EXACT CLAIM

Fix a faithful one-mode Gaussian model at a local point, with two independent covariance derivatives, zero first-moment derivatives, and a positive-definite real weight W. The following classifies equality C_G=C_H, including equality of infima over limiting Gaussian measurements. Symplectically whiten V to vI, v>1/2. A rotation and invertible parameter change put the tangent plane into either (i) D1=Z,D2=X or (ii) D1=I+aZ,D2=X, a≥0, where Z=diag(1,−1) and X=[[0,1],[1,0]]. Use the correspondingly transformed weight. In case (i), equality holds exactly when W is scalar. In case (ii), set s=v²−1/4 and t=v²+1/4. Equality holds exactly when a>t/s and W12=0, W22/W11=m^−2, where k=sqrt((a+1)/(a−1)), x=(2v−k)/(4kv−2), h=x+1/(4x), e=x−1/(4x), and m=ah−e. Equality is attained by the pure Gaussian seed diag(x,1/(4x)) in case (ii), and I/2 in case (i). In every other case C_G>C_H. This is a scoped theorem, not a solution for arbitrary modes or displacement derivatives.

FULL FROZEN PROOF

All quantities and measurements below are local, with measurement settings fixed in differentiation. Center the quadratures at the specified mean.

1. Canonicalization. Put v=sqrt(det V) and S=sqrt(v)V^−1/2. Then det S=1, hence S is symplectic in two dimensions, and SVS^T=vI. A two-dimensional plane of symmetric matrices either is the traceless plane or intersects it in a line. In the latter case rotate and rescale that line's generator to X. Normalize another generator's identity coefficient to 1, then subtract its X component: the result is I+aZ. A rotation by π/2 and a sign change of the X parameter allow a≥0. This includes every trace-bearing plane. If original increments equal A times canonical increments, the canonical derivatives are D'_j=Σ_k D_k A_kj and the canonical weight is A^TWA. Both costs are invariant under this simultaneous transformation.

2. Gaussian reduction and attainment. Before classical processing, a fixed Gaussian measurement has normal output with mean Kd and covariance KVK^T+N, where N−iKΩK^T/2≥0. This follows directly by expressing the measured commuting quadratures as linear combinations of system and independent Gaussian-ancilla quadratures. The sign convention for the inequality is immaterial.

When K has rank two, invertible output transformations put it into the form [I;0]. Regress the first two noise coordinates on the remaining coordinates. The residual and the remaining coordinates are independent; the latter have a parameter-independent distribution. Thus the sufficient observation has covariance V+N_eff. The Schur complement gives N_eff−iΩ/2≥0, so N_eff>0 and det N_eff≥1/4. For singular regression blocks use their support: positivity makes the cross block vanish on their kernel. The pure seed N0=N_eff/(2sqrt(det N_eff)) satisfies N0≤N_eff. Adding independent normal noise realizes the original sufficient observation, so its Fisher information is no greater than that of N0. Rank-one K similarly reduces to one noisy quadrature, whose covariance-only Fisher matrix has rank at most one. Classical processing cannot improve Fisher information. Conversely every positive seed of determinant 1/4 is realizable by a pure Gaussian ancilla and commuting quadrature detection. No randomization over settings is used.

It therefore suffices to optimize over positive symmetric seeds N with det N=1/4. Write C=vI+N. Their Fisher matrix is F_jk=tr(C^−1D_jC^−1D_k)/2>0. Their efficient centered estimators are f_j(z)=z^TB_jz−tr(B_jC), with B_j=(1/2)Σ_k(F^−1)_jk C^−1D_kC^−1. In particular tr(B_jD_k)=δ_jk and span{B_j}=C^−1 span{D_j} C^−1.

The cost is continuous. At any noncompact end the seed eigenvalues are x and 1/(4x), with x→0 after relabeling. In that eigenbasis,
F_jk=D_j11D_k11/[2(v+x)²]+D_j12D_k12/[(v+x)(v+1/(4x))]+D_j22D_k22/[2(v+1/(4x))²].
Uniformly over rotations the last two terms vanish; the first has rank at most one. Thus the smallest Fisher eigenvalue tends to zero and tr(WF^−1) tends to infinity. Compactness of the rotation angles proves that the Gaussian minimum is attained at a finite pure seed. Homodyne limits cannot supply additional equality cases.

3. A necessary noise condition. Dilate a pure seed using independent ancillary quadratures r with covariance N and commutator −iΩ. Then z=R+r commutes. The system observables associated with the efficient estimators are Q_j=R^TB_jR−tr(B_jvI). They are centered and locally unbiased. On the product state define E_j=f_j−Q_j and K_jk=⟨E_jE_k⟩. Ancillary centering gives ⟨Q_jE_k⟩=0, hence F^−1=Z+K, where Z_jk=⟨Q_jQ_k⟩ and K≥0. Therefore Im K=−Im Z.

For any Hermitian positive semidefinite two-by-two K and real W>0,
tr(W Re K)≥2sqrt(det W)|Im K12|.
Indeed apply tr B≥2|Im B12| to B=sqrt(W)Ksqrt(W)≥0. Consequently the measurement cost is at least the Holevo objective of Q, and hence at least C_H. Equality forces equality in this matrix inequality, which implies rank K≤1.

This rank condition forces tr(NB1)=tr(NB2)=0. To see it, split E_j into 2R^TB_jr and r^TB_jr−tr(B_jN). These have orthogonal Gram matrices because the system mean vanishes. The ancillary linear Gram matrix is rank one, N−iΩ/2=uu†. The first summand's Gram rank is the rank of the pair B1u,B2u: the system linear Gram matrix vI+iΩ/2 is strictly positive by faithfulness. Thus rank K≤1 forces these vectors to be complex dependent. Write N=SS^T/2 with real symplectic S. Multiplying both vectors by S^T reduces the assertion to symmetric real matrices B'_j=S^TB_jS acting on u0=(1,±i)^T/sqrt(2). Such a matrix satisfies B'u0=r u0+z conjugate(u0), with r=tr(B')/2 real. If dependent vectors have either r nonzero, their proportionality factor is real and the symmetric matrices are real multiples, contradicting independence of B1,B2. Both traces therefore vanish. Since tr(B'_j)=2tr(NB_j), the stated condition follows.

4. Solving the seed condition. It is equivalent to tr(HD_j)=0 for H=N(vI+N)^−2. If the seed eigenvalues are x,y with xy=1/4, then
x/(v+x)²−y/(v+y)²=(x−y)(v²−1/4)/[(v+x)²(v+y)²].
Thus H is scalar exactly when N is scalar; otherwise they share their distinct eigendirections. For the trace-free plane, the two constraints force N=I/2. For the trace-bearing plane, tr(HX)=0 forces N diagonal, and scalar N fails the other constraint. Write N=diag(x,1/(4x)). The remaining equation is
(1+a)x/(v+x)²+(1−a)/(4x(v+1/(4x))²)=0.
It requires a>1 and, taking positive square roots, becomes 2(v+x)/(4vx+1)=k, with k=sqrt((a+1)/(a−1)). Its unique positive solution is x=(2v−k)/(4kv−2), and exists exactly when k<2v, equivalently a>t/s. It satisfies 0<x<1/2. With h=x+1/(4x), e=x−1/(4x), direct expansion of the same equation gives th+ase+v=0.

5. Weight and unrestricted optimality. Define T=(q²+p²)/2−v, A0=(q²−p²)/2 and S0={q,p}/2. Gaussian fourth moments give their complex Gram matrix with diagonal (s,t,t), off-diagonal ⟨A0 S0⟩=iv, and all other upper off-diagonal entries zero. Their expectation derivatives in the trace-bearing canonical parameters are (1,a,0) and (0,0,1).

At the seed just determined, put m=ah−e>0. The efficient system observables are Q1=(−eT+hA0)/m and Q2=S0. Indeed these are unbiased and their matrices are trace-orthogonal to N. That orthogonal space has dimension two and, by the seed condition, equals the efficient span from step 2. For explicit checking, their matrices are B1=diag(y,−x)/m and B2=X/2, with y=1/(4x). Classical Gaussian fourth moments, subtracting the system Gram matrix, give
K=vh [[m^−2,−i/m],[i/m,1]].
For example K22=(v+x)(v+y)−t=vh; K11=vh/m² follows from xy=1/4 and h²−e²=1; the imaginary entry follows from ⟨Q1Q2⟩=ivh/m.

The noise inequality is saturated precisely when
W11/m²+W22=2sqrt(det W)/m.
The arithmetic–geometric mean inequality and det W≤W11W22 show that this is equivalent to W12=0 and W22/W11=m^−2.

It remains essential to check the unrestricted Holevo optimum. Use the exact linear/quadratic invariant-subspace restriction supplied in the problem. Linear terms have zero constraint derivatives and are complex-orthogonal to centered quadratics by Gaussian odd-moment identities. Their Gram contribution is positive semidefinite and cannot lower the Holevo objective: apply the noise inequality above and the reverse triangle inequality. Thus only T,A0,S0 need be retained.

Every unbiased pair in the trace-bearing case is P1=(1−aβ)T+βA0, P2=−aγT+γA0+S0. For W=diag(w,w/m²), its Holevo objective divided by w is
s−2asβ+(t+a²s)β²+[(t+a²s)γ²+t]/m²+2v|β|/m.
The minimum has γ=0 and β=(as−v/m)_+/(t+a²s). The seed identity yields (t+a²s)h=asm−v, so this minimizer is β=h/m>0, exactly the pair Q. Noise saturation therefore proves sufficiency for the trace-bearing equality conditions.

For the trace-free plane, unbiased quadratics are A0+cT and S0+dT. The additional real Gram contribution is s(c,d)^T(c,d), while the imaginary entry remains v. Thus C_H=t tr W+2v sqrt(det W), attained by A0,S0. The only possible equality seed, N=I/2, gives noise v[[1,−i],[i,1]]. Its noise inequality is saturated exactly when tr W=2sqrt(det W), or W is scalar; then the measurement attains C_H.

Finally, Gaussian attainment from step 2 makes failure of any necessary condition a strict gap, not merely nonattainment by individual settings. The proof uses the supplied quadratic restriction to certify the unrestricted optimum. The supplementary certificate is an exact arithmetic instance and is not needed for the general symbolic argument.

REMAINING GAP

The original classification for arbitrary modes and nonzero displacement derivatives remains unresolved. Primary-source comparison establishing literature novelty has not been completed; no priority or external confirmation is asserted.
