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A Julia package turns partial commutations into faster, smaller SDP relaxations for quantum information problems.

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2026-07-14 15:13 UTC pith:2HXG4ZR3

load-bearing objection Solid, usable software paper: specialized partial-commutativity data structures that measurably shrink and speed up NPA-style SDPs for the QI problems that actually use them. the 2 major comments →

arxiv 2607.09339 v2 pith:2HXG4ZR3 submitted 2026-07-10 quant-ph math.OC

PCPOP.jl: A Julia package for partially commutative polynomial optimization

Mois\'es Bermejo Mor\'an , Abhishek Mishra This is my paper
classification quant-ph math.OC MSC 90C2213P1081P45 PACS 03.65.Ud03.67.-a
keywords partially commutative polynomialsnoncommutative polynomial optimizationsemidefinite relaxationsquantum informationBell inequalitiesstate polynomialstrace polynomialsJulia package
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.

PCPOP is a Julia package that builds and solves the standard hierarchies of semidefinite relaxations for polynomial optimization, covering ordinary non-commutative, tracial, trace and state polynomials. Its distinguishing mechanism is an arithmetic that works directly with partially commuting variables: words are stored by their projections onto maximal cliques of the dependence graph, so multiplication, involution, equality and cyclic equivalence become cheap operations that automatically absorb projector, unitary, unipotent and orthogonality relations common in quantum information. The same representation yields smaller moment matrices than generic Gröbner or substitution methods. The package also automates Wedderburn symmetry reduction, Jordan-algebra dimension reduction and exact-arithmetic Gröbner bases, and is illustrated on Bell inequalities, contextuality, networks, uncertainty relations and device-independent cryptography. Benchmarks against existing tools show both faster set-up and smaller SDPs on the problems that exploit partial commutativity.

Core claim

The central claim is that a clique-representation arithmetic for partially commutative monoids, together with built-in reductions for projectors, unitaries, unipotents and cyclic equivalence, produces valid and substantially smaller semidefinite relaxations for the non-commutative, tracial, trace and state polynomial optimization problems that arise in quantum information.

What carries the argument

Clique representation of partially commutative words: each word is stored as its projections onto the maximal cliques of the dependence graph; multiplication, involution, division and cyclic equivalence then reduce to ordinary non-commutative operations on those projections, automatically incorporating the algebraic relations that appear in Bell and contextuality scenarios.

Load-bearing premise

That the clique projections plus the built-in projector/unitary reductions correctly identify every pair of words that are equivalent under the given partial commutations, so the resulting moment matrix is still a valid relaxation of the original problem.

What would settle it

On any of the benchmark problems (n-cycle CHSH, bilocal Mermin, conditional entropy bounds) construct the same level of the hierarchy with a complete Gröbner basis or an independent package and check whether the optimal values differ beyond solver tolerance, or whether an intermediate product of two clique words fails to equal the product computed by free monoid reduction.

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

If this is right

  • Standard Bell, contextuality and network hierarchies can be built at higher levels or with more parties before the SDP becomes intractable.
  • Device-independent entropy and key-rate bounds that previously required custom substitution rules become one-line calls that exploit the native partial-commutativity arithmetic.
  • Exact-arithmetic pipelines (Gröbner + Wedderburn + rational SDP rounding) become routine for the same quantum-information problems.
  • Graph-product monoids give a compact algebraic language for multi-partite scenarios with overlapping measurement supports.

Where Pith is reading between the lines

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

  • The same clique arithmetic could be grafted onto existing sparsity or chordal-completion techniques to obtain hybrid reductions that neither package currently offers.
  • Because cyclic equivalence is decided in linear time, tracial and free-probability hierarchies become practical for alphabets that previously forced exponential Gröbner bases.
  • If the companion theory paper’s correctness proof holds, any future quantum-information SDP that can be written with partial commutations and projector relations can safely drop its hand-crafted substitution tables.

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

2 major / 7 minor

Summary. The manuscript presents PCPOP.jl, a Julia package for building and solving moment/SOS semidefinite relaxations of non-commutative, tracial, trace, and state polynomial optimization problems. Its distinguishing contribution is a specialized partially commutative monoid implementation (clique/graph-product representations with built-in projector, unitary, unipotent, and orthogonality reductions, plus a linear-time cyclic-equivalence test) aimed at quantum-information problems. The paper supplies mathematical background (§1), feature and implementation descriptions (§2–3), a tutorial with runnable examples (§4), a broad applications chapter (Bell scenarios, contextuality, conditional entropies, networks, uncertainty relations, almost qudits, information capacity; §5), and benchmarks against Ncpol2sdpa, QuantumNPA, Moment, and OSCAR (§6).

Significance. If the reported normal forms are correct and the package is maintained, this is a useful, practical contribution for the quantum-information community: it unifies several NPA-style hierarchies in one Julia interface, adds exact-arithmetic and symmetry/Jordan reductions, and—most importantly—exploits partial commutativity in a way that yields smaller, faster SDPs on standard QI benchmarks (notably n-cycle contextuality and related scenarios in Tables 3–4). Strengths include reproducible tutorial code, recovery of known tight values (CHSH, KCBS, bilocal Mermin, etc.), side-by-side SDP-size and timing tables against independent packages, and an open feature comparison (§2.7). Correctness of the core clique arithmetic is appropriately deferred to the companion theory paper [40] and the cyclic algorithm of [34]; the present manuscript’s role is software design, interface, and empirical performance.

major comments (2)
  1. The central performance claim rests on the clique/graph-product normal forms (and built-in projector/unitary/unipotent reductions) producing valid, complete equivalence classes for the moment matrices (§2.1, §3.2). Correctness is deferred to [40] and [34]. The manuscript should state this dependency more explicitly near the main claim (abstract/§2.1) and list, in one place, the known tight values recovered in §5 as empirical validation that the resulting SDPs remain valid relaxations. Without that short, load-bearing clarification, a reader cannot assess the risk that a missed or spurious identification would invalidate a relaxation.
  2. Reproducibility of the software claim: the installation snippet (§4.1) uses Pkg.add("PCPOP"), but the manuscript does not give a version pin, repository URL/DOI, or commit corresponding to the §6 timings (20-core i7-12700, 64 GB). For a package paper whose main result is comparative performance (Tables 1–4), a fixed release identifier and a short note on solver versions (Mosek) and floating-point vs exact modes used in the tables are needed so that the reported speed-ups can be re-run.
minor comments (7)
  1. Several tutorial/application snippets leave variables undefined or unused (e.g. §5.4 prints val without assigning objective_value(model); §5.3 mixes npa_dual/model_new_obj with the pcpop API of §4.2). Align examples with the documented public API or mark internal helpers.
  2. §4.3 CHSH code uses a[2,0] b[2,0] with Unipotent and objective a[1]*b[1]+… while the displayed problem (25) indexes a0,a1,b0,b1; the indexing is consistent in 1-based Julia but the prose/equation labels should match the code arrays to avoid confusion.
  3. Table 1 caption says SDP size, #cons, and #vars are the same for all packages, yet the narrative in §6.2 notes implementation differences in how equivalence constraints are counted (PCPOP vs others). Clarify when counts are forced equal vs when PCPOP’s single-PSD formulation reduces #cons (as in Table 3).
  4. §2.7 compares features with NCTSSOS and SumOfSquares but §6 benchmarks only Ncpol2sdpa, QuantumNPA, Moment, and OSCAR. A one-sentence note that NCTSSOS sparsity was out of scope for this release would prevent a missing-baseline objection.
  5. Typos and polish: “Gr¨ obner” spacing throughout; “semidefinitze” (§4.10); “optimzation” (§5.1); “Paw/suppress lowski” and similar bibliography encoding artifacts; “Beno ˆ ıt” in acknowledgements. Clean UTF-8 and LaTeX accents.
  6. Figure 1–3 axes use Time (ns) with log scale; state the number of random trials and whether times include monoid build or only multiply+reduce, so the OSCAR vs PCPOP comparison is unambiguous.
  7. §1.6 lists three SDP encodings; default choices (primal=true, canonical matrix variables) are stated later in §4.2. Cross-reference defaults once in §1.6 for readers who skip the tutorial.

Circularity Check

1 steps flagged

No significant circularity: software-performance claims rest on independent third-party benchmarks; self-citation of companion theory paper [40] is legitimate background, not a definitional loop.

specific steps
  1. self citation load bearing [Abstract; §2.1; §3.2; reference [40]]
    "As a distinguished feature, PCPOP implements a specialized framework for polynomial computations in partially commutative variables that provides significant computational advantages for problems appearing in quantum information. ... PCPOPimplements a specialized framework for partially commutative polynomial optimization [40] ... The clique representation is specially suitable for our computational implementation ..."

    Correctness of the clique projections, multiplication/division algorithms, and internal projector/unitary reductions is not proved in the present manuscript; it is imported wholesale from the authors’ companion theory paper [40]. This is a mild self-citation dependency for the algebraic validity of the resulting moment matrices. It is not circular for the performance claims, which are independently measured against third-party solvers on standard problems.

full rationale

PCPOP.jl is a package paper whose central claim is empirical computational advantage of clique-representation arithmetic (plus built-in projector/unitary/unipotent reductions and the cyclic test of [34]) on quantum-information problems. Correctness of the normal forms is deferred to the authors’ companion theory paper [40] and is not re-derived here; that is ordinary software-engineering practice, not circular construction of a result. The reported speed-ups and smaller SDPs are measured against independent packages (Ncpol2sdpa, QuantumNPA, Moment) on publicly stated problems (CHSH, n-cycle contextuality, conditional entropy, etc.) that recover known tight values. There are no fitted parameters renamed as predictions, no uniqueness theorems imported solely from the authors to forbid alternatives, and no self-definitional identities. The single self-citation of [40] is load-bearing only for the algebraic correctness premise, which is externally falsifiable by the package’s exact-arithmetic tutorials and recovery of standard bounds; it does not force the performance numbers. Hence score 1 (minor self-citation that is not load-bearing for the headline claim).

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The work is a software realization of existing mathematical hierarchies plus a specialized representation developed in the companion paper. No free parameters are fitted; the axioms are the standard convergence results for non-commutative moment-SOS hierarchies and the algebraic theory of free partially commutative monoids. The only new entities are the concrete data structures and algorithms that implement those theories.

axioms (3)
  • standard math Under mild boundedness assumptions the NPA / tracial / state / trace moment-SOS hierarchies converge to the true optimum of the corresponding polynomial optimization problem.
    Invoked throughout §1.2–1.5; classical results of Pironio–Navascués–Acín, Burgdorf–Klep–Povh, Klep–Magron–Volčič et al.
  • standard math Clique projections furnish a canonical form for free partially commutative monoids, and the linear-time algorithm of Liu–Wrathall–Zeger decides cyclic equivalence.
    Used as the foundation of all arithmetic in §2.1 and §3.2; cited from the free partially commutative monoid literature and [34].
  • domain assumption The specialized reductions for projectors, unitaries, unipotents and orthogonality preserve the ideal membership and therefore yield valid SDP relaxations.
    Assumed throughout the implementation (§3.2) and justified by reference to the companion theory paper [40].
invented entities (1)
  • GraphProductMonoid / PCMonomial clique-representation data structures with edge_l / edge_r fields no independent evidence
    purpose: Efficient storage and multiplication of partially commutative words that automatically enforce commutation and algebraic constraints.
    Concrete engineering realization of the abstract clique representation; no independent physical existence outside the package.

pith-pipeline@v1.1.0-grok45 · 38682 in / 2302 out tokens · 27411 ms · 2026-07-14T15:13:37.662531+00:00 · methodology

0 comments
read the original abstract

Here we present PCPOP, a Julia package for polynomial optimization that supports non-commutative optimization, tracial polynomial optimization, trace polynomial optimization and state polynomial optimization. PCPOP fully supports exact arithmetic computations and incorporates convenient functionalities such as algebraic reductions based on Gr\"obner basis methods, automatized symmetrization via Wedderburn decompositions, and Jordan algebra reductions. As a distinguished feature, PCPOP implements a specialized framework for polynomial computations in partially commutative variables that provides significant computational advantages for problems appearing in quantum information.

Figures

Figures reproduced from arXiv: 2607.09339 by Abhishek Mishra, Mois\'es Bermejo Mor\'an.

Figure 1
Figure 1. Figure 1: Benchmarking building time of the partially commutative monoid in PCPOP (black) and the Gr¨obner basis in OSCAR (blue) truncated to degree d for the scenarios with n parties in Example (3). ations of a given problem and the size of the resulting relaxations, which automat￾ically reflects in the cost of solving the semidefinite program. All the results displayed in this chapter are obtained with 20 cores 64… view at source ↗
Figure 2
Figure 2. Figure 2: Benchmarking computing canonical forms with respect to the commutation relations in Example 3 in PCPOP (black), OSCAR (blue) and QuantumNPA (red). The figure shows the time in nanoseconds of computing the canonical form of wv for random words w and v of length d in 3n = 6 variables (a,b, c). 5 10 15 20 103 104 d Time (ns) PCPOP OSCAR QuantumNPA [PITH_FULL_IMAGE:figures/full_fig_p043_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Benchmarking computing canonical forms with respect to the commuta￾tion relations and projection constraints in Example 3 in PCPOP (black), OSCAR (blue) andQuantumNPA (red). The figure shows the time in nanoseconds of computing the canon￾ical form of wv for random words w and v of length d in 3n = 6 variables (a,b, c). 6.2 Polynomial optimization We benchmark the performance of PCPOP against the polynomial… view at source ↗

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