Interface Capacity and Architectural Replenishment Determine Entanglement-Generation Speed in Quantum Networks

Abstract: We show that entanglement-generation speed across a fixed network interface is governed by two distinct resources: the entangling capacity of the interface itself and the ability of the surrounding architecture to replenish it with fresh degrees of freedom. For fermionic Gaussian dynamics, we derive the coefficient-sharp bound $\sum_k|\dotθ<em>k|\leq\frac12|K</em>{AB}|_*$ on the collective speed of the canonical entanglement angles. Explicit Ising-chain rematching trajectories saturate this bound, thereby certifying exact minimum interaction times under the stated control model. Beyond the Gaussian setting, exhaustive optimization of the complete $N=8$ tree--tree family shows that, at fixed interface capacity, first-layer entanglement, connectedness, and edge budget, the saturation depth is exactly classified by rooted architecture. With higher-resolution $x$-only control, variational entanglement-enhancing-field (VEEF) optimization reaches the numerically resolved fast-$X$ optimum in a two-channel benchmark. Across all 21 symmetry-reduced rooted orbits, a pre-specified two-time VEEF growth diagnostic recovers the complete replenishment partition directly from optimized dynamics. Interface capacity therefore sets how much entangling flux is available, whereas architecture determines whether fresh degrees of freedom can continually replenish the interface and sustain repeated use of that capacity.

• The paper establishes a sharp Gaussian speed limit, showing that the sum of canonical entanglement-angle rates cannot exceed half the nuclear norm of the cross-boundary coupling, with exact saturation in controlled Ising-chain protocols.
• The paper demonstrates that, with the interface fixed, rooted internal topology determines architectural replenishment: in the 256-graph tree–tree family, 64 architectures reach four ebits in two layers while 192 require three.
• Continuous-control simulations reduce the three-layer stall to approximately 2.88 interface times and show that entanglement-growth rates distinguish two-layer from three-layer architectures across all 21 symmetry-reduced topology classes.

Overview and central claim

This paper addresses a controlled question in entanglement distribution: when the physical interaction crossing a bipartition is held fixed, what determines how quickly entanglement can be generated across that repeatedly used interface? The author organizes the answer around two distinct resources. Interface capacity is the entangling flux supplied by the fixed cross-boundary Hamiltonian, quantified for fermionic Gaussian dynamics by $\Lambda_\partial(t)=\frac12\|K_{AB}(t)\|_*$, half the Schatten 1-norm of the cross-boundary Majorana block. Architectural replenishment is the ability of the internal network to keep presenting fresh degrees of freedom to that interface so that its capacity can be reused. The paper's central claim is that these two resources are separable: capacity bounds how much entangling flux exists, while rooted internal topology determines whether that flux can be sustained.

Exact Gaussian speed limit

The first main result is a coefficient-sharp bound on the collective motion of canonical entanglement angles. For a pure fermionic Gaussian state under a quadratic Majorana Hamiltonian $H=\frac{i}{4}\gamma^T K\gamma$, with cross-boundary block $K_{AB}$, the canonical angles $\theta_k$ obey

$$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$

The proof decomposes $K_{AB}$ into singular channels and applies Cauchy–Schwarz per channel; local quadratic frame changes preserve singular values, making the bound basis-independent. The integrated form gives a many-mode quantum speed limit: reaching $q$ maximally entangled mode pairs at constant capacity requires $T\geq q\pi/(4\Lambda_\partial)$. The paper distinguishes this from prior entropy-rate, fast-local Schmidt-variable, and boundary-current bounds by noting it resolves all canonical angles collectively while retaining the physical boundary generator.

Attainability is established constructively via a Majorana-rematching protocol on uniform Ising chains under arbitrarily fast on-site $X$ rotations. Because every operation in the kicked protocol is a signed Majorana permutation, the state remains a perfect-matching Gaussian state whose entropy is exactly half the number of matching edges crossing the cut. The open-chain protocol achieves unit interface utilization and reaches maximal entanglement at $T=mt_0$ with $H=\frac{i}{4}\gamma^T K\gamma$0, while the periodic chain attains $H=\frac{i}{4}\gamma^T K\gamma$1. The factor-of-two periodic-chain speedup thus follows directly from doubled interface capacity plus exact saturation — these are certified minimum interaction times within the stated control model.

Rooted topology classifies replenishment

With the Gaussian bound saturated, the remaining question is whether saturation persists when only the internal architecture varies. The paper fixes the cross-cut Hamiltonian $H=\frac{i}{4}\gamma^T K\gamma$2 on an $H=\frac{i}{4}\gamma^T K\gamma$3 system with $H=\frac{i}{4}\gamma^T K\gamma$4, $H=\frac{i}{4}\gamma^T K\gamma$5, giving identical cut-rank ($H=\frac{i}{4}\gamma^T K\gamma$6), layer capacity, and first-layer entanglement (2 ebits) for every member of the family. Varying the twelve internal edges yields 4096 labeled graphs; restricting to connected halves with six internal edges yields a complete tree–tree family of 256 architectures.

Exhaustive optimization over binary $H=\frac{i}{4}\gamma^T K\gamma$7 kick patterns gives an exact depth split: 64 graphs reach the maximal four ebits in two Ising layers, while 192 require three layers, with no intermediate values. The strong result is that this split is classified without exception by rooted topology:

$H=\frac{i}{4}\gamma^T K\gamma$8

where $H=\frac{i}{4}\gamma^T K\gamma$9 is the largest distance from any vertex on side $K_{AB}$0 to its nearest interface vertex. Notably, distance alone is insufficient: any $K_{AB}$1 side forces $K_{AB}$2, and among the 144 $K_{AB}$3 architectures the domination criterion separates 64 two-layer cases from 80 three-layer cases. The paper is explicit that this is an exact classification of a finite enumerated family, not an analytic formula derived from the distance descriptor alone. Enlarging the kick alphabet to arbitrary quarter-turns leaves both the depth split and the rule unchanged, confirming robustness within the discrete setting.

Continuous control softens the layer stall but preserves the partition

Higher-resolution $K_{AB}$4-only control reveals that the three-layer stall reflects finite temporal resolution rather than a fundamental limit. On a calibrated $K_{AB}$5 two-channel graph, independent optimizations — a nested fast-$K_{AB}$6 kicked hierarchy with variable segment durations ($K_{AB}$7) and continuous variational entanglement-enhancing-field (VEEF) optimization — converge to threshold times between $K_{AB}$8 and $K_{AB}$9 for reaching 3.99 ebits, below the exact restricted-layer time $\theta_k$0. The agreement of two independent parameterizations identifies $\theta_k$1 as the numerically resolved optimum, though no finite-$\theta_k$2 result or VEEF trajectory is claimed as an analytic upper bound on the continuous-control envelope.

The more consequential finding concerns whether the rooted classes remain distinguishable once the layer restriction is relaxed. Applying VEEF to all 21 symmetry-reduced rooted orbits, with the optimizer blind to class labels, a pre-specified single-time threshold at $\theta_k$3 resolves 18 of 21 orbits, leaving three boundary cases. A second pre-specified diagnostic — the finite-difference growth rate $\theta_k$4 between $\theta_k$5 and $\theta_k$6, with frozen threshold $\theta_k$7 ebit/$\theta_k$8 — separates the classes completely:

• $\theta_k$9: $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$0 ebit/$$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$1
• $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$2: $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$3 ebit/$$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$4

This separation spans nearly three orders of magnitude and recovers the full 21/21 partition without threshold adjustment. Architectures in the two-layer class have effectively exhausted their entangling opportunity by $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$5, while every three-layer architecture remains actively replenishing the interface. The rooted classification is therefore not merely a property of a restricted layer count but a directly resolvable signature of optimized continuous dynamics.

Limitations and open questions

Several scope restrictions are stated plainly. The Gaussian theorem applies only to quadratic dynamics; non-nearest-neighbor Ising couplings acquire Jordan–Wigner strings, so the architectural results rest on exhaustive enumeration rather than analytic proof. The rooted-topology classification is exact for the 256-graph $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$6 tree–tree family but is not generalized beyond it, and the paper identifies turning replenishment structure into a general non-Gaussian bound on repeated interface utilization as the central open problem. The two-layer obstruction is explicitly not a no-go theorem for arbitrary continuous controls, and the continuous-envelope value $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$7 is bounded only numerically. The two-time diagnostic was validated on a symmetry-reduced 21-orbit family with a representative (not orbit-exhaustive) regularization scan, and the post-scoring single-time gap of 0.286 ebit does not meet the conservative pre-specified $$\sum_{k=1}^{m}|\dot\theta_k(t)| \leq \frac12\|K_{AB}(t)\|_* = \Lambda_\partial(t).$$8 criterion — the dynamical statistic, not the snapshot, carries the classification claim. Peak amplitude, bandwidth, and wall-clock duration are acknowledged as separate resource dimensions not optimized here.

Conclusion

The paper establishes a clean decomposition of entanglement-generation speed into an exactly characterized capacity term and an architecture-dependent replenishment term. In the Gaussian sector, the nuclear-norm bound with explicit chain saturation converts interface capacity into certified minimum interaction times. Holding the interface fixed then isolates replenishment as a genuine architectural resource, exactly classified by rooted topology in a finite family and recoverable dynamically through a time-resolved VEEF growth statistic across all reduced orbits. The progression — capacity, rooted replenishment, dynamical identification — provides both a rigorous baseline and a numerical probe of how internal architecture governs sustained use of a fixed quantum interface.