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Generic Schmidt Rank of Weighted Phase Networks: Matching Laws, Parameter Tying, and Higher-Order Union Obstructions

Byoungwoo Lee

Published Aug 9, 2026• 0 citations
Publisher

Abstract

This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: $$r_{\mathrm{gen}} = 2^{\nu(G_\times)},$$ where $\nu(G_\times)$ is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network $K_{p,q}$, the exact rank is $$\operatorname{rank} K_\theta = \min\left\{ p+1,\, q+1,\, \operatorname{ord}(e^{i\theta}) \right\},$$ with infinite multiplicative order interpreted in the natural way. The $K_{2,2}$ example therefore exhibits a rank stratification ($4 \to 3 \to 2$) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with $$r_{\mathrm{str}}=4, \qquad r_{\mathrm{sym}}=3.$$ Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3

DOI
10.5281/zenodo.21857066

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Generic Schmidt-Rank Width of Fixed Weighted Phase Networks: Maximum-Matching Width, Template Diversity, and Single-Parameter Global Saturation

Byoungwoo Lee

This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph $G=(V,E)$ with a fixed real edge-weight template $W=(W_e)_{e\in E}$, we consider the commuting weighted-phase evolution $$U_W(t) = \exp\left( i t \sum_{e=\{u,v\}\in E} W_e n_u n_v \right),$$ with a single scalar evolution parameter $t$. For every bipartition $A\vert{}A^c$, let $\nu_G(A)$ be the maximum matching number of the crossing graph $G[A,A^c]$. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates $W$ such that, outside one locally finite exceptional set of evolution times, the same scalar $t$ simultaneously saturates the matching capacity of every nontrivial bipartition: $$\log_2 \operatorname{SR}_{A\vert{}A^c}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \nu_G(A)$$ for all nontrivial cuts $A\vert{}A^c$. Consequently, $$\chi_{\mathrm{wd}}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \operatorname{mmw}(G)$$ for generic fixed weighted-phase networks, where $\chi_{\mathrm{wd}}$ is Schmidt-rank width and $\operatorname{mmw}(G)$ is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support $K_n$, $$\operatorname{mmw}(K_n) = \left\lceil\frac{n}{3}\right\rceil,$$ so almost every nonuniform fixed template attains $$\chi_{\mathrm{wd}} = \left\lceil\frac{n}{3}\right\rceil,$$ which is the largest Schmidt-rank width permitted for any $n$-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is $$\chi_{\mathrm{wd}}^{\mathrm{unif}} = \log_2 \min\left\{ \left\lceil\frac{n}{3}\right\rceil+1,\, \operatorname{ord}(e^{i\theta}) \right\},$$ with $\operatorname{ord}(e^{i\theta})=\infty$ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: $$\text{Clifford} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=1,$$ $$\text{generic uniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(\log n),$$ $$\text{generic nonuniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(n).$$ This separation is not restricted to dense complete graphs. Since $$\operatorname{rw}(G) \le \operatorname{mmw}(G),$$ graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse $O(n)$-coupler networks with generic Schmidt-rank width $\Theta(n)$. The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension $$D_{\mathrm{unif}} = \left\lceil\frac{n}{3}\right\rceil+1 = \Theta(n),$$ whereas generic nonuniform complete templates require $$D_{\mathrm{gen}} = 2^{\left\lceil n/3\right\rceil} = 2^{\Theta(n)}.$$ Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.

•Aug 9, 2026•0 citations

Restriction-Based Certificate of Bipartite Schmidt Rank in Hypergraph States

C. Fajardo, M. Paraschiv

We investigate bipartite entanglement in qubit hypergraph states across an arbitrary fixed bipartition. Using the real equally weighted (REW) representation, the Schmidt rank across the cut can be computed as the real rank of a phase-cleaned cross-cut sign matrix. Whereas graph states admit an exact cut-rank rule, because the cross-cut phase is purely bilinear, hypergraph states typically contain higher-degree cross-cut interactions, for which the cut-rank rule fails. Our approach certifies entanglement by fixing a single computational-basis assignment on a subset of qubits, thereby selecting a submatrix on an active slice. When this restriction removes all higher-degree cross-cut residues, the remaining cross-cut phase becomes bilinear up to cut-local terms. We call the resulting submatrices residual-free bilinear cores and show that they yield an exponential Schmidt-rank lower bound in terms of the $\mathbb{F}_2$-rank of an exposed core matrix. We further give a combinatorial sufficient condition, phrased as a disjoint bridge matching, that guarantees the existence of large full-rank cores for broad families of CCZ-type bridge patterns, and we present a search-and-verify procedure that constructs and certifies such cores directly from the hyperedge description.

quant-pharXiv:2602.21016•Physical Review A•Feb 25, 2026•0 citations

Tail-Resolved Schmidt-Rank Width of Weighted Phase Networks: Noncommuting Activation Limits, Gauge-Stable Perturbations, and Approximate Tree Complexity

Byoungwoo Lee

This work develops a finite-resolution counterpart of Schmidt-rank width for weighted phase networks and uses it to distinguish algebraic Schmidt-rank activation from spectrally resolved entanglement complexity. For a bipartition $A|A^c$ with Schmidt coefficients $s_1 \ge s_2 \ge \cdots$, define the discarded Schmidt-tail mass $$\tau_r^A(\psi)=\left(\sum_{j>r}s_j^2\right)^{1/2},$$ and the corresponding $\epsilon$-resolved Schmidt rank $$R_\epsilon^A(\psi)=\min\left\{r:\tau_r^A(\psi)\le\epsilon\right\}.$$ The global tail-resolved Schmidt-rank width is then $$\chi_{\mathrm{res}}^{(\epsilon)}(\psi)=\min_T\max_{e\in E(T)}\log_2R_\epsilon^{A_e}(\psi),$$ where $T$ ranges over subcubic decomposition trees and $A_e|A_e^c$ is the bipartition induced by deleting the tree edge $e$. At zero resolution, $\epsilon=0$, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: $$\chi_{\mathrm{res}}^{(\epsilon)}(|G\rangle)=\log_2\left\lceil(1-\epsilon^2)\,2^{\operatorname{rw}(G)}\right\rceil,\qquad0\le\epsilon<1,$$ where $\operatorname{rw}(G)$ is the graph rank-width. Consequently, graph families with linear rank-width retain linear tail-resolved Schmidt-rank width at every fixed finite resolution $\epsilon<1$. The paper next establishes a gauge-stable perturbation theory. After optimization over cut-local diagonal phases, Schmidt truncation tails satisfy the Lipschitz estimate $$\left|\tau_r^A(\psi)-\tau_r^A(\phi)\right|\led_{\mathrm{diag}}^A(\psi,\phi),$$ which yields a global resolution-shift sandwich for $\chi_{\mathrm{res}}^{(\epsilon)}$. For balanced pairwise weighted phase networks with crossing phase errors $\eta_e$, $$d_{\mathrm{diag}}^A\le\frac14\left\|\eta_{E(A,A^c)}\right\|_2.$$ If a fixed coupling template $W$ is implemented with coupling error $\Delta W$ and timing error $\Delta t$, then $$\eta=t\,\Delta W+\Delta t\,W+\Delta t\,\Delta W,$$ giving an explicit certificate that converts finite calibration and timing uncertainty into a guaranteed resolved-width bound. The tail-resolved width also controls approximate subcubic tree-tensor-network complexity. If $D_{\mathrm{TTN}}^{(\epsilon)}$ denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error $\epsilon$, then $$2^{\chi_{\mathrm{res}}^{(\epsilon)}}\leD_{\mathrm{TTN}}^{(\epsilon)}\le2^{\chi_{\mathrm{res}}^{\left(\epsilon/\sqrt{2n-3}\right)}}.$$ The central application concerns the Clifford point of the complete weighted phase network. Consider $$|\Psi_V(\lambda)\rangle=\exp\left[i\sum_e\left(\pi+\lambda V_e\right)n_u n_v\right]|+\rangle^{\otimes n}.$$ For every fixed resolution $$0<\epsilon<\frac{1}{\sqrt{2}},$$ the tail-resolved width remains equal to $1$ for all sufficiently small $|\lambda|$, independently of the perturbation direction $V$. In contrast, at exact zero resolution an arbitrarily small generic nonuniform perturbation activates the maximal complete-support Schmidt-rank width $$\left\lceil\frac{n}{3}\right\rceil,$$ whereas a uniform perturbation activates only $$\log_2\left(\left\lceil\frac{n}{3}\right\rceil+1\right).$$ Thus the zero-resolution and zero-perturbation limits do not commute. For an open, dense, full-measure class of nonuniform perturbation directions, $$\lim_{\epsilon\downarrow0}\lim_{\lambda\to0}\chi_{\mathrm{res}}^{(\epsilon)}=1,$$ while $$\lim_{\lambda\to0,\ \lambda\neq0}\lim_{\epsilon\downarrow0}\chi_{\mathrm{res}}^{(\epsilon)}=\left\lceil\frac{n}{3}\right\rceil.$$ This establishes a sharp distinction between algebraic Schmidt activation and finite-resolution activation: infinitesimal symmetry breaking can create maximal exact Schmidt-rank width while remaining invisible at any fixed finite spectral resolution sufficiently close to the Clifford point. The results connect weighted phase and fixed $ZZ$/Ising networks, graph width parameters, Schmidt-spectrum stability, and approximate tensor-network complexity. The global phase-error certificate is intentionally conservative: it does not by itself establish thermodynamic robustness against a size-independent coherent error on every edge.

•Aug 9, 2026•0 citations

Knot your average qutrit: Measurement-induced entanglement splitting and the cabling dictionary for GHZ and W States

Sougata Bhattacharyya, Sovik Roy

Multipartite entanglement is conventionally classified by state families viz. family of GHZ and W class of states, with each family expected to behave differently under measurement. We show that, at least for the question of how entanglement splits after a single-particle measurement, this is not the division that matters for qutrits. Extending the Aravind's correspondence (which models entanglement as topological linking, and projective measurement as physically cutting a ring from an interlinked configuration\cite{aravind1997}) from qubits to qutrits, we derive the complete measurement-induced entanglement splitting of the GHZ type qutrit state i.e |GHZ_3> and of the full family of symmetric W class qutrit states, six two same - one different states i.e |{W_{p,p,q}^{sym}}> and one all - different state i.e. |W_{0,1,2}>, under both the computational basis (CB) and the mutually unbiased bases (MUBs), obtaining exact eigenvalues and Schmidt ranks for every outcome in every case. We see that the |W_{0,1,2}> state behaves similarly as |GHZ_3> state, a single, outcome-independent residual rank in each basis, while the |{W_{p,p,q}^{sym}> states alone show probability-weighted, outcome-dependent behaviour. The relevant structural line is therefore repeated-index versus all-different-index bag structure, not GHZ class versus $W$ class. We express this classification using a \textit{two-strand cabling} extension of Aravind's \textit{ring-and-link} picture. This is needed because the qutrit residual Schmidt rank (R) takes three values, R belonging to {1,2,3}, rather than the qubit binary (i.e. R belonging to {1,2}). We are explicit throughout that this cabling dictionary is a labeling convention built to reproduce an independently computed Schmidt rank, not a topological invariant derived from the link diagrams themselves, and we discuss what would be needed to close that gap

quant-pharXiv:2608.09076•Aug 11, 2026•0 citations

Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs

Ethan Dickey, Sabre Kais

We study when block-coupled regular graphs can realize prescribed complex quantum-like (QL) bit states as exact synchronized eigenstates. Two regular subgraphs $G_A$ and $G_B$ supply normalized all-ones eigenvectors $V_A$ and $V_B$, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a $2\times 2$ effective block on $\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}$. Within this reduction we prove that two natural symmetric complexifications are not universal for state synthesis under a real-spectrum requirement: complex symmetric coupling with real diagonal regularities forces the target computational basis amplitude ratio $r=ω_2/ω_1$, for $\lvert ψ\rangle = ω_1\lvert 0\rangle + ω_2\lvert 1\rangle$, to satisfy $r^2\in\R$, while real symmetric coupling with complex diagonal regularities forces $r+1/r\in\R$. Replacing complex symmetry by Hermitian coupling removes this phase obstruction. For any nonbasis target state, any prescribed real eigenvalue, and any prescribed nonzero signed spectral gap, a Hermitian weighted coupling realizes the target exactly. Additionally, an independently tuned directed-coupling model gives a second universality mechanism. We then pass from continuous effective parameters to finite weighted graphs with entries in $\{0, \pm1, \pm i\}$ (the fourth roots of unity and zero), characterize the balanced discrete coupling lattice by perfect matchings, and show that exact discrete Hermitian realizations are dense in the synchronized pure-state space. These results give a state-synthesis universality taxonomy for complex QL-bits and identify Hermitian conjugate pairing as the robust structural mechanism that supports arbitrary complex amplitudes with real two-level spectra.

quant-pharXiv:2604.23991•Apr 28, 2026•0 citations

Maximal Operator-Schmidt Rank in a Two-Fold Abelian Wreath-Product Permutation Module

Patrick Reymond

Let A be a finite abelian group, B=A×A, and W=B⋊⟨τ⟩=A wr S₂, where τ swaps the two factors. For an arbitrary subgroup S≤B, consider the induced permutation module C[W/S] with its natural sheet/base tensor factorization C²⊗C[B/S]. We derive a closed formula for the maximum operator-Schmidt rank attained by the image of the complex group algebra C[W]. The formula depends only on three orbit statistics (d,p,u) describing how the annihilator Q=S⊥ intersects the fixed points and two-cycles of the swap involution. The non-swap-invariant case introduces half-orbits, encoded by u. We also prove that the same maximum is attained by unitary elements of the image algebra. This yields, in Nielsen's exact one-use coherent communication model, a symmetry-determined Hartley lower bound on the communication cost of unitaries in the class. The magnitude is necessarily at most two qubits for the present C² sheet factor; the point is the exact symmetry-forced value, not a large asymptotic bound. Supplementary files: bilingual (English/French) README and two Python verification scripts covering 131 exhaustive subgroup cases for the theorem and 390 for the unitary corollary.

•Aug 16, 2026•0 citations
This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen=2ν(G×),r_{\mathrm{gen}} = 2^{\nu(G_\times)},rgen​=2ν(G×​), where ν(G×)\nu(G_\times)ν(G×​) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,qK_{p,q}Kp,q​, the exact rank is rank⁡Kθ=min⁡{p+1, q+1, ord⁡(eiθ)},\operatorname{rank} K_\theta = \min\left\{ p+1,\, q+1,\, \operatorname{ord}(e^{i\theta}) \right\},rankKθ​=min{p+1,q+1,ord(eiθ)}, with infinite multiplicative order interpreted in the natural way. The K2,2K_{2,2}K2,2​ example therefore exhibits a rank stratification (4→3→24 \to 3 \to 24→3→2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rstr=4,rsym=3.r_{\mathrm{str}}=4, \qquad r_{\mathrm{sym}}=3.rstr​=4,rsym​=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3
This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph G=(V,E)G=(V,E)G=(V,E) with a fixed real edge-weight template W=(We)e∈EW=(W_e)_{e\in E}W=(We​)e∈E​, we consider the commuting weighted-phase evolution UW(t)=exp⁡(it∑e={u,v}∈EWenunv),U_W(t) = \exp\left( i t \sum_{e=\{u,v\}\in E} W_e n_u n_v \right),UW​(t)=exp(it∑e={u,v}∈E​We​nu​nv​), with a single scalar evolution parameter ttt. For every bipartition A∣AcA\vert{}A^cA∣Ac, let νG(A)\nu_G(A)νG​(A) be the maximum matching number of the crossing graph G[A,Ac]G[A,A^c]G[A,Ac]. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates WWW such that, outside one locally finite exceptional set of evolution times, the same scalar ttt simultaneously saturates the matching capacity of every nontrivial bipartition: log⁡2SR⁡A∣Ac(∣ΨW(t)⟩)=νG(A)\log_2 \operatorname{SR}_{A\vert{}A^c}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \nu_G(A)log2​SRA∣Ac​(∣ΨW​(t)⟩)=νG​(A) for all nontrivial cuts A∣AcA\vert{}A^cA∣Ac. Consequently, χwd(∣ΨW(t)⟩)=mmw⁡(G)\chi_{\mathrm{wd}}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \operatorname{mmw}(G)χwd​(∣ΨW​(t)⟩)=mmw(G) for generic fixed weighted-phase networks, where χwd\chi_{\mathrm{wd}}χwd​ is Schmidt-rank width and mmw⁡(G)\operatorname{mmw}(G)mmw(G) is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support KnK_nKn​, mmw⁡(Kn)=⌈n3⌉,\operatorname{mmw}(K_n) = \left\lceil\frac{n}{3}\right\rceil,mmw(Kn​)=⌈3n​⌉, so almost every nonuniform fixed template attains χwd=⌈n3⌉,\chi_{\mathrm{wd}} = \left\lceil\frac{n}{3}\right\rceil,χwd​=⌈3n​⌉, which is the largest Schmidt-rank width permitted for any nnn-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is χwdunif=log⁡2min⁡{⌈n3⌉+1, ord⁡(eiθ)},\chi_{\mathrm{wd}}^{\mathrm{unif}} = \log_2 \min\left\{ \left\lceil\frac{n}{3}\right\rceil+1,\, \operatorname{ord}(e^{i\theta}) \right\},χwdunif​=log2​min{⌈3n​⌉+1,ord(eiθ)}, with ord⁡(eiθ)=∞\operatorname{ord}(e^{i\theta})=\inftyord(eiθ)=∞ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: Clifford⟶χwd=1,\text{Clifford} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=1,Clifford⟶χwd​=1, generic uniform⟶χwd=Θ(log⁡n),\text{generic uniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(\log n),generic uniform⟶χwd​=Θ(logn), generic nonuniform⟶χwd=Θ(n).\text{generic nonuniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(n).generic nonuniform⟶χwd​=Θ(n). This separation is not restricted to dense complete graphs. Since rw⁡(G)≤mmw⁡(G),\operatorname{rw}(G) \le \operatorname{mmw}(G),rw(G)≤mmw(G), graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse O(n)O(n)O(n)-coupler networks with generic Schmidt-rank width Θ(n)\Theta(n)Θ(n). The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension Dunif=⌈n3⌉+1=Θ(n),D_{\mathrm{unif}} = \left\lceil\frac{n}{3}\right\rceil+1 = \Theta(n),Dunif​=⌈3n​⌉+1=Θ(n), whereas generic nonuniform complete templates require Dgen=2⌈n/3⌉=2Θ(n).D_{\mathrm{gen}} = 2^{\left\lceil n/3\right\rceil} = 2^{\Theta(n)}.Dgen​=2⌈n/3⌉=2Θ(n). Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.
We investigate bipartite entanglement in qubit hypergraph states across an arbitrary fixed bipartition. Using the real equally weighted (REW) representation, the Schmidt rank across the cut can be computed as the real rank of a phase-cleaned cross-cut sign matrix. Whereas graph states admit an exact cut-rank rule, because the cross-cut phase is purely bilinear, hypergraph states typically contain higher-degree cross-cut interactions, for which the cut-rank rule fails. Our approach certifies entanglement by fixing a single computational-basis assignment on a subset of qubits, thereby selecting a submatrix on an active slice. When this restriction removes all higher-degree cross-cut residues, the remaining cross-cut phase becomes bilinear up to cut-local terms. We call the resulting submatrices residual-free bilinear cores and show that they yield an exponential Schmidt-rank lower bound in terms of the F2\mathbb{F}_2F2​-rank of an exposed core matrix. We further give a combinatorial sufficient condition, phrased as a disjoint bridge matching, that guarantees the existence of large full-rank cores for broad families of CCZ-type bridge patterns, and we present a search-and-verify procedure that constructs and certifies such cores directly from the hyperedge description.
This work develops a finite-resolution counterpart of Schmidt-rank width for weighted phase networks and uses it to distinguish algebraic Schmidt-rank activation from spectrally resolved entanglement complexity. For a bipartition A∣AcA|A^cA∣Ac with Schmidt coefficients s1≥s2≥⋯s_1 \ge s_2 \ge \cdotss1​≥s2​≥⋯, define the discarded Schmidt-tail mass τrA(ψ)=(∑j>rsj2)1/2,\tau_r^A(\psi)=\left(\sum_{j>r}s_j^2\right)^{1/2},τrA​(ψ)=(∑j>r​sj2​)1/2, and the corresponding ϵ\epsilonϵ-resolved Schmidt rank RϵA(ψ)=min⁡{r:τrA(ψ)≤ϵ}.R_\epsilon^A(\psi)=\min\left\{r:\tau_r^A(\psi)\le\epsilon\right\}.RϵA​(ψ)=min{r:τrA​(ψ)≤ϵ}. The global tail-resolved Schmidt-rank width is then χres(ϵ)(ψ)=min⁡Tmax⁡e∈E(T)log⁡2RϵAe(ψ),\chi_{\mathrm{res}}^{(\epsilon)}(\psi)=\min_T\max_{e\in E(T)}\log_2R_\epsilon^{A_e}(\psi),χres(ϵ)​(ψ)=minT​maxe∈E(T)​log2​RϵAe​​(ψ), where TTT ranges over subcubic decomposition trees and Ae∣AecA_e|A_e^cAe​∣Aec​ is the bipartition induced by deleting the tree edge eee. At zero resolution, ϵ=0\epsilon=0ϵ=0, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: χres(ϵ)(∣G⟩)=log⁡2⌈(1−ϵ2) 2rw⁡(G)⌉,0≤ϵ<1,\chi_{\mathrm{res}}^{(\epsilon)}(|G\rangle)=\log_2\left\lceil(1-\epsilon^2)\,2^{\operatorname{rw}(G)}\right\rceil,\qquad0\le\epsilon<1,χres(ϵ)​(∣G⟩)=log2​⌈(1−ϵ2)2rw(G)⌉,0≤ϵ<1, where rw⁡(G)\operatorname{rw}(G)rw(G) is the graph rank-width. Consequently, graph families with linear rank-width retain linear tail-resolved Schmidt-rank width at every fixed finite resolution ϵ<1\epsilon<1ϵ<1. The paper next establishes a gauge-stable perturbation theory. After optimization over cut-local diagonal phases, Schmidt truncation tails satisfy the Lipschitz estimate $$\left|\tau_r^A(\psi)-\tau_r^A(\phi)\right|\led_{\mathrm{diag}}^A(\psi,\phi),$$ which yields a global resolution-shift sandwich for χres(ϵ)\chi_{\mathrm{res}}^{(\epsilon)}χres(ϵ)​. For balanced pairwise weighted phase networks with crossing phase errors ηe\eta_eηe​, ddiagA≤14∥ηE(A,Ac)∥2.d_{\mathrm{diag}}^A\le\frac14\left\|\eta_{E(A,A^c)}\right\|_2.ddiagA​≤41​​ηE(A,Ac)​​2​. If a fixed coupling template WWW is implemented with coupling error ΔW\Delta WΔW and timing error Δt\Delta tΔt, then η=t ΔW+Δt W+Δt ΔW,\eta=t\,\Delta W+\Delta t\,W+\Delta t\,\Delta W,η=tΔW+ΔtW+ΔtΔW, giving an explicit certificate that converts finite calibration and timing uncertainty into a guaranteed resolved-width bound. The tail-resolved width also controls approximate subcubic tree-tensor-network complexity. If DTTN(ϵ)D_{\mathrm{TTN}}^{(\epsilon)}DTTN(ϵ)​ denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error ϵ\epsilonϵ, then $$2^{\chi_{\mathrm{res}}^{(\epsilon)}}\leD_{\mathrm{TTN}}^{(\epsilon)}\le2^{\chi_{\mathrm{res}}^{\left(\epsilon/\sqrt{2n-3}\right)}}.$$ The central application concerns the Clifford point of the complete weighted phase network. Consider ∣ΨV(λ)⟩=exp⁡[i∑e(π+λVe)nunv]∣+⟩⊗n.|\Psi_V(\lambda)\rangle=\exp\left[i\sum_e\left(\pi+\lambda V_e\right)n_u n_v\right]|+\rangle^{\otimes n}.∣ΨV​(λ)⟩=exp[i∑e​(π+λVe​)nu​nv​]∣+⟩⊗n. For every fixed resolution 0<ϵ<12,0<\epsilon<\frac{1}{\sqrt{2}},0<ϵ<2​1​, the tail-resolved width remains equal to 111 for all sufficiently small ∣λ∣|\lambda|∣λ∣, independently of the perturbation direction VVV. In contrast, at exact zero resolution an arbitrarily small generic nonuniform perturbation activates the maximal complete-support Schmidt-rank width ⌈n3⌉,\left\lceil\frac{n}{3}\right\rceil,⌈3n​⌉, whereas a uniform perturbation activates only log⁡2(⌈n3⌉+1).\log_2\left(\left\lceil\frac{n}{3}\right\rceil+1\right).log2​(⌈3n​⌉+1). Thus the zero-resolution and zero-perturbation limits do not commute. For an open, dense, full-measure class of nonuniform perturbation directions, lim⁡ϵ↓0lim⁡λ→0χres(ϵ)=1,\lim_{\epsilon\downarrow0}\lim_{\lambda\to0}\chi_{\mathrm{res}}^{(\epsilon)}=1,limϵ↓0​limλ→0​χres(ϵ)​=1, while lim⁡λ→0, λ≠0lim⁡ϵ↓0χres(ϵ)=⌈n3⌉.\lim_{\lambda\to0,\ \lambda\neq0}\lim_{\epsilon\downarrow0}\chi_{\mathrm{res}}^{(\epsilon)}=\left\lceil\frac{n}{3}\right\rceil.limλ→0, λ=0​limϵ↓0​χres(ϵ)​=⌈3n​⌉. This establishes a sharp distinction between algebraic Schmidt activation and finite-resolution activation: infinitesimal symmetry breaking can create maximal exact Schmidt-rank width while remaining invisible at any fixed finite spectral resolution sufficiently close to the Clifford point. The results connect weighted phase and fixed ZZZZZZ/Ising networks, graph width parameters, Schmidt-spectrum stability, and approximate tensor-network complexity. The global phase-error certificate is intentionally conservative: it does not by itself establish thermodynamic robustness against a size-independent coherent error on every edge.
Multipartite entanglement is conventionally classified by state families viz. family of GHZ and W class of states, with each family expected to behave differently under measurement. We show that, at least for the question of how entanglement splits after a single-particle measurement, this is not the division that matters for qutrits. Extending the Aravind's correspondence (which models entanglement as topological linking, and projective measurement as physically cutting a ring from an interlinked configuration\cite{aravind1997}) from qubits to qutrits, we derive the complete measurement-induced entanglement splitting of the GHZ type qutrit state i.e |GHZ_3> and of the full family of symmetric W class qutrit states, six two same - one different states i.e |{W_{p,p,q}^{sym}}> and one all - different state i.e. |W_{0,1,2}>, under both the computational basis (CB) and the mutually unbiased bases (MUBs), obtaining exact eigenvalues and Schmidt ranks for every outcome in every case. We see that the |W_{0,1,2}> state behaves similarly as |GHZ_3> state, a single, outcome-independent residual rank in each basis, while the |{W_{p,p,q}^{sym}> states alone show probability-weighted, outcome-dependent behaviour. The relevant structural line is therefore repeated-index versus all-different-index bag structure, not GHZ class versus WWW class. We express this classification using a \textit{two-strand cabling} extension of Aravind's \textit{ring-and-link} picture. This is needed because the qutrit residual Schmidt rank (R) takes three values, R belonging to {1,2,3}, rather than the qubit binary (i.e. R belonging to {1,2}). We are explicit throughout that this cabling dictionary is a labeling convention built to reproduce an independently computed Schmidt rank, not a topological invariant derived from the link diagrams themselves, and we discuss what would be needed to close that gap
We study when block-coupled regular graphs can realize prescribed complex quantum-like (QL) bit states as exact synchronized eigenstates. Two regular subgraphs GAG_AGA​ and GBG_BGB​ supply normalized all-ones eigenvectors VAV_AVA​ and VBV_BVB​, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a 2×22\times 22×2 effective block on S=span⁡{∣0⟩,∣1⟩}\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}S=span{∣0⟩,∣1⟩}. Within this reduction we prove that two natural symmetric complexifications are not universal for state synthesis under a real-spectrum requirement: complex symmetric coupling with real diagonal regularities forces the target computational basis amplitude ratio r=ω2/ω1r=ω_2/ω_1r=ω2​/ω1​, for ∣ψ⟩=ω1∣0⟩+ω2∣1⟩\lvert ψ\rangle = ω_1\lvert 0\rangle + ω_2\lvert 1\rangle∣ψ⟩=ω1​∣0⟩+ω2​∣1⟩, to satisfy r2∈Rr^2\in\Rr2∈R, while real symmetric coupling with complex diagonal regularities forces r+1/r∈Rr+1/r\in\Rr+1/r∈R. Replacing complex symmetry by Hermitian coupling removes this phase obstruction. For any nonbasis target state, any prescribed real eigenvalue, and any prescribed nonzero signed spectral gap, a Hermitian weighted coupling realizes the target exactly. Additionally, an independently tuned directed-coupling model gives a second universality mechanism. We then pass from continuous effective parameters to finite weighted graphs with entries in {0,±1,±i}\{0, \pm1, \pm i\}{0,±1,±i} (the fourth roots of unity and zero), characterize the balanced discrete coupling lattice by perfect matchings, and show that exact discrete Hermitian realizations are dense in the synchronized pure-state space. These results give a state-synthesis universality taxonomy for complex QL-bits and identify Hermitian conjugate pairing as the robust structural mechanism that supports arbitrary complex amplitudes with real two-level spectra.
Let A be a finite abelian group, B=A×A, and W=B⋊⟨τ⟩=A wr S₂, where τ swaps the two factors. For an arbitrary subgroup S≤B, consider the induced permutation module C[W/S] with its natural sheet/base tensor factorization C²⊗C[B/S]. We derive a closed formula for the maximum operator-Schmidt rank attained by the image of the complex group algebra C[W]. The formula depends only on three orbit statistics (d,p,u) describing how the annihilator Q=S⊥ intersects the fixed points and two-cycles of the swap involution. The non-swap-invariant case introduces half-orbits, encoded by u. We also prove that the same maximum is attained by unitary elements of the image algebra. This yields, in Nielsen's exact one-use coherent communication model, a symmetry-determined Hartley lower bound on the communication cost of unitaries in the class. The magnitude is necessarily at most two qubits for the present C² sheet factor; the point is the exact symmetry-forced value, not a large asymptotic bound. Supplementary files: bilingual (English/French) README and two Python verification scripts covering 131 exhaustive subgroup cases for the theorem and 390 for the unitary corollary.