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arxiv: 2605.16249 · v1 · pith:27TZK7X3new · submitted 2026-05-15 · 🪐 quant-ph · cs.CC

The Collapse of Unentangled Stoquastic Merlin-Arthur Proof Systems

William Gay , Fernando Granha Jeronimo This is my paper

Pith reviewed 2026-05-20 19:01 UTC · model grok-4.3

classification 🪐 quant-ph cs.CC
keywords stoquastic Merlin-Arthurde Finetti theoremunentanglementquantum proof systemssymmetric extensionsnonnegative tensors
2
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The pith

Stoquastic Merlin-Arthur verification with any polynomial number of unentangled provers equals the single-prover case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that stoquastic Merlin-Arthur proof systems with a polynomial number of unentangled provers have exactly the same power as the single-prover version. Stoquasticity rules out destructive interference, so the requirement that witnesses are product states can be folded into a polynomially larger single witness without losing soundness or completeness. The argument rests on a positive value-based de Finetti theorem that replaces the search over product inputs with the largest eigenvalue of a symmetrized and projected operator whose size grows polynomially with the number of provers and the inverse error. This places the resulting class inside AM intersect PP and therefore inside PSPACE.

Core claim

For every polynomial k the classes satisfy StoqMa(k) equals StoqMa. The proof proceeds by showing that when M is an entrywise nonnegative positive semidefinite contraction on the tensor product of k spaces, its nonnegative product value is approximated to additive error epsilon by the largest eigenvalue of the operator obtained by replacing the first k-1 factors with R symmetric copies, tensoring with the last factor, and projecting onto the product of symmetric subspaces, where R is O(k squared times the sum of log dimensions over epsilon cubed). The spectral quantity is then realized directly by a one-witness stoquastic verifier after approximating the symmetric projectors by dyadic invers

What carries the argument

A positive value-based de Finetti theorem for separately symmetric extensions that approximates the nonnegative product value of an entrywise nonnegative PSD contraction by the largest eigenvalue of a projected symmetrized operator on polynomially many copies.

If this is right

  • StoqMa with any polynomial number of unentangled provers reduces to ordinary single-prover StoqMa.
  • StoqMa is contained in AM and in PP.
  • StoqMa is contained in PSPACE.
  • The de Finetti approximation technique applies directly to other nonnegative tensor optimization problems under stoquastic constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetrized approximation may yield new upper bounds for classical nonnegative tensor optimization without quantum states.
  • The technique could be tested on small explicit stoquastic Hamiltonians to see whether the polynomial blow-up in witness size remains practical.
  • If the error dependence on k and dimension can be improved, the result would extend to super-polynomial numbers of provers.

Load-bearing premise

The nonnegative product value of an entrywise nonnegative positive semidefinite contraction can be approximated to small additive error by the largest eigenvalue of a symmetrized and projected version of the operator built from polynomially many copies of the input spaces.

What would settle it

An explicit small-dimensional entrywise nonnegative PSD contraction M for which the difference between its true nonnegative product value and the largest eigenvalue of the claimed symmetrized projected operator exceeds the stated additive error bound.

Figures

Figures reproduced from arXiv: 2605.16249 by Fernando Granha Jeronimo, William Gay.

Figure 1
Figure 1. Figure 1: Complexity landscape around the result. The highlighted equality is proved here for inverse [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Entanglement and interference are among the most fundamental properties of quantum mechanics. In this work, we investigate the role and power of interference in the context of detecting entanglement. We do so from a computational complexity lens by proving that unentanglement gives no additional power to stoquastic Merlin-Arthur verification. For every polynomial number of provers $k=k(n)$, \[ \text{StoqMa}(k)=\text{StoqMa} . \] Conceptually, the proof separates the role of entanglement from the role of interference: once destructive interference is ruled out by stoquasticity, the product-state constraint can be absorbed into a polynomially larger one-witness stoquastic verification. The main analytic ingredient is a positive, value-based de Finetti theorem for separately symmetric extensions. If $M$ is an entrywise nonnegative positive semidefinite contraction on $A_1\otimes\cdots\otimes A_k$, then the nonnegative product value of $M$ is approximated to additive error $\epsilon$ by the largest eigenvalue of \[ \Pi_R^{<k} (M_{A_{1,1}\cdots A_{k-1,1}A_k}\otimes I) \Pi_R^{<k}, \qquad R=O\!\left(\frac{k^2\sum_i\log\dim A_i}{\epsilon^3}\right), \] where $\Pi_R^{<k}$ is the operator on $A_1^{\otimes R} \otimes \cdots \otimes A_{k-1}^{\otimes R} \otimes A_k$ projecting to the subspace $\mathrm{Sym}^R(A_1) \otimes \cdots \otimes \mathrm{Sym}^{R}(A_{k-1}) \otimes A_k$. The spectral relaxation is then realized as an actual one-witness stoquastic verifier. After replacing the uniform permutation averages in the symmetric projectors by inverse-polynomially close dyadic inverse-invariant averages. Consequently, \[ \text{StoqMa}(k)=\text{StoqMa}\subseteq\text{AM}\cap\text{PP}\subseteq\text{PSPACE} . \] The positive de Finetti theorem is isolated as a standalone technique and may be useful in other nonnegative tensor-optimization and stoquastic-verification settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that StoqMA(k) = StoqMA for every polynomial number of provers k. It does so by establishing a positive value-based de Finetti theorem for separately symmetric extensions: for an entrywise nonnegative PSD contraction M on A1 ⊗ ⋯ ⊗ Ak, the nonnegative product value is approximated to additive error ε by λ_max of Π_R^{<k} (M_{A_{1,1}⋯A_{k-1,1}A_k} ⊗ I) Π_R^{<k} with R = O(k² ∑_i log dim A_i / ε³). This spectral quantity is then realized as the acceptance probability of a single-witness stoquastic verifier after replacing uniform permutation averages in the symmetric projectors by inverse-polynomially close dyadic inverse-invariant averages, yielding the collapse and the inclusion StoqMA ⊆ AM ∩ PP ⊆ PSPACE.

Significance. If the result holds, it shows that once destructive interference is ruled out by stoquasticity, the product-state constraint from unentanglement can be absorbed into a polynomially larger one-witness stoquastic verification. This separates the roles of entanglement and interference in quantum verification. The explicit polynomial bounds in the de Finetti theorem and its isolation as a standalone technique for nonnegative tensor optimization are strengths; the paper also supplies a concrete reduction path to classical classes.

major comments (1)
  1. [Abstract and realization section] Abstract and the section realizing the spectral relaxation as a stoquastic verifier: the manuscript asserts that replacing the uniform permutation averages inside Π_R^{<k} by inverse-polynomially close dyadic inverse-invariant averages yields an operator whose largest eigenvalue is the acceptance probability of a one-witness stoquastic verifier. Uniform averages preserve entrywise nonnegativity because M is entrywise nonnegative and permutations act by basis permutation. No argument is supplied, however, that the dyadic construction preserves entrywise nonnegativity. If any matrix element becomes negative, the constructed verifier is no longer stoquastic and the reduction StoqMA(k) → StoqMA fails. This step is load-bearing for the central equality.
minor comments (2)
  1. [Notation and de Finetti theorem statement] The notation M_{A_{1,1}⋯A_{k-1,1}A_k} in the projected operator could be defined more explicitly, including the precise identification of the tensor factors and the action of the partial trace or embedding.
  2. [Introduction] A short comparison paragraph relating the new positive de Finetti theorem to existing quantum de Finetti results (e.g., those based on symmetric extensions or information-theoretic bounds) would help readers assess novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying the need for an explicit argument regarding entrywise nonnegativity preservation under the dyadic approximation. We address the comment below and will revise the manuscript to include the requested clarification.

read point-by-point responses

  1. Referee: [Abstract and realization section] Abstract and the section realizing the spectral relaxation as a stoquastic verifier: the manuscript asserts that replacing the uniform permutation averages inside Π_R^{<k} by inverse-polynomially close dyadic inverse-invariant averages yields an operator whose largest eigenvalue is the acceptance probability of a one-witness stoquastic verifier. Uniform averages preserve entrywise nonnegativity because M is entrywise nonnegative and permutations act by basis permutation. No argument is supplied, however, that the dyadic construction preserves entrywise nonnegativity. If any matrix element becomes negative, the constructed verifier is no longer stoquastic and the reduction StoqMA(k) → StoqMA fails. This step is load-bearing for the central equality.

    Authors: We agree that an explicit argument for this preservation property is necessary for the clarity of the reduction. The dyadic inverse-invariant averages are constructed explicitly as convex combinations of permutation operators with strictly positive dyadic-rational coefficients (of the form m/2^l with m, l positive integers). Because the original matrix M is entrywise nonnegative in the computational basis and each permutation operator merely reorders basis indices, the action of any such permutation on M yields a matrix that remains entrywise nonnegative. Any convex combination with positive weights therefore also remains entrywise nonnegative. We will add a short lemma (or a self-contained paragraph) immediately after the definition of the dyadic averages in the realization section, stating this fact and verifying that the resulting operator has nonnegative matrix elements, thereby ensuring the verifier remains stoquastic. This revision strengthens the presentation without changing any claims or bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation relies on new de Finetti theorem

full rationale

The paper establishes StoqMa(k)=StoqMa by first proving a positive value-based de Finetti theorem that approximates the nonnegative product value of an entrywise nonnegative PSD contraction M via the largest eigenvalue of a projected symmetric operator with R = O(k² ∑ log dim A_i / ε³). This theorem is presented as a standalone analytic ingredient with its own proof. The spectral quantity is then realized as a one-witness stoquastic verifier by replacing uniform permutation averages in Π_R^{<k} with inverse-polynomially close dyadic inverse-invariant averages. The chain uses newly derived results plus standard linear-algebra facts rather than reducing the target equality to any fitted parameter, self-defined quantity, or load-bearing self-citation by construction. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts about positive semidefinite operators and symmetric subspaces plus the newly introduced de Finetti theorem; no free parameters or invented physical entities appear.

axioms (2)
  • domain assumption Entrywise nonnegative positive semidefinite contractions admit symmetric extensions whose eigenvalues approximate product values
    Invoked as the main analytic ingredient in the abstract.
  • domain assumption Stoquasticity rules out destructive interference
    Used to separate entanglement from interference so that product-state constraints can be absorbed.

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Reference graph

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