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REVIEW 1 major objections 1 minor 21 references

Higher local dimensions up to 8 yield no improvement over the d=5 winning probability of approximately 0.6218 in the Guess-Your-Neighbour's-Input game.

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2026-06-26 16:36 UTC pith:VALYZHP7

load-bearing objection The GPU SCS implementation is a usable engineering step for process-matrix SDPs, but the claim that d=6-8 brings no gain over d=5 rests on see-saw runs whose closeness to the global optimum is not shown. the 1 major comments →

arxiv 2606.20519 v1 pith:VALYZHP7 submitted 2026-06-18 quant-ph math.OC

GPU-accelerated semidefinite programming for causal games

Emanuel-Cristian Boghiu , Kyrylo Simonov This is my paper
classification quant-ph math.OC
keywords causal gamesprocess matrixsemidefinite programmingGuess-Your-Neighbour's-Input gamelocal dimensionquantum correlationsGPU acceleration
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tests whether increasing the local dimension in process-matrix strategies can raise the winning probability in the Guess-Your-Neighbour's-Input causal game above the known value of about 0.6218 at dimension 5. An upper bound of 0.7592 exists for any dimension, so the authors run a see-saw optimization procedure formulated as semidefinite programs to check if higher dimensions close the gap. They implement a custom solver with GPU acceleration for the positive-semidefinite projection step, achieving a six-fold speedup that allows exploration up to dimension 8. No meaningful gains appear, which matters because it either points to the need for qualitatively different strategies or indicates the upper bound may not be tight. A reader would care as the result constrains the possible performance of quantum correlations without definite causal order.

Core claim

A custom GPU-accelerated implementation of the SCS solver is used to run see-saw optimization on the Guess-Your-Neighbour's-Input game in the process-matrix formalism. At each step the optimization is cast as a semidefinite program whose dominant cost, projection onto the positive-semidefinite cone, is offloaded to the GPU. Exploration of local dimensions up to d=8 produces no significant improvement over the value of approximately 0.6218 attained at d=5. The authors conclude that either qualitatively different strategies are required to approach the known upper bound of 0.7592 or that the bound itself is not tight.

What carries the argument

The see-saw optimization scheme in which each alternation is formulated as a semidefinite program whose positive-semidefinite projection is computed on a GPU.

Load-bearing premise

The see-saw optimization procedure, when run at dimensions 6-8, reaches a value sufficiently close to the global optimum that the absence of improvement can be taken as evidence that higher dimension does not help.

What would settle it

A concrete process-matrix strategy at dimension 9 or higher that achieves a winning probability noticeably above 0.63 in the Guess-Your-Neighbour's-Input game would falsify the reported plateau.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The known process-matrix value of approximately 0.6218 for the Guess-Your-Neighbour's-Input game remains unchanged even when local dimension is raised to 8.
  • Dimension 5 is already sufficient to reach the performance level currently attainable by see-saw methods.
  • Closing the gap to the dimension-independent upper bound of 0.7592 requires either new families of strategies or a tighter bound.
  • The computational advantage of the GPU implementation makes exhaustive checks at still higher dimensions feasible in principle.

Where Pith is reading between the lines

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

  • If the plateau persists under alternative optimization methods, the 0.7592 bound may itself be loose.
  • The result suggests that scaling dimension alone is not the route to better indefinite-causal-order correlations in this game.
  • Similar GPU-accelerated see-saw loops could be applied to other causal games to test whether dimension saturation is generic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a custom GPU-accelerated implementation of the SCS solver, offloading the positive-semidefinite projection to achieve a reported six-fold speedup. It applies this to a see-saw SDP procedure for maximizing the winning probability in the Guess-Your-Neighbour's-Input causal game under the process-matrix formalism. The authors explore local dimensions up to d=8 and report no significant improvement over the previously known d=5 value of approximately 0.6218 (with the dimension-independent upper bound at 0.7592), concluding that either qualitatively different strategies or a tighter bound may be required.

Significance. The GPU acceleration constitutes a practical engineering contribution that enables larger-scale SDP optimizations in quantum causal inference. If the numerical results hold, the negative finding at d=6–8 is a useful constraint on the search space for process-matrix strategies in this game. The work also supplies an explicit, reproducible implementation path for similar see-saw problems.

major comments (1)
  1. [Abstract] Abstract (and the numerical-results section): the headline claim that 'we do not find significant improvements' for d=6,7,8 is load-bearing for the paper's interpretation, yet the manuscript supplies no information on the number of random initializations, convergence tolerances, achieved objective values at each dimension, or any cross-validation against the known d=5 optimum. Because see-saw is a local method whose feasible-set dimension scales as O(d^6), the absence of these diagnostics leaves open the possibility that the reported values are suboptimal fixed points rather than evidence that higher dimension is unhelpful.
minor comments (1)
  1. The abstract states that 'no significant improvement was observed' without quoting the actual optimized values or any error estimates at d=6–8; adding these numbers (even if only in a table) would make the claim easier to assess.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for identifying the need for additional numerical diagnostics. We address the single major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the numerical-results section): the headline claim that 'we do not find significant improvements' for d=6,7,8 is load-bearing for the paper's interpretation, yet the manuscript supplies no information on the number of random initializations, convergence tolerances, achieved objective values at each dimension, or any cross-validation against the known d=5 optimum. Because see-saw is a local method whose feasible-set dimension scales as O(d^6), the absence of these diagnostics leaves open the possibility that the reported values are suboptimal fixed points rather than evidence that higher dimension is unhelpful.

    Authors: We agree that the current manuscript does not report the number of random initializations, convergence tolerances, per-dimension objective values, or explicit cross-validation against the d=5 optimum. We will revise the numerical-results section (and update the abstract if needed) to include these diagnostics. The see-saw procedure is indeed local and the feasible-set dimension grows rapidly, so documenting multiple random starts and recovery of the known d=5 value is necessary to support the interpretation that higher dimensions yield no significant improvement. We will also note the O(d^6) scaling and the practical limits this imposes. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical see-saw SDP against external bound

full rationale

The paper reports direct numerical optimization of process-matrix strategies for the GYNI game via see-saw SDP, using a custom GPU implementation of the SCS solver. The central claim (no significant gain for d=6–8 over the d=5 value) is obtained by running the optimizer and comparing the attained values to a dimension-independent upper bound taken from prior literature. No quantity is defined in terms of itself, no fitted parameter is renamed as a prediction, and no load-bearing step reduces to a self-citation chain. The work is self-contained computational exploration; the absence of a global-optimality certificate is a limitation on the strength of evidence, not a circularity in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard process-matrix SDP formulation and the convergence properties of see-saw iteration; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption See-saw iteration converges to a value close enough to the global optimum that absence of improvement at higher dimension can be interpreted as a genuine negative result.
    Invoked when the authors conclude that higher dimension does not help.

pith-pipeline@v0.9.1-grok · 5773 in / 1316 out tokens · 26031 ms · 2026-06-26T16:36:33.142336+00:00 · methodology

0 comments
read the original abstract

The process matrix formalism describes quantum correlations in scenarios without a fixed causal order between local laboratories. Operational signatures of such correlations can be investigated through causal games. A paradigmatic example is the Guess-Your-Neighbour's-Input game, in which two parties attempt to guess each other's inputs. Correlations compatible with any definite, or probabilistically mixed, causal order cannot achieve a winning probability exceeding $1/2$. The best process-matrix strategy currently known attains a value of approximately $0.6218$ using local dimension $d=5$, while the strongest known dimension-independent upper bound is $0.7592$. In this work, we investigate whether increasing the local dimension beyond $d = 5$ can narrow this gap. To this end, we employ a see-saw optimization scheme in which each step is formulated as a semidefinite program. For scalability, we develop a custom implementation of the SCS solver in which the dominant computational cost, the projection onto the positive-semidefinite cone, is offloaded to a GPU, yielding a six-fold speedup. Using this implementation, we explore local dimensions up to $d = 8$, and we do not find significant improvements over the value at $d=5$. Our results suggest that either qualitatively different strategies are required to approach the known upper bound, or that the bound itself is not tight.

Figures

Figures reproduced from arXiv: 2606.20519 by Emanuel-Cristian Boghiu, Kyrylo Simonov.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the bipartite scenario [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Residual evolution for a representative process-matrix [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

discussion (0)

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Reference graph

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