One generating measurement does not make pseudo-broadcasting equivalent to non-contextuality

P. Jokinen, M. Weilenmann, M. Plávala, J.-P. Pellonpää, J. Kiukas and R. Uola, No-Broadcasting Characterizes Operational Contextuality, Phys. Rev. Lett. 133, 240201 (2024), arXiv:2406.07305v1, Appendix D, state:

One can also see that $\mathcal S(\mathcal A)$ is a Cartesian product of $k$ simplexes if $\mathcal A$ is generated by $k$ independent measurements. This leads to a conjecture that if $\mathcal A$ is generated by $k$ measurements, then $1\mapsto k$-pseudobroadcasting is equivalent to non-contextuality. This is only a conjecture since even if $\mathcal A$ is generated by $k$ measurements, $\mathcal S(\mathcal A)$ in general is only a subset of a Cartesian product of $k$ simplexes. We leave resolving this issue and proving or disproving the conjecture for future work.

This quotation is from Arxiv.tex, line 482, in the arXiv v1 source archive (534 lines; md5 cc2fa845c0b0f4155ed18dcb76e4c92e). The source macros \hS, \hA and \btn{k} are expanded to $\mathcal S$, $\mathcal A$ and $1\mapsto k$; the wording is unchanged. The main-text conjecture is on line 348. The definitions and further quotations are in the literature note.

Issue #21862 fixes the reading of generation: $\mathcal A$ is the real convex hull of the binary classical post-processings $\sum_o c_oM_j(o)$, with $0\le c_o\le1$, of $k$ finite POVMs. A subtheory has a convex set $\mathcal T$ of density matrices and a convex set $\mathcal A$ of effects, contains $I$ among its effects, and is closed under $E\mapsto I-E$. The conjectured equivalence is refuted by a qubit subtheory generated by one four-outcome POVM.

The source's Theorem 5 (Arxiv.tex, lines 336–338) identifies measurement and preparation non-contextuality with $1\mapsto n$ pseudo-broadcastability for every $n\in\mathbb N$. The conjecture would replace this entire hierarchy by its $k$-copy condition whenever $k$ measurements generate the effects. It would make feasibility at that single level a certificate of non-contextuality. The source's finite preparation and measurement tests impose linear conditions on the candidate pseudo-broadcasting map.

Issue #21862 records the bounded literature check:

• The preregistration's orchestrator reports that the arXiv listing contains only v1 and that the published University of Turku reprint, PhysRevLet-Pellonpää-2025-FINAL.pdf, retains the conjecture on pages 6 and 12. The v1 conjecture is present in the source archive identified above.
• The orchestrator reports no settlement from a web search for pseudo-broadcasting, contextuality and “generated by k measurements”, or from Zenodo. The related arXiv:2601.14067 text was reported to contain neither this conjecture nor pseudo-broadcasting in subtheories.
• The search contributor reports 8 OpenAlex citations and 10 Semantic Scholar citing works, with no settlement in the screened texts arXiv:2601.16952, 2512.04173, 2511.19266, 2509.17967, 2504.02944, 2503.05884, 2409.13024, 2408.01368, 2407.09624 and 2211.14351. MathDB and formal-conjectures searches were also reported to have no match.
• The preregistration records no earlier repository issue, module, Library note or dossier for arXiv:2406.07305.

The literature status is not-found-in-searched-scope.

Use outcome order $(x+,x-,z+,z-)$ and state order $(++,+-,-+,--)$. Define

$M_{x\pm}=\tfrac14\begin{pmatrix}1&\pm1\\pm1&1\end{pmatrix}$, $M_{z+}=\operatorname{diag}(\tfrac12,0)$ and $M_{z-}=\operatorname{diag}(0,\tfrac12)$,

and $\rho_j=v_jv_j^*/10$, where $v_0=(3,1)$, $v_1=(1,3)$, $v_2=(3,-1)$ and $v_3=(1,-3)$. Let $\mathcal T=\operatorname{conv}{\rho_j:j\in\mathrm{Fin},4}$ and $\mathcal A=\operatorname{conv}{\sum_o c_oM_o:0\le c_o\le1}$.

1. The subtheory and its generator. rho_density establishes that $\rho_j$ is a density matrix for every $j\in\mathrm{Fin},4$. measurement_povm establishes positivity of all four $M_o$ and $\sum_oM_o=I$. witness_subtheory establishes Subtheory for these fixed $\mathcal T,\mathcal A$; witness_generated establishes GeneratedBy with $k=1$. The post-processing set equals its convex hull: postprocessings_convex applies to every finite matrix family. For a family satisfying IsPOVM and an effect belonging to its post-processing set, postprocessings_positive gives positivity and postprocessings_complement gives membership of the complement.

2. One copy. For every natural dimension $d$ and sets $t,a$ satisfying Subtheory t a, one_copy gives PseudoBroadcastable t a 1. It uses the identity map with the tensor indices reindexed by $(\mathrm{Fin},1\to\mathrm{Fin},d)\simeq\mathrm{Fin},d$. The one-copy partial trace is the input matrix. Nonnegative product expectations follow from the positivity of the allowed density matrix and effect.

3. Relations forced on a model. For the fixed witness, witness_relations gives $\rho_0+\rho_3=\rho_1+\rho_2=I$ and $M_0+M_1=M_2+M_3=I/2$. Assume Noncontextual for these $\mathcal T,\mathcal A$, with any hidden-label type $\iota$ and matrix families $g_\lambda,s_\lambda$ satisfying all its positivity, normalization and HasSum reproduction conditions. Set $\mu_{j\lambda}=\operatorname{tr}(\rho_jg_\lambda)$ and $r_{o\lambda}=\operatorname{tr}(s_\lambda M_o)$. The four preparations belong to $\mathcal T$, and measurement_member places $M_o$ in $\mathcal A$ for every $o\in\mathrm{Fin},4$. Thus these weights and responses are real and nonnegative, $\mu_{0\lambda}+\mu_{3\lambda}=\mu_{1\lambda}+\mu_{2\lambda}$, and $r_{0\lambda}+r_{1\lambda}=r_{2\lambda}+r_{3\lambda}=1/2$. The sums over $\lambda$ of each $\mu_j$ equal one.

4. The separator. For real $u,v,w,z\ge0$ with $u+z=v+w$ and real $0\le x,y\le1/2$, corner_bound states

   $\big[u(x+y)+v(x+1/2-y)+w(1/2-x+y)+z(1-x-y)\big]/4\le3(u+z)/8$.

   In the proof of witness_contextual, substitute the real parts of $(\mu_0,\mu_1,\mu_2,\mu_3)$ for $(u,v,w,z)$ and of $(r_0,r_2)$ for $(x,y)$, for each hidden label. The conditions in item 3 supply every hypothesis. With $i_x=(0,0,1,1)$ and $i_z=(2,3,2,3)$, summing gives

   $S:=\tfrac14\sum_{j=0}^3\big[\operatorname{tr}(\rho_jM_{i_x(j)})+\operatorname{tr}(\rho_jM_{i_z(j)})\big]\le\tfrac38\sum_\lambda(\mu_{0\lambda}+\mu_{3\lambda})=\tfrac34$.

   born_sum gives $\operatorname{tr}(\rho_jM_{i_x(j)})+\operatorname{tr}(\rho_jM_{i_z(j)})=17/20$ for every $j\in\mathrm{Fin},4$. The proof uses convergent HasSum families and hasSum_le, so this contradiction also covers arbitrary summable hidden-label models. witness_contextual concludes $\neg\operatorname{Noncontextual}(\mathcal T,\mathcal A)$ for the fixed witness.

5. Conclusion. result applies the alleged universal equivalence at $d=2$, $k=1$, these $\mathcal T,\mathcal A$, witness_subtheory and witness_generated. Its pseudo-broadcasting side is supplied by one_copy witness_subtheory; its non-contextual side contradicts witness_contextual.

The refutation depends on matching the source's subtheory definitions and the registered meaning of generation:

• GeneratedBy uses arbitrary finite POVMs and all their binary classical post-processings, followed by real convex hull. It imposes no independence, distinctness or norm-one condition on the generating measurements.
• PseudoBroadcastable requires a complex linear map that preserves the trace of every input matrix, reproduces every allowed one-copy marginal probability, and gives nonnegative expectations for every product of allowed effects on every allowed preparation. It requires no complete positivity or positive semidefinite output matrix.
• Noncontextual allows any label type and summable families. The preparation weights are nonnegative and normalized on $\mathcal T$; each response matrix has trace one and nonnegative evaluations on $\mathcal A$; their weighted responses reproduce all allowed Born probabilities. Global positivity of $g_\lambda$ or $s_\lambda$ is not assumed. Excluding only finite-label models or only positive semidefinite response matrices would not suffice.
• claim quantifies over all natural $d,k$, including zero, but its proof is contradicted at $d=2,k=1$. Neither a zero-dimensional system nor the $k=0$ case supplies the counterexample.

Adding independence of the generators would change the conjecture being settled. The one-POVM witness has an operational relation among its outcomes, and that relation must also be respected by the response functionals.

The canonical source is D5/S3/QuantumContext/SingleMeasurementPseudobroadcastingRefutation.lean (305 lines).

• Definitions. It defines Subtheory, postprocessings, GeneratedBy, partialTraceKeep, PseudoBroadcastable, Noncontextual and claim.
• Proof. It proves result : ¬ claim.
• Axioms. It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.
• Escape registrations. result is the designated refutation result of the utility header and is exempt. The module has no other public theorem.

Frozen state: Golden/Frozen/state/D5/S3/QuantumContext/SingleMeasurementPseudobroadcastingRefutation.lean.json, statement sha256:06522447de852e967990efa31d12ab0adb20ba81cb41d8358079981618bd0c6e.

Declaration statements:

• result: sha256:5bca5902bddbd51c7c79153da9f854fda256b7428ec4c3a4d0e3d9453e1e35c3;
• claim: sha256:64fe4bb41ea0f534e5913fbb2d19b8f54041a5791f3e58fcd1176cc62f03f6c0.

Tier 1 conjecture of a 2024 journal letter, preregistered in issue #21862 before any Lean. theorem; resolution refuted.

The module is admitted under the open-problem-resolution basis of CLAUDE.md §3.2. The refutation is a certified instance with basis refutes. There is no digestion atom.

What the settlement shows

Proved by result: it is false that for every natural dimension $d$, natural $k$, and sets $\mathcal T,\mathcal A$ of complex $d\times d$ matrices, Subtheory and GeneratedBy with $k$ imply the equivalence between $k$-copy PseudoBroadcastable and Noncontextual.

Computed, not formalized (exact rational arithmetic):

• The Born probability table, in the state and outcome orders used above, has rows $(2/5,1/10,9/20,1/20)$, $(2/5,1/10,1/20,9/20)$, $(1/10,2/5,9/20,1/20)$ and $(1/10,2/5,1/20,9/20)$. Each selected score is $17/20$, exceeding $3/4$ by $1/10$.
• The four outcome matrices have real linear rank three. Their relation $M_0+M_1-M_2-M_3=0$ is verified exactly.
• The 16 corner evaluations with $u,v\in{0,1}$, $z=1-u$, $w=1-v$ and $x,y\in{0,1/2}$ have largest separator contribution $3/8$.
• For $a,b\in{\pm1}$, the four trace-one matrices $\sigma_{ab}=(I+aX+bZ)/2$ have determinant $-1/4$ and response tuples $((1+a)/4,(1-a)/4,(1+b)/4,(1-b)/4)$. All entries of these four tuples are nonnegative.

Argued, not formalized.

• Why the measurement count misses the factors. Here $X=\begin{pmatrix}0&1\1&0\end{pmatrix}$ and $Z=\operatorname{diag}(1,-1)$. The real span of the effects is $\operatorname{span}_{\mathbb R}{I,X,Z}$, since $I=\sum_oM_o$, $X=2(M_0-M_1)$ and $Z=2(M_2-M_3)$. These three matrices are linearly independent, so the displayed relation among the four $M_o$ is the only one. A normalized functional $\varphi$ positive on $\mathcal A$ therefore corresponds exactly to four numbers $q_o=\varphi(M_o)\ge0$ with $q_0+q_1=q_2+q_3=1/2$: these conditions respect the sole relation, give $\varphi(I)=1$, and ensure positivity on every nonnegative post-processing. Consequently $\mathcal S(\mathcal A)\simeq[0,1]^2$ under $(q_o)\mapsto(2q_0,2q_2)$. It is a square, the product of two segments, rather than a simplex. One four-outcome generator gives a constrained slice of its outcome simplex; it does not give one simplex factor. The relevant state space consists of all allowed positive functionals, not only quantum density matrices.

• The counterexample is not two-copy pseudo-broadcastable. Suppose a map $\Xi$ satisfied the module's two-copy conditions for these fixed $\mathcal T,\mathcal A$. For $a,b\in{\pm1}$ put $w_{ab}(\rho)=4\operatorname{tr}[\Xi(\rho)(M_{xa}\otimes M_{zb})]$. These are nonnegative, linear in $\rho$, and sum to one on $\mathcal T$, because each outcome pair sums to $I/2$ and $\Xi$ preserves trace. Use the trace-one response matrices $\sigma_{ab}=(I+aX+bZ)/2$. Their evaluations on $M_{xc}$ and $M_{zd}$ are respectively $\delta_{ac}/2$ and $\delta_{bd}/2$, hence are nonnegative on all of $\mathcal A$. Summing $w_{ab}\delta_{ac}/2$ over $a,b$ gives $\operatorname{tr}[\Xi(\rho)(M_{xc}\otimes I)]=\operatorname{tr}(\rho M_{xc})$; the other marginal similarly reproduces $M_{zd}$. Linearity extends reproduction to every allowed effect. Nondegeneracy of the finite-dimensional trace pairing represents each linear weight $w_{ab}$ as $\operatorname{tr}(\rho G_{ab})$. This is a four-label Noncontextual model for the fixed witness, contrary to witness_contextual. Thus the example separates one copy from two copies; it does not supply a contextual two-copy pseudo-broadcastable subtheory.

• A corrected statement using simplex factors. Let $r\ge1$, let $\mathcal T$ be a nonempty compact convex set of density matrices in finite dimension, and let $\mathcal A$ be the convex hull of finitely many effects, contain $I$, and be closed under complements. If $\mathcal S(\mathcal A)$ is affinely isomorphic to a Cartesian product of at most $r$ finite simplexes, the surviving criterion is equivalence of $1\mapsto r$ pseudo-broadcastability and non-contextuality. This uses the number of factors of the actual dual state space. G. Aubrun, A. Müller-Hermes and M. Plávala, Monogamy of entanglement between cones (2022 preprint, arXiv:2206.11805v2, DOI 10.1007/s00208-024-02935-4), Theorem 2 (main.tex, lines 259–266), gives the corresponding finite-level criterion for finite-dimensional closed proper cones, with the normalization functional in the interior of the dual cone. Its product-of-simplexes implication is established on lines 572–605. Taking the preparation and effect cones in their real linear spans supplies these hypotheses here: compact normalized preparations give a closed proper preparation cone, while finite effect generation and complement closure give a closed effect cone with order unit $I$.

  More explicitly, write the product as $\prod_{j=1}^q\Delta_j$ with $q\le r$. Its barycentric coordinate effects $h_{j,o}$ are positive on $\mathcal S(\mathcal A)$, so finite-dimensional cone duality places them in the closed cone generated by $\mathcal A$; for each $j$, $\sum_oh_{j,o}=I$. Apply a putative $r$-copy map to one such coordinate effect per factor, with $I$ in unused copies. The resulting joint weights are nonnegative by conic linearity and sum to one. Their $j$th marginal reproduces every coordinate of factor $j$, by the map's marginal identities. The mixture of the product vertices with these weights therefore reproduces the entire input functional. The weights are linear preparation functionals and each product vertex is a normalized positive response functional, giving a finite non-contextual model. The reverse implication is the source's Theorem 5, since non-contextuality supplies pseudo-broadcasting for every $n$, in particular $n=r$. For the square $q=2$, so two copies suffice for this implication; their existence is excluded by the preceding argument.

• Effect on the source. The all-$n$ equivalence in Jokinen et al.'s Theorem 5 does not depend on the conjecture. Its implication from non-contextuality to each finite level survives, so infeasibility at any finite level still certifies contextuality. What the counterexample excludes is inferring non-contextuality merely from feasibility at the number of generating measurements. The reported numerical agreement at three copies for the particular Mazurek scenario (Arxiv.tex, lines 348–350) is not contradicted by this different one-generator example; it does not establish the universal conjecture. No verdict for every unrestricted $k\ge2$ is supplied by this settlement.

The bounded literature check does not establish exhaustive worldwide novelty or priority. The reprint's pages 6 and 12, the arXiv version listing, arXiv:2601.14067, the citation counts and ten citing texts, and the web, Zenodo, MathDB, formal-conjectures and repository searches reported in #21862 are not independently reproduced. The interpretation of “generated” is the binary post-processing and convex-hull reading fixed by that issue; the source does not give a separate formal definition of that word.

The exact computations, the identification of the dual square, the exclusion of two-copy pseudo-broadcasting, the simplex-factor criterion under its stated additional hypotheses, and the consequences for the source discussed under “Argued, not formalized” are not additional Lean declarations in this module. The source's all-$n$ theorem and Aubrun et al.'s cone theorem are literature inputs, not formalized here. Their application does not assert the original generator-count conjecture for arbitrary $k\ge2$, or certify the source's numerical Mazurek regions. The correspondence between the source's operational definitions and the Lean definitions requires the explicit reading stated above.