Gaussian holonomy beyond engineered weak couplings

A Gaussian holonomy beyond engineered weak couplings

This appendix proves the broader Gaussian statement used to place the principal theorem's restriction correctly. It also makes explicit why recurrence or a Lie-algebra calculation alone is insufficient for the exact claim.

Theorem A.1 (Connected fixed quadratic networks)

Let a positive harmonic holding matrix have nonzero off-block quadratic couplings on a connected graph of distinguishable particles. Assume full finite block-local quadratic controls, with no uniform amplitude ceiling. For any specified centered real positive Gaussian preparation with precision $A>0$, the physically implemented Gaussian ray returns, with the original holding Hamiltonian restored, have configuration group $SO(A)$. In whitened coordinates this is $SO(d)$. The original coupling strengths and frequencies need not be weak or commensurate.

A.1 An exact recurrent auxiliary seed

Retain all prescribed interparticle entries. Choose fixed within-particle off-diagonal trap entries so that the graph of nonzero scalar-coordinate entries is connected, and call the resulting off-diagonal matrix $C$. Diagonal local traps remain free. We need integer frequencies with suitable modal overlaps for $\diag(\lambda)+\delta C$.

Lemma A.2 (Nonvanishing path coefficients)

For a connected scalar-coordinate graph and a fixed root $r$, positive integers $n_1,\ldots,n_d$ can be chosen so that the root frequencies of lemma 4.1(i) are distinct and each eigenvector of $\diag(n_i^2)+\delta C$ has a nonzero component at $r$ for all sufficiently small nonzero $\delta$.

Proof

For distinct diagonal values $\lambda_i$, perturb the eigenvector of $\lambda_j$ with its $j$ component fixed to one. Iteration of the other component equations shows that the first possibly nonzero coefficient at $r$ has order $\delta^{\operatorname{dist}(r,j)}$ and equals

$$\sum_{\substack{\pi:r\leadsto j\\\text{shortest paths}}} \frac{\prod_{\{a,b\}\in\pi}C_{ab}} {\prod_{v\in\pi,\ v\ne j}(\lambda_j-\lambda_v)} .$$(A.1)

It is a nonzero rational function of the diagonal values. To verify nonvanishing without assuming signs of the edges, fix one shortest path and its distinct vertex diagonals, and send all off-path diagonal values to infinity. A shortest path has no chord. Thus the only path term involving exclusively its vertices is the selected path; its nonzero product survives, whereas every other term vanishes in this limit.

Clear denominators in the finitely many expressions (A.1). Their nonzero numerator polynomials, the distinct-diagonal factors, and the finitely many forbidden frequency equalities have a nonzero product after the substitution $\lambda_i=n_i^2$. A nonzero polynomial cannot vanish on the entire positive integer lattice: induction on the number of variables reduces this to the finiteness of the roots of a nonzero univariate polynomial. Hence some integer tuple avoids every zero. The leading nonzero coefficients then prove the assertion for small $\delta$.

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Choose these integers. The diagonal retuning argument used in section 5 gives analytic functions $d_i(\delta)=n_i^2+O(\delta^2)$ for which $\diag(d_i(\delta))+\delta C$ has exactly the eigenvalues $n_i^2$. The $O(\delta^2)$ diagonal changes leave every leading $\delta^{\operatorname{dist}(r,j)}$ path coefficient unchanged, so the nonzero modal overlaps persist after retuning. For a sufficiently large finite positive integer $L$ set

Its interparticle entries are the original fixed entries of $C$, its frequencies are exactly $Ln_i$, and it is positive. All required changes from the original holding matrix are one-body trap changes.

In the auxiliary variables $\tau=Lt$ and $x=\sqrt L\,q$, the Schrödinger equation for this seed has kinetic term $-\Delta_x/2$ and stiffness $\diag(d_i(\delta))+\delta C$ with $\delta=L^{-2}$. A dimensionless local control $\widehat D(\tau)$ is the actual physical control $D(t)=L^2\widehat D(Lt)$. This is a coordinate calculation specifying finite physical controls, not a change of the kinetic term or an operation on the pair interaction.

By lemma A.2, the diagonal control at $r$ couples every modal pair. The squared-overlap matrix remains near the identity. Thus lemma 4.1 gives an identity neighborhood of exact symplectic endpoints at time $2\pi/L$. For each edge of a spanning tree of $C$, formula (5.1) gives a nonzero leading elementary rotation in these scaled coordinates. Its bracket-basis determinant remains nonzero for large finite $L$. The correction and exact group arguments of proposition 4.2, lemma 4.3 therefore prove $SO(d)$ configuration holonomy at the real ground Gaussian of $K_*$.

A.2 Transport back to the original preparation

Proof of theorem A.1

Join the original positive holding matrix to $K_*$ by a smooth positive interpolation of its local blocks; convexity of the positive cone permits this. Let $\mathsf S_P$ be the known symplectic propagator of this history. At $K_*$ every symplectic matrix is exactly reachable with endpoint holding restored, by the identity-neighborhood argument after theorem 5.2. Choose a compensating endpoint $\mathsf S_{\mathrm{tar}}\mathsf S_P^{-1}$, where $\mathsf S_{\mathrm{tar}}=\diag(Q,Q^{-T})$ and $Q=A_*^{-1/2}A^{1/2}$, with $A_*=K_*^{1/2}$. The combined history takes the original real Gaussian to the real ground Gaussian of $K_*$ modulo phase.

Its actual guided map is some global linear $L_P$ satisfying $L_P^TA_*L_P=A$, by (4.6). The real-endpoint time reversal gives a physical inverse map $L_P^{-1}$, restoring the original holding Hamiltonian. Conjugating the seed return group therefore gives $SO(A)$. Conversely every Gaussian ray return belongs to $SO(A)$ by (4.6). These transported loops return the designated ray; no claim that their full propagator is a phase times the identity is needed.

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For clarity, a useful general endpoint criterion is

It makes the endpoint differential onto on any nonempty time interval: an annihilating covector kills the analytic orbit $e^{-t\operatorname{ad}_{\mathsf A}}\mathsf B(D)$, hence all its Taylor coefficients; Cayley–Hamilton bounds the needed powers. This is a criterion for the derivative along the chosen reference, stronger than a bare dynamical Lie-algebra assertion.

If that reference is merely recurrent, submersion at time $T_0$ gives an endpoint neighborhood $\mathsf S_0(T_0)\exp B_\eta$. Append a holding wait to time $\tau>T_0$ for which $\mathsf S_0(\tau)$ is sufficiently close to $I$. The reachable set then contains $\mathsf S_0(\tau)\exp B_\eta$, which contains an exact neighborhood of $I$. This is how recurrence plus an open endpoint image can give exact reachability. Recurrence by itself supplies only approximations. In (A.2) the recurrence is already exact, so no limiting protocol is used.