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∣Ac with Schmidt coefficients s1≥s2≥⋯, define the discarded Schmidt-tail mass τrA(ψ)=(∑j>rsj2)1/2, and the corresponding ϵ-resolved Schmidt rank RϵA(ψ)=min{r:τrA(ψ)≤ϵ}. The global tail-resolved Schmidt-rank width is then χres(ϵ)(ψ)=minTmaxe∈E(T)log2RϵAe(ψ), where T ranges over subcubic decomposition trees and Ae∣Aec is the bipartition induced by deleting the tree edge e. At zero resolution, ϵ=0, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: χres(ϵ)(∣G⟩)=log2⌈(1−ϵ2)2rw(G)⌉,0≤ϵ<1, where 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. 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). For balanced pairwise weighted phase networks with crossing phase errors ηe, ddiagA≤41ηE(A,Ac)2. If a fixed coupling template W is implemented with coupling error ΔW and timing error Δt, then 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) denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error ϵ, 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. For every fixed resolution 0<ϵ<21, the tail-resolved width remains equal to 1 for all sufficiently small ∣λ∣, 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 ⌈3n⌉, whereas a uniform perturbation activates only 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, limϵ↓0limλ→0χres(ϵ)=1, while limλ→0,λ=0limϵ↓0χres(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 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.
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2 ⊗ V and I 2 ⊗ W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.