The Collapse of Unentangled Stoquastic Merlin-Arthur Proof Systems

TL;DR

This paper proves that unentangled stoquastic Merlin-Arthur proof systems have no more power than their entangled counterparts, showing a collapse in their computational complexity class.

Contribution

It introduces a positive de Finetti theorem for nonnegative matrices and demonstrates that entanglement does not enhance the power of stoquastic verification.

Findings

Unentangled and entangled stoquastic proof systems are equivalent in power.

The positive de Finetti theorem approximates nonnegative matrix values via spectral relaxation.

The proof establishes that StoqMa equals StoqMa(k) and is contained in PSPACE.

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.