For every faithful n-mode, p-parameter Gaussian model in the question, with nonsingular SLD Fisher information and real W>0, the following finite feasibility criterion is necessary and sufficient for C_G=C_H. All data are evaluated locally. Write k=2n, r=R-d, m_j=∂_j d, D_j=∂_j V, and Γ=V+iΩ/2. Choose a real trace-orthonormal basis B_α of symmetric k×k matrices. Define h=k+k(k+1)/2 centered observables T=(r_a,r^T B_α r−tr(B_α V)). Define the Hermitian h×h matrix S and real h×p matrix J by
S_ab=Γ_ab, S_aα=S_αa=0, S_αβ=2tr(B_α Γ B_β Γ^T), J_aj=(m_j)_a, J_αj=tr(B_α D_j).
The transpose Γ^T, not an adjoint, is intentional. Set P=I_h−J(J^T J)^{-1}J^T.
The unknowns are real symmetric G of size k, real symmetric E of size p, and Hermitian Y of size p. Form
F_jl=m_j^T Gm_l+tr(GD_jGD_l)/2,
L_aj=(Gm_j)_a, L_αj=tr(B_α GD_jG)/2,
A=LE, Q=E−A^TSA.
The criterion is exactly
G>=0, Γ^{-1}−G>=0, FE=I_p, Q>=0, Y>=0, ReY=W, YQ=0, P Re(SAY)=0.
All complex inequalities mean Hermitian positive semidefiniteness. No optimal cost is an input. When feasible, put H=G^{1/2}. An attaining measurement has commuting output y=Hr+z, where independent ancillary Gaussian quadratures z have covariance N=I−HVH^T and commutator [z_a,z_b]=−i(HΩH^T)_ab. Such an exact finite Gaussian-ancilla/homodyne realization exists even for singular G. Its Fisher matrix is F and the common cost is tr(WE). Indeed C_G always has an attaining measurement under the operational definition in the question. This is a finite semialgebraic optimality certificate, not a claimed efficient algorithm or closed-form geometric classification.

1. Polynomial moment space. Wick contraction gives S_uv=Tr(ρT_uT_v), with the entries stated above. Faithfulness implies Γ>0 and S>0: a nonzero linear combination of the independent centered polynomials cannot have zero squared norm in a faithful state. J contains their expectation derivatives, with T held fixed. Gaussian SLDs belong to this real linear/quadratic space and have coefficients (ReS)^{-1}J. Consequently the nonsingular-QFI assumption implies that J has full column rank.
We use the exact invariant-polynomial-space restriction supplied in the CGA26 progress. Its mechanism also explains its applicability here. In Williamson coordinates a faithful Gaussian density is proportional to exp(−Σ_a ε_a a_a†a_a), with ε_a>0. Modular action preserves the linear/quadratic monomials. The bounded real commutation operator defined through ReTr(ρU D(V))=ImTr(ρUV) therefore preserves their real Hermitian span: on each monomial/adjoint pair its coefficients are hyperbolic tangents of half the modular frequency. Real orthogonal projection onto this span is thus orthogonal also for the imaginary inner product. It preserves the derivative constraints, since the SLDs lie in the span. Projecting any admissible tuple splits its complex Gram matrix into the projected Gram matrix plus a PSD residual. The Holevo functional is monotone under such additions by the representation in step 4. Hence the exact Holevo problem uses real coefficient matrices A satisfying J^TA=I and Gram matrix A^TSA. The invariant-space reduction and Gaussian SLD structure are background results, not claimed as new here.

2. Exact compact precision domain. Before classical processing, every measurement in the specified Gaussian-ancilla/unitary/homodyne class has Gaussian outcome mean Kd+c and covariance C=KVK^T+N, where N−iKΩK^T/2>=0. Indeed its measured commuting quadratures are linear combinations of system and independent ancillary quadratures. Remove deterministic redundant outputs, whose support is parameter independent, so C>0. Set H=C^{-1/2}K. Then I−HΓH^T>=0. The Schur complement gives Γ^{-1}−H^TH>=0. Thus its precision G=H^TH belongs to the compact set K={G real symmetric: G>=0, Γ^{-1}−G>=0}. Differentiating its Gaussian distribution, with measurement settings held fixed, gives exactly F(G) above. Arbitrary classical processing cannot increase Fisher information and therefore cannot improve the cost for W>0.
Conversely, given any G in K, choose H=G^{1/2} and N=I−HVH^T. The same Schur complement yields N−iHΩH^T/2>=0. This uncertainty inequality permits an exact finite ancillary realization, including degeneracy. Put C_z=−HΩH^T. Positivity of N+iC_z/2 implies ker N⊆ker C_z. On the support of N whiten its covariance to I and orthogonally reduce the transformed C_z into blocks cΩ and zero blocks. Positivity gives |c|≤2. A nonzero block is realized using the two quadratures of one independent Gaussian mode with covariance I_2/|c|, scaled by sqrt(|c|), with orientation chosen for the sign of c. That mode is physical because 1/|c|≥1/2. Each remaining commuting unit-variance coordinate is a suitably scaled position quadrature of its own vacuum mode. Zero-variance coordinates are zero operators. Undo the transformations. This constructs z with exactly the required covariance and commutator, without infinitely squeezed ancillary states.
The rows defining y=Hr+z commute. They are independent because their covariance is I. Every finite independent isotropic row system extends to a symplectic basis; a Gaussian unitary followed by homodyne therefore detects y exactly.
The set K is closed and bounded. It contains a sufficiently small G=εI>0. For this G, F>0: its quadratic form is a sum of a squared mean norm and a squared symmetric covariance norm, and J has full rank. F is continuous on K. Furthermore tr(WF^{-1})≥λ_min(W)/λ_min(F), so bounded-cost sequences cannot approach singular F. Compactness proves that the Gaussian infimum is attained, including all singular-precision boundaries.

3. Score pullback. For the normalized output, centered at the specified point, its local score is
s_j(y)=(Hm_j)^Ty+[y^THD_jH^Ty−tr(HD_jH^T)]/2.
Its covariance is F. Taking ancillary expectation pulls s_j back to Σ_u T_u L_uj: the ancillary covariance cancels its constant term, leaving the centered system quadratic with coefficient GD_jG/2. If E=F^{-1}, the efficient local estimator t=Es has covariance E, is locally unbiased, and pulls back to observables with coefficient matrix A=LE. Direct substitution gives J^TL=F, hence J^TA=I. The measurement pullback is completely positive; its matrix Schwarz inequality gives E−A^TSA>=0. These assertions concern fixed local estimators and fixed measurement settings, not differentiated settings.

4. Holevo SDP and complementarity. For every Hermitian Z, the Holevo functional equals
min{tr(WT): T real symmetric, T>=Z}
=max{tr(YZ): Y>=0, ReY=W}.
To verify the primal formula, subtract ReZ and conjugate by sqrt W. The remaining problem is min tr U with U real symmetric and U>=iB for a real skew B. Conjugation gives U>=−iB as well. Positive and negative spectral projectors of iB give tr U≥||iB||_1; equality is attained by the real symmetric U=|iB|. The displayed maximization is its SDP dual, with strict primal feasibility.
Consequently the polynomial Holevo problem is the finite SDP
min tr(WT), J^TA=I, M(T,A)=[[T,A^T],[A,S^{-1}]]>=0,
where A and T are real. It is strictly feasible: choose A_0=J(J^TJ)^{-1} and T=tI with t>λ_max(A_0^TSA_0). Its objective is bounded below by zero, so SDP strong duality supplies an attained dual optimum. A primal optimum also exists: bounded objective bounds T>=0, and T>=A^TSA bounds A since ReS>0. A bounded minimizing sequence thus has a convergent feasible subsequence.
Let D>=0 be the Hermitian block multiplier for M. Stationarity in real T gives ReD_11=W. Stationarity in real A, along J^TΔ=0, gives P ReD_21=0. Complementarity is DM=0. At T=E set Y=D_11. The upper-right block equation gives D_12=−YA^TS; Hermiticity gives D_21=−SAY. The lower-right equation gives D_22=SAYA^TS, and the upper-left equation gives YQ=0. The remaining stationarity equation is precisely P Re(SAY)=0. Conversely the resulting block multiplier is PSD, since D=[I;−SA]Y[I,−A^TS]. This derives the signs and uses strict feasibility rather than assuming dual attainment.

5. Equivalence. Suppose C_G=C_H. Step 2 supplies an optimal Gaussian precision G with F>0. Step 3 supplies E=F^{-1} and A=LE, making (E,A) feasible for the Holevo SDP with the common optimal objective. Step 4 supplies an attained dual multiplier, hence Y and all the stated equations.
Conversely suppose the equations hold. F is PSD by its formula; FE=I implies F>0 and E=F^{-1}. Also J^TA=I. For any other real B with J^TB=I, put Δ=B−A. Then
tr(YB^TSB)−tr(YA^TSA)=2tr(Δ^T Re(SAY))+tr(YΔ^TSΔ)>=0.
The first term vanishes because PΔ=Δ and P Re(SAY)=0. The second is nonnegative because Y,S>=0. Moreover YQ=0 gives tr(YA^TSA)=tr(YE)=tr(WE), since E is real and ReY=W. The dual representation implies that every admissible polynomial Holevo cost is at least tr(WE), whereas E>=A^TSA gives the reverse bound. By step 1 this is the unrestricted Holevo value. The physically constructed measurement has cost tr(WE), proving equality and attainment.

6. Scope of the criterion. Complex PSD constraints can be replaced by real block PSD constraints. All displayed conditions then become finitely many polynomial equations and PSD inequalities in G,E,ReY,ImY, with coefficients determined by the local moments and derivatives. Thus the result is an algebraic feasibility certificate with no unexplained optimal values, not a restatement asking to compute and compare C_G and C_H. Its additional content beyond the supplied Holevo reduction is the compact exact physical precision parametrization, boundary attainment, and the explicit efficient-score restriction A=L(G)E combined with dual complementarity. It remains an implicit optimality classification and makes no claim of computational efficiency or a simpler geometric classification. External novelty has not been independently confirmed. The retained symbolic checker illustrates special instances; the proof does not depend on numerical checks.
