EXACT CLAIM

Two strict-gap theorems, with [q,p]=i and locally fixed Gaussian-measurement settings, including infima over limiting measurements.

(A) Let a smooth faithful one-mode Gaussian model have three parameters, zero first-moment derivatives at the specified point, and three linearly independent symmetric covariance derivatives. For every real positive-definite 3×3 weight W, C_G>C_H.

(B) Let a smooth faithful one-mode two-parameter Gaussian model, after centering its local mean, have V=vI₂ with v>1/2, ∂₁V=τI₂, ∂₂V=0, ∂₁d=(1,0)ᵀ and ∂₂d=(0,1)ᵀ, where τ≠0. For every real positive-definite 2×2 weight W, C_G>C_H.

Measurements are the Gaussian-ancilla, fixed Gaussian-unitary and homodyne measurements in the question, with arbitrary subsequent parameter-independent classical processing. The conclusions do not assume that an optimal measurement has finite squeezing. Part (A) closes the missing parameter-count case when combined with the attributed p=1 and p=2 covariance-only project baselines; those baselines and the published two-parameter Holevo formulas are not claimed as new results.

FULL FROZEN PROOF

All statements concern one fixed local point. Write R for the centered system quadratures. A fixed displacement used for centering does not alter any local derivative. In (A), let v=√det V and S=√v V^(−1/2). Since det S=1, S is symplectic in dimension two; replacing R by SR gives V=vI. Faithfulness gives v>1/2. Covariance derivatives transform as D_j↦SD_jSᵀ and remain independent. No parameter change is needed. Under any parameter change θ=Aη, derivatives transform by A and the weight by AᵀWA.

1. General PSD noise equality lemma.
Let K=A+iB≥0 be a Hermitian p×p matrix, with A real symmetric and B real skew-symmetric. Then tr A≥||B||₁. Moreover, equality implies rank K≤floor(p/2).
Indeed put H=iB and let sign(H) vanish on its kernel. Since sign(H) is purely imaginary and skew-symmetric, tr(sign(H)A)=0. Therefore tr(sign(H)K)=tr|H|. As I−sign(H)≥0, the inequality follows. Equality implies that K is supported on ker(I−sign(H)), the positive spectral subspace of H. Nonzero eigenvalues of H occur in opposite pairs, so this subspace has dimension rank(B)/2≤floor(p/2). Applying this argument to √W K√W proves
tr(W Re K)≥||√W Im K√W||₁,
and saturation has the same rank consequence. This is a p-dimensional statement, not a transfer of the two-dimensional determinant argument.

2. Gaussian measurement reduction.
Before classical processing a Gaussian measurement has a normal outcome with mean K₀d and covariance K₀VK₀ᵀ+N₀, where N₀−iK₀ΩK₀ᵀ/2≥0. This follows by writing its commuting measured quadratures as system quadratures plus independent Gaussian-ancilla quadratures.
If K₀ has rank two, an invertible output change makes it [I;0]. Regressing the first two noise coordinates on the remaining coordinates leaves a sufficient observation z with covariance C=V+N. The remaining coordinates have parameter-independent distribution. The Schur complement gives N−iΩ/2≥0, hence N>0 and det N≥1/4. Singular regression blocks are handled on their support; positivity annihilates cross blocks on the kernel.
The seed N_p=N/(2√det N) satisfies det N_p=1/4 and N_p≤N. Adding independent Gaussian noise realizes the original observation from this pure-seed measurement, so Fisher information cannot improve. Rank-one K₀ similarly reduces to a noisy quadrature, dominated by noiseless homodyne. Rank-zero observations have zero information. Subsequent classical processing cannot increase information.
Conversely, every positive N with det N=1/4 is realized by a pure Gaussian ancilla: take ancillary quadratures r with covariance N and commutator −iΩ; then z=R+r commutes. Consequently, it suffices to consider pure seeds and homodyne boundaries.

3. Dilation noise and the Holevo comparison.
At a pure seed, let F be the nonsingular classical Fisher matrix. Its efficient locally unbiased centered estimators f_j are F^(−1) times the normal scores, so their covariance is F^(−1). Write
f_j(z)=a_jᵀz+zᵀB_jz−tr(B_jC),
with real a_j and symmetric real B_j. Define system observables
Q_j=a_jᵀR+RᵀB_jR−tr(B_jV).
Taking the ancillary expectation shows that these are centered and locally unbiased. With E_j=f_j−Q_j on the product system–ancilla state, ancillary centering gives ⟨Q_jE_k⟩=0. Thus
F^(−1)=Z+K,  Z_jk=⟨Q_jQ_k⟩,  K_jk=⟨E_jE_k⟩≥0.
In particular Im K=−Im Z. The lemma gives
tr(WF^(−1))≥tr(W Re Z)+||√W Im Z√W||₁≥C_H.
If the first inequality is strict, the measurement cost is strictly above C_H.
The noise decomposes as
E_j=a_jᵀr+2RᵀB_jr+[rᵀB_jr−tr(B_jN)].
These three families have mutually orthogonal Gram contributions: use zero system mean and vanishing odd Gaussian moments. The system linear Gram matrix M=vI+iΩ/2 is positive definite. The ancillary linear Gram matrix N−iΩ/2 has rank one and can be written uu†, where Re u and Im u are linearly independent real vectors. The bilinear contribution therefore has rank equal to the complex rank of the vectors B_j u (or their conjugates, depending on the Gram convention). The positive definiteness of M ensures that no additional kernel is introduced.

4. Proof of (A) at finite seeds.
With zero mean derivatives, the normal scores are quadratic and
B_j=(1/2)Σ_k(F^(−1))_jk C^(−1)D_kC^(−1).
The three D_k form a basis of Sym₂(R); hence F>0 and the three B_j also form such a basis. Their vectors B_j u span C² over C. For example, the complex span of real symmetric matrices contains diag(1,0), diag(0,1) and the off-diagonal symmetric matrix; applying these to a vector whose real and imaginary parts are independent produces two independent vectors. Thus the bilinear noise Gram matrix has rank two. The full K, being its sum with other PSD Gram matrices, has rank at least two.
For p=3, saturation in Step 1 would require rank K≤1. Therefore every finite pure-seed cost is strictly above C_H.

5. Compactification and equality of infima in (A).
Parametrize pure seeds by rotations O and eigenvalues x,1/(4x), with 0<x≤1/2. Adjoin x=0; together with the compact rotation circle this is a compact parameter space. In the seed eigenbasis, for covariance-only tangents,
F_jk=D_j11 D_k11/[2(v+x)²]+D_j12 D_k12/[(v+x)(v+1/(4x))]+D_j22 D_k22/[2(v+1/(4x))²].
As x→0 this converges uniformly in rotation to a matrix of rank at most one. Thus tr(WF^(−1))→∞ for p=3: the smallest eigenvalue of F tends to zero and W>0. The finite-seed cost is continuous and finite somewhere, so its minimum is attained away from this boundary. Step 4 then proves C_G>C_H, not just nonattainment.

6. Proof of (B) at finite seeds.
At V=vI, the normal scores have linear coefficients C^(−1)e₁,C^(−1)e₂ and only the first score has a quadratic coefficient, namely (τ/2)C^(−2). Therefore the efficient estimators have linearly independent real linear coefficients a₁,a₂, and
B_j=β_j B₀,  β_j=(F^(−1))_j1,  B₀=(τ/2)C^(−2).
Here F>0, β is a nonzero real vector, and B₀ is invertible. The bilinear noise Gram matrix is a strictly positive multiple of ββᵀ. The ancillary-linear contribution is rank one, with support generated by the vector whose components are a_jᵀu, up to conjugation. This vector cannot be a complex multiple of β: its real and imaginary parts are linearly independent because the real coefficient matrix with rows a_jᵀ is invertible and Re u, Im u are independent. These two rank-one contributions consequently have distinct supports and their sum has rank two. Hence rank K=2, whereas saturation for p=2 requires rank at most one. Every finite pure seed therefore has cost strictly greater than C_H.

7. Informative homodyne boundaries in (B).
A boundary observation is y=lᵀR, with l=(c,s)ᵀ, c²+s²=1. Locally its mean derivatives are c,s and variance derivatives τ,0. Its scores are
c y/v+τ(y²−v)/(2v²),  s y/v.
The Fisher matrix is nonsingular exactly when s≠0. In that case the efficient estimators can be written
f_j=α_j y+β_j(y²−v),
where α=(0,1/s)ᵀ and β=(1/τ,−c/(sτ))ᵀ; these coefficients follow directly from local unbiasedness and span the two scores.
Set T=(q²+p²)/2−v and replace these estimators by system observables Q_j=α_j y+β_jT. Since ∂₁⟨T⟩=τ and ∂₂⟨T⟩=0, this pair is centered and locally unbiased. Gaussian moments give Var(y)=v, Var(y²−v)=2v² and Var(T)=v²−1/4. The complex cross moment of y and T vanishes. Both the original estimator Gram matrix and the Q Gram matrix have zero imaginary part, since within each family every component is a linear combination of the same y and the same centered quadratic.
Thus the Holevo objective of Q is strictly smaller than the homodyne cost by the exact quantity
(v²+1/4) βᵀWβ>0.
The comparator is zero; strictness follows from v²+1/4>0, W>0 and β₁=1/τ≠0. This proves strictness at every informative homodyne boundary without presuming divergent boundary cost.

8. Compactification for (B).
For finite seeds the Fisher matrix is
F_jk=e_jᵀC^(−1)e_k+(τ²/2)δ_j1δ_k1 tr(C^(−2)).
On the compactified seed space of Step 5, C^(−1) extends continuously to llᵀ/v. The displayed F therefore extends continuously to exactly the homodyne Fisher matrix of Step 7. Define the cost as infinity at singular F. It is lower semicontinuous on this compact space, because approaching singular F makes tr(WF^(−1)) diverge. A finite minimum exists. At an interior minimizer Step 6 is strict; at a boundary minimizer Step 7 is strict. Hence C_G>C_H.

These proofs construct Holevo-feasible observables directly and need neither a numerical optimization nor the unrestricted quadratic-subspace reduction. The new p=3 conclusion combines with, rather than changes, the supplied scalar and two-parameter covariance-only criteria. No statement about arbitrary mixed tangent spaces or multiple modes follows.

REMAINING GAP

The original arbitrary-n, arbitrary mixed-tangent necessary-and-sufficient classification remains open. The p=1 and p=2 covariance-only classifications remain attributed project baselines; their proofs are not resubmitted as new results here.
