What reset preserves: full laws and exact counterexamples

In normal coordinates $q_+=(x+r)/\sqrt2$, $q_-=(x-r)/\sqrt2$, the plus vacuum has zero velocity. The exact minus Gaussian has width $b(t)$ and phase gradient giving $\dot q_-=(\dot b/b)q_-$. Thus $q_+(t)=q_+(0)$ and $q_-(t)=b(t)q_-(0)$, proving the identity endpoint. This calculation uses the complete wave during the pulse, not its endpoint matrix alone. The oscillator spectral phases give the separate quantum SWAP statement proved in the reset construction.

Use the standard Gaussian CDF $\Phi$ and density $\gamma$ defined above, and set $u=\Phi(x)$, $v=\Phi(r)$. Against the entrance product Gaussian use

Both marginals are uniform in quantiles, $0\leq f\leq2$, and the conditional distance is

The square root $\sqrt2|\cos\pi(u-v)|$ has genuine weak derivatives. Its physical Gaussian-relative Fisher information is finite; indeed

For the equality first integrate the unused quantile, then use $\int_{\mathbb R}\gamma^3=1/(2\pi\sqrt3)$. Zeros of $f$ cause no missing singular derivative: the displayed square root is locally Lipschitz, and the derivative formula holds almost everywhere. The identity endpoint retains the entire joint density and hence its conditional discrepancy.

Push the inequality of entrance measures through the common measurable path map; this proves domination on path space, not just at each time. For the second statement push both measures through $\mathrm{id}\times F$. Total variation contracts under measurable pushforward, and the image of the comparison is $\gamma_W\otimes F_\#\mu_N$. Apply the inverse map for equality in the invertible case.

The classical matrix equation has coefficient $A_0(t)+\lambda A_1(t)$. Its Dyson series converges uniformly on every compact complex gain set, bounded by the exponential of the integrated coefficient norm. Every entry of the finite-time symplectic matrix is therefore entire in $\lambda$. Write the final scratch annihilator in terms of both initial mode coordinates and momenta. Reset for every scratch state, with an initial archive vacuum, requires both coefficients of the initial scratch position and momentum to vanish. One can see this by displacing arbitrary scratch inputs: otherwise the supposedly fixed output vacuum would have a variable first moment. Vanishing on an open real interval forces both entire coefficients to vanish also at zero gain. There the scratch evolves by its own invertible two-dimensional symplectic block. A nonzero final annihilator row cannot annihilate that entire block, a contradiction.

Put $\nu=\sqrt{1+e}$. The exact phase-space evolution is $q(t)=q\cos\nu t+p\sin\nu t/\nu$ and $p(t)=p\cos\nu t-\nu q\sin\nu t$. Substitution into $a(t)$ gives (10.2). The resulting squeezed vacuum has vacuum probability $(1+|\beta|^2)^{-1/2}$: solving the transformed annihilator equation gives a normalized Gaussian, whose overlap with the original Gaussian evaluates to this value. If $0\leq v\leq1$, then $1-(1+v)^{-1/2}\geq v/4$, by multiplying out positive denominators or integrating $\tfrac12(1+s)^{-3/2}$ on $[0,v]$. Hence the off-vacuum norm is at least $|\beta|/2$. At the stated quarter-period this is $e/(4\sqrt{1+e})\geq e/5$.