Monotonicity of the negativity of randomized graph states

J.-Y. Wu, M. Rossi, H. Kampermann, S. Severini, L. C. Kwek, C. Macchiavello, D. Bruß, Randomized Graph States and their Entanglement Properties, Phys. Rev. A 89, 052335 (2014), arXiv:1403.3828v3, Section V:

However, even though this conjecture is supported by numerical evidence, it is an open question whether the monotonic behavior of the negativity in terms of the randomness $p$ is a common feature to all RG states.

The randomized graph state of a graph $G$ is $\rho^p_G=\sum_{F\subseteq E(G)}p^{|F|}(1-p)^{|E(G)\setminus F|} |F\rangle\langle F|$, where $|F\rangle=\prod_{ab\in F}\mathrm{CZ}_{ab} |+\rangle^{\otimes n}$, and the negativity across a bipartition $A|B$ is $N(\rho)=(|\rho^{\Gamma_A}|_1-1)/2$, evaluated "with respect to all possible bipartitions of the qubits". The conjecture is that $N$ increases monotonically in $p$; the paper reports it numerically for the complete graphs $K_n$ and the star graphs $S_n$ with $n\le4$.

A randomized graph state models the preparation of a graph state with probabilistic entangling gates: every controlled-Z gate succeeds with probability $p$ and the record of which gates succeeded is lost. At $p=0$ the state is the product state $|+\rangle^{\otimes n}$ and at $p=1$ it is the graph state of $G$, so one expects more entanglement for more reliable gates. The question asks whether this holds for every graph and every cut. The frozen declaration D5/S3/Quantum/Entanglement/RandomizedGraphNegativityRefutation.result answers it negatively.

Issue #12382 classifies the question as Tier 1 and records the checks made before any Lean:

• arXiv:1403.3828 v3 (2014-06-03) is the latest version and the published version, and the question is printed as quoted;
• Salem–Silva–Andrade, arXiv:2310.20418v2 (Phys. Rev. A 109, 012416, 2024), Sec. IV, report non-monotonic entanglement only for randomized hypergraph states with hyperedges of different orders and write that "such behavior could not be observed in RG states"; their arXiv:2506.20075v2 (2026), Fig. 3 caption, states "Monotonicity holds for hypergraphs with uniform hyperedge orders"; the review Poderini–Bruß–Macchiavello, arXiv:2603.10917v1 (2026), Sec. III.4, attributes non-monotonicity to non-uniform hypergraphs;
• noisy and percolated graph-state results (Aolita et al., arXiv:1006.3152; Kieling–Rudolph–Eisert, quant-ph/0611140) concern other models.

These are orchestrator-reported and seat-reported readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty. Convexity of the trace norm bounds the negativity of the mixture from above by the average negativity of the graph states $|F\rangle$, so the large negativity of some subgraph states does not by itself give a lower bound for the mixture.

$G=K_{3,3}$ on ${0,\dots,5}$ with parts $A={0,2,4}$ and $B={1,3,5}$, $p=97/100$ and $q=1$.

1. Expanding the product over the edges, $\rho^p_G(x,y)=2^{-n}\prod_{ab\in E}(p,c_{ab}+1-p)$ with $c_{ab}=1$ if $x_ax_b=y_ay_b$ and $c_{ab}=-1$ otherwise; so the partial transpose $X_p=(\rho^p_G)^{\Gamma_A}$ has entries $(1-2p)^{d}/64$, where $d$ counts the edges on which the exchanged labels disagree.
2. At $p=1$, with $u_{ij}(x)=,|x_A|\equiv i\ (2),^{j|x_B|}$, $w=u_{01}-u_{10}$ and $v=u_{01}+u_{10}$, the entrywise identity $X_1+\tfrac1{128}ww^{\mathsf T}=\tfrac1{64}(u_{00}u_{00}^{\mathsf T} +u_{11}u_{11}^{\mathsf T})+\tfrac1{128}vv^{\mathsf T}$ writes $X_1$ as a difference of two positive semidefinite matrices with traces $3/2$ and $1/2$; the triangle inequality gives $|X_1|_1\le2$, so $N(\rho^1_G)\le1/2$.
3. At $p=97/100$, fourteen pairwise orthogonal integer vectors $u_j\in\mathbb Z^{64}$ give the projection $P=\sum_ju_ju_j^{\mathsf T}/|u_j|^2$ and the unitary $U=I-2P$; by the variational formula $|X|1\ge\operatorname{Re}\operatorname{tr}(UX)$, $|X{97/100}|1\ge1-2\sum_ju_j^{\mathsf T}X{97/100}u_j/|u_j|^2 =2.0280536\ldots>2$.
4. Hence $N(\rho^{97/100}_G)>1/2\ge N(\rho^1_G)$ with $97/100\le1$.

The kernel-checked result is the negation of the universal statement over finite simple graphs on Fin n, subsets $A$ of the vertices and all real $0\le p\le q\le1$. The basis states are the maps Fin n → Bool, the graph state of an edge set is its product of controlled-Z phases times the amplitude $2^{-n/2}$ of $|+\rangle^{\otimes n}$, the mixture runs over the subsets of G.edgeFinset, the partial transposition is the frozen CycleFiveStrongOneResistance.transposePart (it exchanges the $A$ parts of the row and column labels), and the trace norm is the frozen FiniteTraceDistance.traceNorm, $\operatorname{Re}\operatorname{Tr} \sqrt{X^\dagger X}$. A measure that is an increasing function of $N$, such as the logarithmic negativity $\log_2(2N+1)$, is refuted as well; restricting the question to $K_n$ and $S_n$ changes the question.

Exhaustive NumPy scan of the connected graphs on 4–6 vertices, all bipartitions and $p\in{0,0.01,\dots,1}$ (issue #12382): five pairs $(G,A)$, all on 6 vertices, show a decrease; none on 4 or 5 vertices. The largest is this example: $N(\rho^p_{K_{3,3}})$ rises above $1/2$ on $[0.936,0.999]$ with maximum $0.5140464\ldots$ near $p=0.97$, and the derivative at $p=1$ is about $-1$ (finite difference). The exact lower bound of step 3 is $N\ge0.5140268\ldots$. The certificate vectors split by the $S_3\times S_3$ symmetry of the parts (six in the symmetric sector, two in each mixed sector, four in the standard–standard sector).

The canonical source is D5/S3/Quantum/Entanglement/RandomizedGraphNegativityRefutation.lean. Its public declarations are Qubits, czPhase, plusState, graphState, rgState, negativity, claim, k33, partA and result; the partial transposition is the frozen D5/S3/Quantum/Entanglement/CycleFiveStrongOneResistance.transposePart, and the trace norm and its lemmas are frozen in D5/S3/Quantum/Foundation/FiniteTraceDistance; k33 carries a decidable adjacency instance. The frozen module state has statement identity sha256:a40b0f5478e2394807b3cd5c3fe6678251f25ad3e25d09ddd424e32bc01e1f60. The result declaration has statement identity sha256:8699682eb0805d4dcf1355be31c5dac754e3a40e309a3d295ac441cf8ab56906. The Freeze event is sha256:cb386a648738573ec8ccc5fa15cead0984628eb4c316f21e726fda95f4cae172; its project-level frozen prerequisites are the Freeze events of D5/S3/Quantum/Foundation/FiniteTraceDistance and D5/S3/Quantum/Entanglement/CycleFiveStrongOneResistance. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Tier 1 published question; resolution Refuted by D5/S3/Quantum/Entanglement/RandomizedGraphNegativityRefutation.result. proof_shape: content; admission_basis: open-problem-resolution (issue #12382). Utility kind certified-instance, basis refutes the module's claim.

What the settlement shows

• Proved by result: the negativity of randomized graph states is not monotone in $p$: for $K_{3,3}$ across its two parts it is larger at $p=97/100$ than at $p=1$.
• Proved inside the proof of result: the closed form $(1-2p)^d/64$ of the partial transpose for $K_{3,3}$ at every $p$; the decomposition of step 2, giving $N(\rho^1_{K_{3,3}})\le1/2$; and the bound $|X_{97/100}|_1>2$ from the explicit unitary $I-2P$.
• Mechanism (computed, not stated in Lean): a graph state across a cut has negativity $(2^r-1)/2$, where $r$ is the rank over GF(2) of the biadjacency matrix of the cut (checked by singular values for all $512$ spanning subgraphs of $K_{3,3}$). The biadjacency matrix of $K_{3,3}$ is the all-ones $3\times3$ matrix, $r=1$, so the endpoint has $N=1/2$; deleting any one edge raises $r$ to $2$ ($N=3/2$), and among the $512$ spanning subgraphs $462$ have $r\ge2$. Near $p=1$ the mixture puts weight about $9(1-p)$ on the one-edge deletions, and the computed negativity grows. In the scan, all five decreasing pairs have $r=1$ for the full graph and a single-edge deletion with $r=2$; among the $3273$ pairs where no single-edge deletion raises $r$, none decreases, and $551$ of the $556$ pairs where one does are still monotone on the grid.
• Computed, not in Lean: $N(\rho^p_{K_{3,3}})$ also decreases slightly on $[0.405,0.488]$ (from $0.10323$ to $0.10213$); $K_{4,2}$ with $A$ the part of size $2$ decreases by $1.8\times10^{-4}$ between $p=0.41$ and $0.47$ while its maximum $1/2$ is at $p=1$.
• Follows from it (not stated in Lean): the statements of Salem–Silva–Andrade that non-monotonic behaviour "could not be observed in RG states" (2024) and that monotonicity holds for hypergraphs with uniform hyperedge orders (2026) do not extend to all ordinary graphs, since $K_{3,3}$ is 2-uniform; by continuity the decrease also holds against some $q<1$.
• Open here: a characterization of the pairs $(G,A)$ with monotone negativity (the rank condition above is necessary on the scanned graphs but not sufficient, and its necessity in general is not proved); whether the complete graphs $K_n$ and the stars $S_n$, the families of the paper's numerics, are monotone for all $n$; and the sign of $dN/dp$ at $p=1$, which the first-order perturbation of the trace norm ties to the kernel of $(|G\rangle\langle G|)^{\Gamma_A}$.

Mansour et al., Laser Phys. 31, 035201 (2021), "Decay of negativity of randomized multiqubit mixed states", was not read in full; Salem–Silva– Andrade (2024, ref. [40]) describe it as observing monotonic negativity for states generated by random Ising-type entangling operators. The literature checks do not establish worldwide priority. The Lean kernel verifies the encoded statement and its axiom closure; its correspondence to the paper, including the reading of the question as a universal statement over all graphs, bipartitions and pairs $p\le q$, is checked by reading the source and the definitions.