A Gaussian meter breaks the classical entropy-floor witness
Result, 2026-10-28; written derivation, unrefereed. The classical entropy-floor class used as a witness in the canonical-averaging note is not closed under a single selective Gaussian position measurement. The initial body and pointer both obey the same positive entropy and covariance-action floor. The pointer has independent Gaussian error and kick with the minimum action product \(S_0\zeta\), and the body–pointer interaction is a reversible canonical shear. A single record event has probability at least \(9/32\), yet its body posterior can have covariance action arbitrarily close to zero. Both unread one-pair marginals remain in the entropy-floor class.
For a Gaussian body preparation the same meter does preserve the floor, even after conditioning on any measurable record event. This exact contrast identifies the hidden issue: the entropy floor admits non-Gaussian mixtures containing a very narrow component, concealed by a broad component. With \(\hbar=2S_0\zeta\), those mixtures are not Wigner functions of quantum states, although they are positive, have finite covariance and satisfy the same global entropy and covariance inequalities. A coherent-state overlap proves their nonquantumness below.
A narrower classical class—the \(W_2\) closure of finite mixtures of admissible Gaussians—survives this meter, affine canonical changes and copy-index mixers. It retains the supplied floor, but a smooth nonlinear canonical shear takes one of its Gaussian states outside the class. A three-point kernel test proves the failure without invoking a numerical calculation or a general no-go theorem.
The construction tests one specified classical instrument, not all physically possible readouts. Its initial body aperture becomes unbounded along the zero-action sequence. The entropy threshold is supplied, and no positive \(\hbar\) is derived. The underlying Gaussian equivalence has substantial prior art: Bartlett, Rudolph and Spekkens 2012 (publisher abstract read) recover Gaussian quantum statistics from a classical uncertainty condition plus maximum entropy at fixed moments. The two-pointer theorem already defeats a prior density ceiling under explicit instrument-access assumptions. Here the target is the later entropy witness and one meter; no literature priority is claimed.
1. Entropy floor and a canonical meter
Choose calibration units \(L_0,P_0>0\), \(S_0=L_0P_0\), and use dimensionless canonical variables \(Q=q/L_0\), \(P=p/P_0\) and similarly \(Y,\Pi\) for a separate pointer. A covariance matrix \(C\) of \((Q,P)\) has physical covariance action \(S_0\sqrt{\det C}\). Fix a supplied \(\zeta>0\) and write
\[ G_v(z)=\frac{1}{2\pi v}\exp\!\left(-\frac{|z|^2}{2v}\right),\qquad \mathcal E_\zeta=\left\{\rho\,dz\in\mathcal P_2(\mathbb R^2): h(\rho)\ge\log(2\pi e\zeta)\right\}, \tag{1} \]
where \(h(\rho)=-\int\rho\log\rho\,dz\) is dimensionless differential entropy. The Gaussian maximum-entropy inequality implies
\[ \rho\in\mathcal E_\zeta\quad\Longrightarrow\quad S_0\sqrt{\det\operatorname{Cov}_\rho(Q,P)}\ge S_0\zeta. \tag{2} \]
Prepare the pointer independently with density \(G_\zeta(Y,\Pi)\). Thus its error \(Y\) and kick \(\Pi\) have variances \(\zeta\) and their physical standard-deviation product is \(S_0\zeta\). The canonical interaction is
\[ Q'=Q,\quad P'=P-\Pi,\quad R=Y'=Y+Q,\quad \Pi'=\Pi. \tag{3} \]
It is the unit-strength Hamiltonian pulse generated by \(S_0Q\Pi\): the common dimensionless symplectic form is \(S_0(dQ\wedge dP+dY\wedge d\Pi)\). Equation (3) is invertible and preserves its four-dimensional phase volume. We record the pointer coordinate \(R\), retaining either its value or an event \(A\) of positive probability. The instruments and independent pointer preparation are explicit availability assumptions.
2. Exact Gaussian record closure
Theorem 1. Let the body input be a Gaussian with covariance \(C=\begin{pmatrix}a&c\\c&b\end{pmatrix}\), where \(a>0\) and \(\det C\ge\zeta^2\). Under (3), every exact pointer outcome has a Gaussian body posterior with covariance
\[ C_{\rm post}=\begin{pmatrix} a\zeta/(a+\zeta)&c\zeta/(a+\zeta)\\ c\zeta/(a+\zeta)&b+\zeta-c^2/(a+\zeta) \end{pmatrix},\quad \det C_{\rm post}=\zeta\frac{\det C+a\zeta}{a+\zeta}\ge\zeta^2. \tag{4} \]
For every Borel record event \(A\) with positive probability, the postselected body law also belongs to \(\mathcal E_\zeta\).
Proof. The output body covariance before reading is \(\begin{pmatrix}a&c\\c&b+\zeta\end{pmatrix}\), its covariance with \(R\) is \((a,c)^{\mathsf T}\), and \(\operatorname{Var}R=a+\zeta\). Gaussian conditioning gives (4). Its last inequality is equivalent to \(\det C\ge\zeta^2\). The conditional covariance does not depend on the outcome. Conditioning instead on \(A\) mixes those Gaussian conditional laws, all with the same covariance and Gaussian entropy \(\log(2\pi e\sqrt{\det C_{\rm post}})\). Entropy concavity gives the stated floor for the mixture. \(\square\)
Thus this meter is a classical realization of the quantum Gaussian update on the Gaussian sector. Closure for that sector does not settle closure for the larger class (1).
3. A non-Gaussian preparation selected by one ordinary record
For \(0<\delta\le1/2\) put
\[ s=\zeta\delta,\qquad b_\delta=\zeta\delta^{-3},\qquad d=\sqrt\zeta\,\delta^{-2},\qquad \rho_\delta=\tfrac34G_s(Q,P)+\tfrac14G_{b_\delta}(Q-d,P). \tag{5} \]
All variances are finite for each \(\delta>0\). By concavity of entropy,
\[ h(\rho_\delta)\ge\tfrac34\log(2\pi e s) +\tfrac14\log(2\pi e b_\delta) =\log(2\pi e\zeta). \tag{6} \]
The input body and pointer therefore both belong to \(\mathcal E_\zeta\). Take the finite-resolution record event
\[ A=\{|R|\le\sqrt\zeta\}. \tag{7}\]
Theorem 2. Let \(w_\delta\) be the posterior weight of the broad component in (5) after event (7), and let \(\kappa_\delta\) be the physical covariance action of the postselected body. Then
\[ \mathbb P(A)\ge\frac9{32},\qquad 0\le w_\delta\le\delta^{3/2}e^{-1/(4\delta)}, \tag{8} \]
and
\[ \frac{\kappa_\delta}{S_0\zeta}\le \sqrt{(\delta+\delta^2+5w_\delta) (1+\delta+w_\delta\delta^{-3})} \ \longrightarrow\ 0. \tag{9} \]
Consequently the posterior lies outside \(\mathcal E_\zeta\) for all sufficiently small \(\delta\), despite the fixed positive pointer floor.
Proof. In the narrow and broad components respectively,
\[ R_s\sim N(0,\zeta(1+\delta)),\qquad R_b\sim N(d,\zeta(1+\delta^{-3})). \tag{10} \]
Let \(p_s,p_b\) be their probabilities of \(A\). Since \(1/\sqrt{1+\delta}\ge1/\sqrt2\), the normal integral gives
\[p_s\ge \frac{e^{-1/4}}{\sqrt\pi}\ge\frac38.\tag{11}\]
For \(0<\delta\le1/2\), the broad normal density on the interval is at most its value at \(\sqrt\zeta\), so
\[ p_b\le\sqrt{\frac2\pi}\,(1+\delta^{-3})^{-1/2} \exp\!\left[-\frac{(\delta^{-2}-1)^2}{2(1+\delta^{-3})}\right] \le\delta^{3/2}e^{-1/(4\delta)}. \tag{12} \]
The second inequality uses \((1-\delta^2)^2/[2\delta(1+\delta^3)]\ge1/(4\delta)\). The total event probability is \((3p_s+p_b)/4\ge9/32\), and \(w_\delta=p_b/(3p_s+p_b)\le p_b\), proving (8).
For the narrow component, Gaussian conditioning on \(R=r\) gives conditional mean \(\delta r/(1+\delta)\) and variance \(\zeta\delta/(1+\delta)\). Averaging over \(|r|\le\sqrt\zeta\) yields
\[\mathbb E(Q^2\mid A,\text{narrow})\le\zeta(\delta+\delta^2).\tag{13}\]
For the broad component the conditional variance is \(\zeta/(1+\delta^3)\), and the conditional mean is \([\sqrt\zeta\,\delta+r]/(1+\delta^3)\). On \(A\) its absolute value is at most \(2\sqrt\zeta\), so the corresponding second moment is at most \(5\zeta\). The final momentum \(P'=P-\Pi\) is, within either component, independent of \((Q,R)\), centered, and has variance \(s+\zeta\) or \(b_\delta+\zeta\). Hence
\[ \operatorname{Var}(Q\mid A)\le\zeta(\delta+\delta^2+5w_\delta),\quad \operatorname{Var}(P'\mid A)\le\zeta(1+\delta+w_\delta\delta^{-3}). \tag{14} \]
The covariance determinant is no larger than the product of these variances, which proves (9). Since \(w_\delta\delta^{-3}\to0\), the bound tends to zero. Equation (2) excludes the posterior from the entropy-floor class. \(\square\)
The failure is selective. Before reading \(R\), the body law is the input law averaged over independent momentum translations, so entropy concavity gives \(h(Q,P-\Pi)\ge h(Q,P)\). The pointer’s unread law has \(R=Q+Y\) independent of \(\Pi\), and \(h(R)\ge h(Y)\) by the same argument. Both unread marginals therefore remain in \(\mathcal E_\zeta\); the full joint entropy is conserved by the canonical shear. Selective closure is a strictly stronger requirement than this unread evolution.
The counterexample uses no divergent quantity at any fixed \(\delta\). Its broad-component variance and position separation grow as \(\zeta\delta^{-3}\) and \(\sqrt\zeta\,\delta^{-2}\), respectively. Thus it does not defeat a separately imposed finite preparation aperture or energy budget uniformly in \(\delta\); the resource-relative distinction remains.
4. Why the positive classical density is not a quantum state
Set the physical reduced action to \(\hbar=2S_0\zeta>0\), so the commutator of the calibrated variables is \([\widehat Q,\widehat P]=i\hbar/S_0=2i\zeta\). Let \(\sigma_0\) be the pure minimum-uncertainty Gaussian state whose normalized Wigner function is \(G_\zeta\). If (5) were the Wigner function of a density operator \(\rho\), the Wigner trace identity would give
\[ \operatorname{Tr}(\rho\sigma_0) =2\pi\frac{\hbar}{S_0}\int\rho_\delta(z)G_\zeta(z)\,dz \ge \frac{3}{2(1+\delta)}>1,\qquad 0<\delta<\tfrac12. \tag{15} \]
Here \(\int G_sG_\zeta\,dz=[2\pi(s+\zeta)]^{-1}\), and the broad component contributes a nonnegative amount. Equation (15) is impossible because a density operator has expectation at most one in a normalized pure state. The overlap identity is the usual Wigner Hilbert–Schmidt formula; see Bartlett–Rudolph–Spekkens, §IV.B, equation (90) (passage read; the factor in (15) uses our unit-integral Wigner convention). This is an explicit positivity test beyond the entropy and covariance inequalities, not a new characterization of all Wigner functions.
Under the quantum identification above, the meter \(G_\zeta\) itself is a legitimate minimum-uncertainty state and (3) is the Wigner action of the quadratic unitary pulse. The counterexample cannot be imported as a quantum preparation: its concentrated component makes the full input operator nonpositive. In particular, positivity of a classical phase-space density, a covariance floor, entropy closure and Gaussian-meter back-action do not alone supply quantum state positivity.
5. A record-stable classical repair and its nonlinear obstruction
Let \(\mathcal C_\zeta\) be the finite convex mixtures of one-pair Gaussian probabilities with \(\det C\ge\zeta^2\), and let \(\overline{\mathcal C}_\zeta\) be its \(W_2\) closure in \(\mathcal P_2\).
Theorem 3. The class \(\overline{\mathcal C}_\zeta\) has physical covariance-action infimum \(S_0\zeta\) and is closed under affine canonical transformations, independent-copy orthogonal canonical mixing followed by retention of one pair, and the Gaussian meter (3) followed by any positive-probability Borel record event. Nevertheless, for every \(\lambda\ne0\) the classical canonical shear
\[ Q'=Q,\qquad P'=P+\lambda Q^2 \tag{16} \]
sends \(G_\zeta\in\overline{\mathcal C}_\zeta\) to a density outside that class, although the output remains in \(\mathcal E_\zeta\).
Proof of the closed operations. Entropy concavity and (2) put every finite Gaussian mixture in \(\mathcal E_\zeta\), and the latter is \(W_2\)-closed by the central-limit note, §6. Thus the closure has action at least \(S_0\zeta\); \(G_\zeta\) attains it.
An affine canonical map carries every admissible Gaussian to another. For one output of an orthogonal copy-index mixer, the covariance is \(\sum_j a_j^2C_j\) with \(\sum_j a_j^2=1\). Each \(C_j+i\zeta J\succeq0\), \(J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\), so their convex combination also obeys that inequality, equivalent in one pair to \(\det C\ge\zeta^2\). Finite input mixtures therefore give finite output mixtures. These operations are continuous in \(W_2\) on independent product laws, so they extend to the closure.
For a finite Gaussian mixture, Theorem 1 shows that conditioning on a record value gives a finite mixture of admissible Gaussians. Conditioning on a Borel event \(A\) gives an integral mixture over the record values. Its record distribution has finite second moment; approximation by finitely supported distributions makes this an element of the \(W_2\) closure. To extend the result to all input laws in the closure, write the event likelihood as
\[ \ell_A(q)=\int_A\frac{e^{-(r-q)^2/(2\zeta)}}{\sqrt{2\pi\zeta}}\,dr. \tag{17} \]
If \(A\) has positive probability, it has positive Lebesgue measure; \(\ell_A\) is bounded, continuous and strictly positive. The posterior is the input law weighted by \(\ell_A(Q)\), normalized, followed by convolution of \(P\) with the independent Gaussian kick. \(W_2\) convergence of inputs implies weak convergence plus uniformly integrable second moments, so these weighted normalized laws converge in \(W_2\); coupling the same kick preserves that convergence. Exact record values use the bounded positive Gaussian likelihood at that value. Hence all the claimed record outputs remain in \(\overline{\mathcal C}_\zeta\).
Proof of the nonlinear obstruction. Equation (16) is a global canonical map, generated by the pulse \(-S_0\lambda Q^3/3\). Its output is
\[ f_\lambda(q,p)=\frac{1}{2\pi\zeta} \exp\!\left[-\frac{q^2+(p-\lambda q^2)^2}{2\zeta}\right]. \tag{18} \]
The shear preserves entropy, so \(h(f_\lambda)=\log(2\pi e\zeta)\); it has finite covariance and belongs to \(\mathcal E_\zeta\).
Every member of \(\mathcal C_\zeta\) is a Wigner function of a positive quantum state when \(\hbar=2S_0\zeta\). For every Schwartz test state \(\psi\), its Wigner function is bounded and continuous, and the Wigner overlap with any member of \(\mathcal C_\zeta\) is nonnegative. This necessary condition is preserved by weak convergence, hence by \(W_2\) closure. We now violate it for (18).
The inverse Wigner kernel corresponding to (18) is continuous and equals
\[ K_\lambda(x,y)=\frac{e^{-(x^2+y^2)/(4\zeta)}}{\sqrt{2\pi\zeta}} \exp\!\left[\frac{i\lambda(x+y)^2(x-y)}{8\zeta}\right]. \tag{19} \]
If it came from a positive operator, every finite matrix \([K_\lambda(x_i,x_j)]\) would be positive semidefinite; smooth test bumps approximating point masses justify this necessary condition. Normalize the diagonal of the three-point matrix at \(x_1=-a,x_2=0,x_3=a\). The normalized off-diagonal entries have modulus one and cyclic phase \(-\lambda a^3/(4\zeta)\), so its determinant is
\[ 2\left[\cos\!\left(\frac{\lambda a^3}{4\zeta}\right)-1\right]. \tag{20} \]
Choosing \(a=(4\pi\zeta/|\lambda|)^{1/3}\) makes (20) equal \(-4\). A finite combination of smooth compactly supported bumps therefore has negative expectation against \(K_\lambda\). Its Wigner function is a bounded continuous test violating the necessary condition above, so \(f_\lambda\notin\overline{\mathcal C}_\zeta\). \(\square\)
This is a concrete boundary between the two classical repairs: the larger \(\mathcal E_\zeta\) survives this nonlinear Liouville shear but fails selective Gaussian recording; the smaller \(\overline{\mathcal C}_\zeta\) survives that record but not the shear. Neither statement excludes every other classical completion. In particular \(\zeta\) is supplied in both constructions.
The distinction also has an exact quantum form. The quantum pulse for \(-S_0\lambda Q^3/3\) sends the minimum Gaussian wavefunction to \(\psi(q)\exp[i\lambda q^3/(6\zeta)]\). Its density kernel is
\[ K_{\rm quantum}(x,y)=K_\lambda(x,y) \exp\!\left[\frac{i\lambda(x-y)^3}{24\zeta}\right]. \tag{21} \]
The additional off-diagonal phase is absent in the purely classical shear. Equation (21) is a direct calculation at fixed physical \(\hbar=2S_0\zeta\). On separations \(|x-y|=O(\sqrt\zeta)\) the phase is \(O(|\lambda|\sqrt\zeta)\) as \(\hbar/S_0=2\zeta\to0\). The three-point test at fixed \(\lambda\) instead uses
\[ \frac a{\sqrt\zeta}= \left(\frac{4\pi}{|\lambda|}\right)^{1/3}\zeta^{-1/6},\qquad \frac{K_\lambda(a,a)}{K_\lambda(0,0)} =\exp\!\left[-\frac12\left(\frac{4\pi}{|\lambda|}\right)^{2/3} \zeta^{-1/3}\right]. \tag{22} \]
Thus the normalized determinant stays at \(-4\) while the test points move into suppressed Gaussian tails. It gives no lower bound on an operational distance that remains finite in the large-action limit. As in the earlier cusp comparison, \(S_0\) must describe a physical family of characteristic actions for \(\hbar/S_0\to0\) to mean more than a change of units.
6. Consequence for STATE
The entropy-floor witness satisfies the existing copy and actual-cusp transport tests, but a single allowed Gaussian meter and a positive- probability coarse record break its selective closure. The Gaussian-mixture repair survives the record and copy tests but fails a nonlinear classical canonical shear. Quantum operator positivity blocks both exhibited nonphysical densities, with \(\hbar\) supplied; quantum nonlinear evolution also differs from the classical shear by (21). Independently deriving a state and dynamics law, its physical scale and closure for actual records remains open.