Gaussian-measurement equality with the Holevo bound

Problem

When does the optimal single-copy Gaussian-measurement cost equal the Holevo bound? Give necessary and sufficient conditions in terms of the local first moments, covariance, their first derivatives, and the weight matrix defined below. Let \(\rho_\theta\) be a smooth faithful \(n\)-mode Gaussian model, with integers \(n,p\geq1\) and \(\theta\in\Theta\subset\mathbb R^p\) for an open set \(\Theta\). Write \(R=(q_1,p_1,\ldots,q_n,p_n)^{\mathsf T}\) for the canonical quadrature vector. Its mean vector and covariance are defined in Eq. (1). The commutation convention is Eq. (2).

\begin{equation} \begin{aligned} (d_\theta)_j&:=\operatorname{Tr}(\rho_\theta R_j),\\ (V_\theta)_{jk}&:=\frac12\operatorname{Tr}(\rho_\theta\{R_j-(d_\theta)_j,R_k-(d_\theta)_k\}). \end{aligned} \tag{1} \end{equation}

\begin{equation} [R_j,R_k]=i\Omega_{jk},\qquad \Omega:=\bigoplus_{j=1}^{n}\begin{pmatrix}0&1\\-1&0\end{pmatrix}. \tag{2} \end{equation}

Assume the symmetric logarithmic derivative quantum Fisher information matrix is nonsingular. Let \(W\) be a real positive-definite \(p\times p\) weight matrix.

Define the optimal single-copy Gaussian-measurement cost by Eq. (3).

\begin{equation} C_G(\theta,W):=\inf_{M\ \mathrm{Gaussian}}\operatorname{tr}(WF_M(\theta)^{-1}), \tag{3} \end{equation}

where \(F_M\) is the classical Fisher information matrix of measurement \(M\), and singular \(F_M\) has infinite cost. Gaussian measurements mean Gaussian-ancilla preparations followed by Gaussian unitaries and homodyne detection with arbitrary classical processing.

Define the Holevo bound \(C_H(\theta,W)\) as the infimum of Eq. (4).

\begin{equation} \operatorname{tr}(W\operatorname{Re}Z)+\|\sqrt W\operatorname{Im}Z\sqrt W\|_1 \tag{4} \end{equation}

The infimum is over Hermitian observables \(X_1,\ldots,X_p\) with finite second moments, subject to Eq. (5).

\begin{equation} \begin{aligned} Z_{jk}&:=\operatorname{Tr}(\rho_\theta X_jX_k),\\ \operatorname{Tr}(\rho_\theta X_j)&=0,\qquad \operatorname{Tr}((\partial_k\rho_\theta)X_j)=\delta_{jk}. \end{aligned} \tag{5} \end{equation}

The measurement is optimized locally at the specified \(\theta\); its setting is held fixed when taking derivatives for \(F_M\). The symmetric logarithmic derivatives \(L_j\) satisfy \(2\partial_j\rho_\theta=\rho_\theta L_j+L_j\rho_\theta\). Their Fisher matrix has entries \(\operatorname{Re}\operatorname{Tr}(\rho_\theta L_jL_k)\). The requested criterion must characterize \(C_G(\theta,W)=C_H(\theta,W)\), rather than merely restate these two optimizations. Equality of the infima is the target. A limiting sequence of Gaussian measurements counts, even if no individual measurement attains the value.

Source

This attainability classification is derived from the Gaussian-measurement restriction identified as an open direction in the Discussion of Chang, Genoni, and Albarelli [CGA26]. It is narrower than asking how to evaluate the Holevo bound itself.

Progress

The Holevo minimization for a faithful Gaussian model can be restricted exactly to centered linear and symmetrized quadratic quadrature observables. Chang, Genoni, and Albarelli prove the required invariant-subspace property in Derivations and give a finite semidefinite program in Eqs. (36) and (72) [CGA26]. This computes \(C_H\) from the local moments and derivatives. It does not show that a Gaussian measurement attains it.
For Gaussian shift models with parameter-independent covariance, Gaussian measurements attain the Holevo bound. Bradshaw, Lam, and Assad construct the measurement in Sec. III.B and extend it to arbitrary finite mode and displacement-parameter counts in Appendix C [BLA18]. A positive weight can be absorbed into an invertible linear reparameterization. The result does not cover general covariance parameters.

Comment

The unresolved question is a general criterion for equality with the restricted single-copy measurement cost. A finite optimization of the Holevo bound already exists. Quadratic optimizing observables can require non-Gaussian detection, and collective asymptotic attainability is a different property.

References

[CGA26]: S. Chang, M. G. Genoni, and F. Albarelli, “Efficiently Evaluating Holevo, RLD and SLD Cramér-Rao Bounds for Multiparameter Quantum Estimation with Gaussian States,” Communications Physics 9, 126 (2026).DOI arXiv
[BLA18]: M. Bradshaw, P. K. Lam, and S. M. Assad, “Ultimate Precision of Joint Quadrature Parameter Estimation with a Gaussian Probe,” Physical Review A 97, 012106 (2018).DOI arXiv

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