bound of $2 \log d$ bits cannot be saturated without biasing the outcomes or using $d^2$ outcomes.

3. Explicit Solution for Qubit ( $d=2$ ) Measurements
The paper provides a complete characterization of unbiased extremal qubit measurements (with 2, 3, or 4 outcomes).

Parameterization: All unbiased extremal qubit MICs (4 outcomes) are shown to be group covariant and can be described by two angles, $\delta$ and $\alpha$ .
Optimal States: The authors identify the specific pure states that minimize the guessing probability for any given unbiased qubit MIC. These states are found to be the normals to the faces of the tetrahedron formed by the measurement vectors.
SIC vs. Others: A counter-intuitive result is presented: among all four-outcome unbiased extremal qubit measurements, the SIC measurement (which is optimal for quantum state tomography) yields the least intrinsic randomness. It has the highest guessing probability ( $P_{\text{guess}} = \frac{1}{4}(1+l)$ , where $l$ is maximized for SICs).

4. The "Scissors" Family of Measurements
To approach the maximal randomness bound of $2 \log d$ bits without biasing the outcomes (which would require non-extremal or skewed measurements), the authors introduce a one-parameter family of "scissors" measurements.

Qubits: In dimension 2, setting $\alpha=0$ in the general parameterization yields a family parameterized by $\delta$ . As $\delta \to 0$ , the measurement approaches a non-extremal limit, and the guessing probability for specific noisy states approaches the theoretical maximum.
Higher Dimensions: The authors extend this concept to dimensions 3 and 4. They define families of unbiased MICs (using Weyl-Heisenberg operators for $d=3$ and a specific generalization for $d=4$ ) that include a SIC measurement as a specific point in the parameter space. As the parameter approaches a limiting value, the measurements approach a non-extremal state that generates arbitrarily close to $2 \log d$ bits of randomness.

Significance and Claims
The paper claims to solve the problem of characterizing intrinsic randomness for unbiased extremal rank-one measurements in dimension two explicitly for any state. It establishes that while SIC measurements are optimal for tomography, they are suboptimal for generating intrinsic randomness among unbiased MICs.

The primary theoretical contribution is the demonstration that maximal randomness ( $2 \log d$ bits) can be generated device-dependently in any dimension where a SIC exists, provided one utilizes the proposed skewed SIC measurements. Furthermore, the "scissors" families provide a pathway to approach this maximal randomness using unbiased measurements, bridging the gap between extremal and non-extremal strategies. The results hold for source-device-independent scenarios due to the informational completeness of the measurements discussed.

The authors conclude modestly, noting that while they have characterized these measurements, it remains an open question whether the higher-dimensional scissors-like families will be useful for device-independent randomness generation, and how these results translate to realistic scenarios involving noise.