Most incompatible measurements and sum-of-squares optimisation

This paper advances the quantification of measurement incompatibility in finite-dimensional quantum systems by deriving analytical universal bounds through novel parent measurements, formalizing their construction via sum-of-squares optimization, and demonstrating their application in certifying high-dimensional quantum steering.

The Gist

Imagine you are trying to bake a cake, but you have a strict rule: you must be able to mix all your ingredients together in one big bowl without them fighting each other. In the quantum world, this "mixing" is called joint measurability. If you can mix your quantum measurements (your ingredients) into one "parent" measurement, they are compatible. If they fight and refuse to mix, they are incompatible.

This paper is about finding the "most stubborn" ingredients—the measurements that are the hardest to mix together. Why does this matter? Because in quantum physics, being incompatible is actually a superpower. It's the fuel that allows for things like "quantum steering," which is a way to prove that a system is truly quantum and not just a classical trick. The more incompatible your measurements are, the more noise (static or errors) your quantum system can handle before the magic disappears.

Here is the breakdown of what the author, Sébastien Designolle, discovered, using simple analogies:

1. The Problem: Finding the "Worst" Mixers

Scientists have known for a while how to find the most incompatible pairs of measurements (like trying to mix two specific spices that hate each other). But what happens when you have a whole spice rack with 5, 10, or 100 measurements? Finding the absolute "worst" mixers for large groups has been a huge mathematical headache.

The author's goal was to build a universal "recipe" (a parent measurement) that works for any group of measurements to prove just how incompatible they can be.

2. The Method: The "Sum-of-Squares" Ladder

To solve this, the author built a mathematical ladder called a Sum-of-Squares (SOS) hierarchy.

• The Analogy: Imagine you are trying to prove a shape is a perfect square.
  • Level 1 (The Basics): You check if the sides are straight. This is like the author's "Degree 2" method. It's a simple, clean formula that works well and improves on what we knew before.
  • Level 2 (Climbing Higher): You check the corners and the diagonals. This is the "Degree 3" and "Degree 4" methods.
  • The Top of the Ladder: The author realized that instead of just checking one specific shape, they could use a computer to check any shape made of "squares" (mathematical polynomials that are always positive). This is the Sum-of-Squares optimization.

By climbing this ladder, the author could construct "parent measurements" that are more flexible and powerful than previous methods.

3. The Big Discovery: The "Anticommuting" Champions

One of the most exciting findings is about a specific type of measurement called anticommuting observables.

• The Analogy: Think of these as measurements that are like "Left" and "Right" or "Up" and "Down" in a quantum sense. They are so fundamentally opposed that if you try to measure one, the other immediately flips or changes.
• The Result: The author proved that for simple "yes/no" (dichotomic) measurements, these "Left/Right" opposites are the most incompatible measurements possible. They are the ultimate "unmixable" ingredients. This confirms that if you want to build the most robust quantum system, you should use these specific types of measurements.

4. The Computer's Role: Beating the Math

While the author found perfect mathematical formulas (analytical results) for many cases, they also used a computer to solve the "Sum-of-Squares" puzzle for more complex situations.

• The Result: The computer found solutions that were even better than the author's own best mathematical formulas. This is like writing a perfect recipe by hand, but then having a super-computer taste-test and tweak the ingredients to make the cake even fluffier.
• The Proof: The paper shows that this computer method works. It successfully improved the known limits of how incompatible measurements can be, proving that the "ladder" approach is a powerful tool.

5. The Real-World Application: The "Dimension Witness"

The paper concludes by explaining how this helps in the real world of quantum technology.

• The Analogy: Imagine you are trying to guess the size of a box (the dimension of a quantum system) without opening it. You can only poke it with your measurements.
• The Application: Because the author found the "most incompatible" measurements, they created a better "ruler" (a dimension witness). If you use these measurements and see a certain amount of "quantum steering" (the system reacting strongly to noise), you can prove with certainty that the system is a high-dimensional quantum object, not a small, simple one. This is done in a "one-sided device-independent" way, meaning you don't have to trust the other person's equipment to know the truth.

Summary

In short, this paper builds a better mathematical toolbox to find the "most stubborn" quantum measurements.

1. It proves that opposing measurements (anticommuting ones) are the champions of incompatibility.
2. It introduces a hierarchy of methods (the Sum-of-Squares ladder) that allows computers to find even better solutions than human formulas alone.
3. It provides a better ruler to certify the size and complexity of quantum systems, which is crucial for building future quantum computers and secure communication networks.

The paper does not claim to have built a new quantum computer or cured a disease; it simply provides the mathematical "blueprints" and "rulers" needed to understand and certify how powerful these quantum systems can be.

Technical Summary

Technical Summary: Most Incompatible Measurements and Sum-of-Squares Optimisation

Problem Statement
Measurement incompatibility (or the lack of joint measurability) is a fundamental resource in quantum theory, underpinning phenomena such as Bell nonlocality and Einstein-Podolsky-Rosen steering. While the incompatibility of pairs of measurements is well-characterized, determining the "most incompatible" sets of measurements for larger collections ($k > 2$) in finite-dimensional systems remains a significant open problem. The core challenge lies in establishing universal bounds valid for all possible sets of measurements. This requires constructing a "universal parent measurement" (a joint measurement from which any specific set can be derived via post-processing) that satisfies semidefinite positivity constraints. Previous approaches relied on combining pairwise information or specific geometric constructions, which often failed to capture the full structure of larger sets or lacked experimental viability.

Methodology
The authors develop a systematic framework based on sum-of-squares (SOS) optimisation to construct universal parent measurements. The methodology proceeds in two main phases:

1. Analytical Constructions (Low-Degree Polynomials):
   The authors first exhibit universal parent measurements defined as positive polynomials in sums of the measurement operators.

   • Degree Two: They construct parent measurements of the form $G \propto (S - \alpha \mathbb{1})^2$, where $S$ is the sum of projectors. This approach yields analytical lower bounds on incompatibility robustness ($\eta$) for both fixed dimension ($d$) and fixed number of outcomes ($n$).
   • Degree Three: By extending the polynomial degree to three ($G \propto S(S - \beta \mathbb{1})^2$), they derive tighter bounds, particularly for the generalised incompatibility robustness ($\eta_g$).
   • Special Cases: For mutually unbiased bases (MUBs), they utilize higher-degree polynomials (up to degree four) and specific properties of MUBs to improve bounds on depolarising robustness ($\eta_d$).

2. Sum-of-Squares Hierarchy (Numerical Optimisation):
   Recognizing that restricting parent measurements to simple polynomials in $S$ is limiting, the authors formalize a general hierarchy.

   • They define normalisable polynomials (where summation over outcomes reduces to the identity) and pinched-marginalisable polynomials (where marginals can be bounded using specific inequalities).
   • This allows the problem to be cast as a Semidefinite Programming (SDP) instance (Eq. 49 and 50), where the parent measurement is sought within the set of SOS polynomials of degree $2t$.
   • A restricted hierarchy (Eq. 50) is implemented numerically to handle computational complexity, utilizing "pinched" constraints to reduce the number of variables.

Key Contributions

• Analytical Bounds: The paper provides new analytical universal parent measurements that improve upon the state-of-the-art bounds for various incompatibility measures ($\eta_d, \eta_r, \eta_g$).
• Characterization of Maximal Incompatibility: The authors prove that sets of dichotomic measurements arising from anticommuting observables are maximally incompatible with respect to random ($\eta_r$) and generalised ($\eta_g$) robustness. Specifically, they show $\chi^r_k(2) = 1/\sqrt{k}$ and $\chi^g_k(2) = \frac{1}{2}(1 + 1/\sqrt{k})$.
• Unified Framework: The work unifies previous analytical methods (from references [10, 13, 20]) and extends them by introducing a hierarchy that allows for both analytical derivation and numerical refinement.
• Numerical Improvements: Through the restricted SOS hierarchy, the authors obtain numerical bounds that strictly improve upon their own analytical values and previous literature in specific cases (e.g., $d=4, k=5$).

Results

• Anticommuting Observables: The study confirms that anticommuting observables constitute the most incompatible dichotomic sets, providing tight bounds for these specific configurations.
• Dimension Witnesses: The derived bounds are applied to demonstrate genuine high-dimensional steering. The authors provide an upper bound on steering robustness (Eq. 42) and a corresponding dimension witness (Eq. 43) that depends on the number of settings ($k$) and the system dimension ($d$).
• Comparative Performance:
  • For $k=2$, the analytical results reproduce known tight bounds.
  • For $k > 2$, the degree-two and degree-three analytical bounds improve upon previous loose bounds derived from pairwise combinations.
  • Numerical results from the SOS hierarchy (Table I) demonstrate that the method can surpass analytical limits, particularly in the regime of fixed dimension and fixed outcomes.
  • Table III summarizes the progression of lower bounds on $\eta_g$ for $d=4, k=5$, showing a clear improvement from cloning-based bounds (0.5200) to the pinched-marginalisable numerical results (~0.5391).

Significance and Claims
The paper claims to establish a pathway to certifying incompatibility in scenarios involving larger collections of measurements, moving beyond the two-measurement regime.

• Theoretical Advance: It provides a rigorous method to construct universal parent measurements, addressing the technical obstacle of proving semidefinite positivity for arbitrary sets.
• Optimality: The authors demonstrate that for relevant robustness measures, anticommuting observables are indeed the most incompatible dichotomic measurements.
• Methodological Power: The SOS hierarchy is presented as a powerful tool that unifies analytical and numerical approaches. While the authors are modest about the convergence of the general hierarchy (noting it is an open question whether it becomes tight for all $k, d, n$), they show it is already tight in edge cases ($k=2$, $n=2$, $d=2$) and capable of improving bounds in more complex scenarios.
• Application: The results offer direct applications for one-sided device-independent dimension witnesses, allowing for the certification of quantum system dimensionality with improved robustness against noise.

The work does not propose new experimental setups but rather provides the theoretical bounds and certification tools necessary to interpret such experiments, particularly those involving high-dimensional steering.