💻 computer science

Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone

An exactly certified counterexample shows that the stationary quantum bootstrap is not tight in two dimensions: a candidate satisfying every stationary constraint, with a strictly positive moment matrix, assigns a negative value to a polynomial that is non-negative everywhere. Adding the eigenstate constraints removes the violation in every setting tested, apparently by bounding the momentum directions the stationary relaxation leaves free. A related gap in the underlying matrix cones first opens at five variables.

Original authors: Daniel Keren

Published 2026-09-10✓ Author reviewed ⓘ

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Daniel Keren

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum systems are described by states, and one way to study them is the bootstrap: instead of solving the equations of motion, you propose a candidate description and test whether it obeys a short list of consistency rules. The most important rule is a positivity requirement. If a physical quantity can never be negative, then its average in any genuine state cannot be negative either. The practical version of this test asks whether the candidate assigns a non-negative value to every quantity that can be written as a sum of squares, because that is a condition a computer can check efficiently. For a single particle moving on a line, this is enough: every non-negative polynomial in one variable is a sum of squares, so the two notions coincide. In two or more dimensions they come apart. There exist polynomials that are non-negative everywhere but cannot be written as a sum of squares, and a candidate state that passes the computable test might still assign one of them a negative value. Whether that abstract gap is ever reached by an actual quantum problem had not been settled. This paper settles it, and the answer depends on which version of the bootstrap is being used. The weaker version, called stationary here, imposes only that the state does not change in time. It applies to thermal states and to mixtures, and it is the version used when the object of study is an equilibrium state rather than a single energy level. The stricter version adds a condition that holds only for states of definite energy. For a particle in two dimensions moving in a quartic potential with four symmetric wells, the author constructs a candidate that satisfies every stationary requirement exactly, in rational arithmetic with no rounding, and whose associated moment matrix is strictly positive. Applied to a particular polynomial that is non-negative across the whole plane but is not a sum of squares, that candidate returns a negative value. No quantum state can behave this way, so the candidate is spurious and the stationary bootstrap is not tight. Because the moment matrix records momentum as well as position, the certificate rules out a sum-of-squares representation in the full operator algebra, not merely in the commuting variables. The stricter eigenstate version behaves differently. Repeating the same search with the energy conditions imposed produced no spurious candidate, across two, three and five degrees of freedom, at two truncation levels, and for several potentials chosen specifically to align with known non-sum-of-squares obstructions. The one exception was a case where the truncation was so coarse that the energy conditions had nothing left to say, which reduces it to the weaker version. The reason is identifiable rather than merely observed. Under the stationary conditions alone, the high-order momentum averages of a candidate can be made arbitrarily large while the position averages stay ordinary. The spurious candidate exploits exactly this: its momentum averages are inflated by orders of magnitude. The energy conditions couple momentum to position and bound those averages, and the paper proves such a bound explicitly in one case. How far the bound reaches depends on the truncation level and the degree of the potential, and when the potential is too rich for the truncation the mechanism has no room to act. Dimension matters for a concrete reason. Under sign-flip symmetry, the question of which candidates can be excluded turns into a question about matrices that are non-negative on the positive orthant. For up to four variables these coincide with the well-behaved matrices the bootstrap can certify; from five variables onward they do not. The paper's five-dimensional example is a ring of coupled sites with cyclic symmetry, and the cyclic structure is essential rather than decorative: averaging over all permutations of the sites instead would collapse the relevant obstruction to something the bootstrap certifies without difficulty. The negative result is a proof. The positive one is not. The searches supporting tightness of the eigenstate bootstrap use a local method, so they establish that no separating polynomial was found in the families examined, not that none exists. The practical recommendation for practitioners is small and cheap: when a bootstrap calculation in two or more dimensions reports something surprising, test the candidate against a known non-negative polynomial that is not a sum of squares. The test costs one linear program.

Technical Summary: Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone

Problem Statement
The paper investigates the "tightness" of the numerical bootstrap method for quantum mechanics. The bootstrap approach determines spectra by testing the positivity of a candidate state against a set of linear constraints derived from the Hamiltonian $H$ and operator algebra, without solving differential equations. A central subtlety in this truncated moment problem is the distinction between two positivity conditions:

Sum of Squares (SOS): Requiring the linear functional $L(A^\dagger A) \geq 0$ for all operators $A$ in a polynomial span.
Pointwise Nonnegativity: Requiring $L(f) \geq 0$ for every polynomial $f$ that is nonnegative everywhere on $\mathbb{R}^d$ .

While these two conditions coincide in one dimension (due to Hilbert's result that every nonnegative univariate polynomial is an SOS), they diverge in two or more dimensions. The paper asks whether this mathematical gap between the SOS cone and the nonnegative cone is ever realized by a physical quantum system (i.e., whether a "spurious" moment vector can satisfy all bootstrap constraints yet fail to correspond to any physical state).

The author distinguishes between two types of relaxations:

Stationary Bootstrap: Imposes $\langle [H, O] \rangle = 0$ (valid for thermal and mixed states).
Eigenstate Bootstrap: Additionally imposes $\langle OH \rangle = E \langle O \rangle$ (valid only for energy eigenstates).

Methodology
The author employs exact rational arithmetic to avoid numerical rounding errors, ensuring all claims are rigorously verified.

Construction of Counterexamples: They formulate the bootstrap constraints as a Semidefinite Program (SDP). For the stationary case, they search for a moment vector that satisfies all constraints up to a specific truncation level but yields a negative expectation value for a polynomial that is globally nonnegative.
Separating Witnesses: To prove a moment vector is unphysical, they construct a "separating witness"—a polynomial $p \geq 0$ on $\mathbb{R}^d$ such that $L(p) < 0$ .
Symmetry Reduction: The analysis utilizes potentials invariant under sign flips ( $q_j \to -q_j$ ) and coordinate permutations. This reduces the feasible set and the search space for witnesses without losing generality regarding the existence of spurious points.
Copositive Cone Analysis: In higher dimensions, the author relates the problem to the copositive cone (matrices $W$ where $x^T W x \geq 0$ for all $x \geq 0$ ). They utilize the known gap between the cone of positive semidefinite matrices plus nonnegative matrices ( $P+N$ ) and the copositive cone, which first appears at dimension $d=5$ .

Key Results

1. The Stationary Bootstrap is Not Tight (Theorem 1)
For a two-dimensional quartic double-well potential ( $d=2$ ) at truncation level $K=3$ , the author exhibits a moment vector $m$ with rational entries that:

Satisfies all stationary constraints $\langle [H, O] \rangle = 0$ exactly.
Has a positive definite moment matrix $M_3$ .
Assigns a negative expectation value ( $L_m(p_c) \approx -0.086$ ) to a specific polynomial $p_c$ that is nonnegative on all of $\mathbb{R}^2$ .
Consequently, $m$ is not the moment sequence of any quantum state (pure or mixed).
Crucial Detail: The polynomial $p_c$ is not a sum of squares. If it were, the positive definiteness of $M_3$ would have guaranteed $L_m(p_c) \geq 0$ . The failure occurs because the SOS cone is strictly smaller than the nonnegative cone in $d \geq 2$ .

2. The Eigenstate Bootstrap Appears Tight (Systematic Search)
When the eigenstate constraints $\langle OH \rangle = E \langle O \rangle$ are added, the violation disappears. The author conducted a systematic search across:

Dimensions $d=2, 3, 5$ .
Truncation levels $K=3, 4$ .
Various potentials (including those designed to exploit known non-SOS forms like Robinson's form and Horn's matrix).
In all tested settings where the Hamiltonian is bounded below with a discrete spectrum, no separating witness was found. The only exception was a case at $K=2$ where the degree budget was so low that the eigenstate constraints effectively vanished, leaving only the energy normalization.

3. The Mechanism: Bounded Momentum Moments
The paper identifies the physical mechanism preventing the gap in the eigenstate case.

In the stationary relaxation, high-order momentum moments (e.g., $\langle p^4 \rangle, \langle p^6 \rangle$ ) are unbounded on the feasible set. The spurious point found in Theorem 1 exploits this by having inflated momentum moments that allow it to satisfy position-based constraints while violating nonnegativity.
The eigenstate constraints $\langle OH \rangle = E \langle O \rangle$ effectively bound these high-order momentum moments.
Proposition 1: For a quartic potential in $d=2$ at $K=3$ , the eigenstate constraints imply an explicit bound on $\langle p^4 \rangle$ in terms of the energy $E$ and the potential's minimum. This bound prevents the moment vector from escaping into the region between the SOS and nonnegative cones.

4. Copositivity and Dimensionality
The author analyzes why violations might appear in higher dimensions.

At $d=5$ , the gap between the SOS cone and the nonnegative cone is linked to the gap between the cone $P+N$ and the copositive cone (Horn's matrix).
At $K=2$ , the bootstrap certifies membership in $P+N$ , allowing a violation if the witness lies in the copositive cone but outside $P+N$ .
At $K=3$ , the bootstrap certifies membership in a higher level of Parrilo's hierarchy ( $K_1$ ). The author shows that Horn's matrix lies inside $K_1$ , explaining why no violation was found at $K=3$ for symmetric potentials in $d=5$ .

Significance and Claims
The paper makes the following claims regarding its contribution:

Resolution of an Open Question: It provides the first rigorous, exact-arithmetic proof that the stationary bootstrap relaxation is not tight in dimensions $d \geq 2$ . It demonstrates that satisfying the stationary constraints and positivity of the moment matrix is insufficient to guarantee a physical state.
Distinction of Relaxations: It highlights that the eigenstate constraints are not merely a refinement but fundamentally alter the geometry of the feasible set by bounding momentum moments, thereby closing the gap between the SOS and nonnegative cones in the tested regimes.
Limitations of Current Theory: The results place the problem outside the scope of existing convergence theories for noncommutative polynomial optimization, which rely on Archimedean conditions (bounded operators) that the Weyl algebra ( $[q, p] = i$ ) does not satisfy.
Conjecture: The author conjectures that the eigenstate constraints imply an Archimedean-type bound on momentum moments ( $\langle p^{2k} \rangle \leq C(E, k)$ ), which would ensure the tightness of the eigenstate bootstrap for sufficiently high truncation levels.
Practical Diagnostic: They propose a diagnostic test for future bootstrap studies: whenever a violation is reported in $d \geq 2$ , one should check if the candidate moment vector yields a negative expectation for a known non-SOS nonnegative polynomial.

The author remains modest about the eigenstate results, characterizing them as "systematic numerical evidence" rather than a proof of tightness, acknowledging that the alternating search method used is local. However, they argue that the identified mechanism (boundedness of momentum moments) provides a strong theoretical basis for the observed tightness.

Technical Summary: Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone

More like this