Operational sealing and quantitative failures of POVM form

POVM form implies sealing by linearity. Conversely, sealing makes $F_A(\rho)=\mu_{\mathfrak E}(A)$ well defined for any decomposition of $\rho$. Randomized mixtures make $F_A$ affine on the density matrices. An affine function on a finite-dimensional convex set with nonempty relative interior extends uniquely to its affine hull: choose an affine basis forming a simplex around a relative-interior point, match the function at the vertices, and extend the equality from that simplex to any other point along a segment from the interior point. Therefore $F_A(\rho)=\operatorname{tr}(E(A)\rho)$ for a unique Hermitian $E(A)$. Since $F_A$ takes values in $[0,1]$ on every pure state, $0\le E(A)\le I$, and $E(\mathcal D)=I$. Countable additivity of each $\mu_\psi$ makes $E$ weakly countably additive: for disjoint events the positive partial sums are bounded by $I$, and their limiting quadratic forms agree with those of $E$ of the union. This is a POVM.

Fix a finite set $F$ of pure projectors $\rho_a$ and values $h_a$. The finite linear program minimizing $t$ subject to $|h_a-\operatorname{tr}(H\rho_a)|\le t$ has dual

Taking traces gives $\sum_av_a=0$, so the positive and negative parts have the same mass, at most $1/2$. Dividing each by that mass gives two equal-density ensembles; conversely half the difference of any two such ensemble weight vectors is dual feasible. Thus finite linear-programming duality gives $\Gamma_F(h)=2d_F(h)$, also when both sides vanish.

Put $D=\sup_Fd_F(h)$. Clearly $D\le d(h)$. Fix a finite tomographically complete pure-state frame $F_0$. For any $\eta>0$, all the constraints on $F_0$ with error at most $D+\eta$ bound the coefficients of $H$; their solution set is compact. The constraint sets obtained by also adding any finite $F$ have the finite-intersection property, since $d_{F\cup F_0}(h)\le D$. Compactness supplies one $H$ satisfying every pure-state constraint with error $D+\eta$. Letting $\eta$ tend to zero gives $d(h)\le D$. Since every pair of finite ensembles is contained in some finite $F$, $\Gamma(h)=\sup_F\Gamma_F(h)=2D=2d(h)$. No continuity of $h$ was needed.

A minimizer for $d(h)$ exists by the same compactness argument. Its spectrum lies in $[-d(h),1+d(h)]$, as follows by evaluating at its eigenvectors. Clipping that spectrum to $[0,1]$ changes $H$ by operator norm at most $d(h)$ and gives an effect with uniform error at most $2d(h)$. This proves the upper effect bound; the lower follows because effects are a subset of Hermitian matrices.

Finally, on any finite $F$ the maximization over pairs of ensemble weights is a linear program with nonnegative variables and constraints $\sum_aq_a=\sum_ar_a=1$ and $\sum_a(q_a-r_a)\rho_a=0$. The rank is at most $d^2+1$, since the trace of the matrix constraint is redundant with the two normalization conditions. A maximizing basic feasible solution uses at most $d^2+1$ nonzero weights in total. Choose $F$ with $\Gamma_F(h)>c$ to obtain the stated witness. The difference of event probabilities lower-bounds total variation.

For a common dominating measure, let $B(P,Q)=\int\sqrt{dP\,dQ}$. Cauchy–Schwarz gives $\TV(P,Q)\le\sqrt{1-B(P,Q)^2}$, while $\TV(P,Q)\ge1-B(P,Q)$. Affinity multiplies under independent products, so $B(P^{\otimes n},Q^{\otimes n})\le(1-\delta^2)^{n/2}$. The optimal equal-prior error is $(1-\TV)/2$.