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No Finite NPA Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

This paper proves that in the simplest bipartite Bell scenario with two binary measurements per party, no finite level of the Navascués–Pironio–Acín (NPA) hierarchy exactly characterizes the complete quantum set, as the hierarchy's outer approximations strictly contain non-quantum behaviors that accumulate at local deterministic points.

Original authors: Anubhav Chaturvedi

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anubhav Chaturvedi

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 map the boundaries of a mysterious, invisible island called "Quantum Reality." This island sits right next to a much larger, foggy continent called "Everything That's Possible." Scientists have long wondered: How close can we get to the true edge of the Quantum Reality island using a specific tool called the NPA hierarchy? Think of the NPA hierarchy as a set of increasingly detailed, high-resolution satellite maps. Each "level" of the map adds more detail, zooming in closer to the island's shoreline. For years, researchers hoped that if they just zoomed in enough—reaching a specific, finite level of detail—they would finally see the exact outline of the island, with no extra fog or missing pieces. This is a big deal because if a finite map worked perfectly, it would mean we could describe all quantum behaviors with a single, simple mathematical rule, making it much easier to verify if a strange new experiment is truly quantum or just a clever trick.

The paper by Anubhav Chaturvedi tackles this question in the simplest possible setting: a scenario with two people (Alice and Bob) making two choices each, with two possible outcomes. It's the "Hello World" of quantum experiments, the smallest non-trivial playground where these rules apply. The author asks a simple but profound question: Is there a "Level 10" or "Level 100" on the NPA map that finally matches the Quantum Reality island perfectly? The answer, surprisingly, is a definitive "no." The paper proves that no matter how many times you zoom in, no finite level of this map ever captures the entire island exactly. There is always a tiny, stubborn sliver of "fake" territory that the map includes but doesn't actually exist in quantum physics.

The Infinite Zoom That Never Lands

To understand why this is a problem, imagine you are trying to draw the perfect circle of a coin using a grid of square pixels. If you use a low-resolution grid, your circle looks blocky and jagged. If you increase the resolution, it looks smoother. You might think, "If I just keep increasing the resolution, eventually the pixels will be so small they perfectly match the curve." In the world of quantum physics, the NPA hierarchy is that grid. For a long time, scientists thought that for the simplest experiments, one of these grids would eventually become perfect.

Chaturvedi's paper shows that for a specific type of quantum experiment involving a "doubly tilted" setup (a fancy way of saying the experiment is slightly skewed to test the edges of quantum behavior), this never happens. As the experiment gets closer and closer to a "local" state—where Alice and Bob stop acting like quantum particles and start acting like ordinary, predictable objects—the error in the NPA map doesn't just get smaller; it behaves in a weird, stubborn way.

The author focuses on a specific point where the quantum advantage (the "extra" power quantum mechanics has over classical physics) is vanishingly small. It's like watching a race where the winner is only a hair's breadth ahead of the loser. The paper calculates exactly how this tiny advantage shrinks as the experiment approaches the limit. It turns out the quantum advantage shrinks at a very specific rate, proportional to the cube of the distance to the limit.

Here is where the magic trick happens. The author constructs a mathematical "test vector"—a specific way of looking at the data—that reveals a hidden shape. When the NPA map tries to approximate the quantum set at this specific limit, the error it produces looks like a famous mathematical object called the Motzkin polynomial.

The Motzkin Monster

The Motzkin polynomial is a mathematical shape that is always positive (it never dips below zero), but it has a secret: it cannot be built by adding up simple squares of other polynomials. In the language of the NPA hierarchy, being able to build something from "sums of squares" is the golden ticket. If a map is perfect, every error it makes must be expressible as a sum of squares.

The paper proves that if any finite level of the NPA hierarchy were exact, it would force the Motzkin polynomial to be a sum of squares. But we know from math that the Motzkin polynomial cannot be a sum of squares. It's like trying to build a perfect circle out of square bricks; no matter how many bricks you use, you can't make a perfect circle if the bricks are strictly square. The paper shows that the NPA hierarchy, no matter how high you go, is stuck with "square bricks" (fixed finite lists of words), and the quantum reality island has a "circular edge" that those bricks can never perfectly trace.

The Ghost in the Machine

The consequence of this is fascinating. Because the NPA map is never exact, it always includes some "ghost" behaviors. These are scenarios that the map says are possible, but which actually violate the laws of quantum physics. The paper shows that these ghosts don't hide in some far-off, weird corner of the universe. Instead, they accumulate right next to the most boring, predictable, classical behaviors.

Imagine a local deterministic behavior as a rock sitting on the ground. The paper proves that no matter how close you get to that rock, there is always a "ghost" behavior floating just a tiny bit away from it that the NPA map thinks is quantum, but isn't. This means that even in the simplest, most basic quantum experiment, you can never fully separate the "truly quantum" from the "fake quantum" using a fixed, finite set of rules.

Why This Matters

This result separates two different questions that scientists had been mixing up. We already knew that for certain specific measurements (like the standard CHSH test), a low-level NPA map gives the exact right answer. We also knew that for "one-sided tilted" experiments, a slightly higher level (Level 1 + AB) works perfectly. It was tempting to think that if these specific slices worked, the whole cake must be perfect.

Chaturvedi's paper puts a stop to that hope. It proves that while the map might be perfect for specific slices or specific points, it is never perfect for the entire set of quantum behaviors in the simplest scenario. The error might be tiny, but it is mathematically guaranteed to exist. The "complete quantum set" is too complex to be captured by any single, finite list of rules.

In the end, the paper tells us that the quantum world is a bit more elusive than we hoped. Even in its simplest form, it refuses to be pinned down by a finite, static description. The NPA hierarchy will always get closer and closer, converging asymptotically, but it will never quite land on the exact target. The Motzkin polynomial stands as a mathematical guardian, ensuring that the boundary between the quantum and the classical remains just out of reach of any finite approximation.

Technical Summary: No Finite NPA Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

Problem Statement
The Navascués–Pironio–Acín (NPA) hierarchy provides a sequence of semidefinite programming (SDP) relaxations that yield outer approximations to the set of quantum correlations ( $Q$ ). While the hierarchy converges asymptotically to the quantum set in the commuting-operator model, a fundamental open question remains: does any single finite level of the hierarchy exactly characterize the complete quantum set in a given Bell scenario?

This paper addresses this question in the minimal nontrivial Bell scenario: the bipartite case with two parties, two binary inputs, and two binary outputs (the $(2,2,2)$ scenario). Previous results have shown that specific finite levels are exact for certain projections or specific Bell functionals. For instance, Level 1 yields the exact Tsirelson bound for the standard CHSH inequality, and the intermediate "Level $1+AB$" (which includes cross-party products) yields the exact quantum maximum for all one-sided tilted CHSH functionals. However, it was unknown whether a finite level could exactly describe the entire set of quantum behaviors in this scenario, particularly near the boundary where quantum advantages vanish.

Methodology
The authors investigate the symmetric doubly tilted CHSH functional:
$h_\alpha = A_0B_0 + A_0B_1 + A_1B_0 - A_1B_1 + \alpha(A_0 + B_0), \quad 0 \le \alpha < 1$
where the local maximum is $2+2\alpha$ . The study focuses on the endpoint behavior as $\alpha \to 1$ (or $T = 1-\alpha \to 0$ ). In this limit, the optimal quantum state approaches a product state, and the measurement pairs become compatible, causing the Bell violation to vanish.

The methodology proceeds through the following steps:

Endpoint Scaling Analysis: Using Jordan's lemma and the reduction to $4\times4$ blocks for two-qubit strategies, the authors derive the exact asymptotic expansion of the quantum maximum $\omega_Q(\alpha)$ as $\alpha \uparrow 1$ . They establish that the quantum advantage over the local bound scales cubically: $\omega_Q(1-T) - (4-2T) \sim \frac{4}{3}T^3$ .
Motzkin Polynomial Identification: By rescaling the measurement angles and the operator gap $\omega_Q(1-t^2)I - H_t$ (where $T=t^2$ ), the authors construct a specific vector $\xi_t$ such that the expectation value converges to the Motzkin polynomial $M(X,Y) = X^4Y^2 + X^2Y^4 - 3X^2Y^2 + 1$ (up to a constant factor) in the limit $t \to 0$ .
Sum-of-Squares (SOS) Obstruction: The paper utilizes the duality of the NPA hierarchy. If a finite level $L$ were exact, the difference between the NPA bound $\omega_L$ and the quantum bound $\omega_Q$ would be representable as a sum of squares of polynomials in the measurement projectors.
Compactness and Contradiction: The authors prove that if the error $\omega_L(\alpha) - \omega_Q(\alpha)$ were bounded by a term of order $(1-\alpha)^3$ (matching the quantum scaling), the limit of the rescaled SOS certificates would force the Motzkin polynomial (plus a nonnegative constant) to be a sum of squares of real polynomials.
Newton Polytope Argument: Citing the properties of the Newton polytope, the authors demonstrate that the Motzkin polynomial is not a sum of squares, and crucially, that adding any nonnegative constant $c \ge 0$ does not make it a sum of squares. This is due to the specific structure of the allowed monomials in the square factors, which forces the coefficient of the $X^2Y^2$ term to be nonnegative, whereas in the Motzkin polynomial it is $-3$.

Key Results

Divergence of Finite-Level Error: For every finite level $L \ge 1$ , the error between the NPA bound and the true quantum maximum diverges relative to the cubic scaling of the quantum advantage:
$\lim_{\alpha \uparrow 1} \frac{\omega_L(\alpha) - \omega_Q(\alpha)}{(1-\alpha)^3} = +\infty$
This implies that the NPA error eventually exceeds the entire quantum advantage by an arbitrarily large factor as one approaches the local deterministic point.
Strict Inclusion of Behavior Sets: Consequently, for any standard NPA relaxation defined by a fixed finite list of words in the measurement projectors (denoted $N_S$ ), the quantum set is strictly contained within the relaxation:
$Q \subsetneq N_S$
This holds even for the "almost quantum" set (derived from the Level $1+AB$ relaxation), which is known to satisfy various information-theoretic principles.
Accumulation of Non-Quantum Behaviors: The paper proves that non-quantum behaviors feasible under any fixed finite NPA level can be constructed to lie arbitrarily close to a local deterministic behavior (specifically $p_{loc}(a,b|x,y) = \delta_{a,1}\delta_{b,1}$ ).

Significance and Claims
The paper establishes that the exactness of finite NPA levels for specific Bell functionals (like CHSH or one-sided tilted CHSH) does not extend to the complete characterization of the quantum set in the minimal scenario. The obstruction is visible arbitrarily close to a local deterministic behavior where the quantum state becomes a product state and measurements become compatible.

The authors emphasize that this result is a uniform statement about the family of symmetric doubly tilted CHSH functionals. It does not preclude certificates where the required NPA level increases as $\alpha \to 1$ , but it proves that no single fixed finite level suffices to describe the complete quantum set. This distinguishes the symmetric branch from the one-sided tilted axes, where Level $1+AB$ remains exact despite the state approaching a product state, highlighting that weak entanglement and small violations alone do not necessitate higher NPA levels unless the specific symmetry of the problem (compatibility of both measurement pairs) is present.

The work relies entirely on analytical proofs involving the Motzkin polynomial and compactness arguments for fixed-level sum-of-squares certificates, without invoking the undecidability results of $MIP^*=RE$ or specific nonlocal games, focusing instead on the elementary $(2,2,2)$ Bell scenario.

The Infinite Zoom That Never Lands

The Motzkin Monster

The Ghost in the Machine

Why This Matters

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