Single-seed post-processing of Gaussian POVM densities
- Field
- Topic
Problem
Is every POVM with Gaussian density a classical post-processing of one displaced Gaussian seed? Fix \(n\geq1\) bosonic modes with \([a_j,a_k^\dagger]=\delta_{jk}\). Let \((X,\Sigma)\) be a measurable outcome space and \(\mu\) a positive measure. Let \(x\mapsto\tau_x\) be a measurable family of Gaussian density operators. Assume \(E(B):=\int_B\tau_x\,\mu(dx)\) defines a POVM and \(E(X)=I\), with integrals in the weak operator sense. Define the displacements by Eq. (1).
The question asks for a Gaussian density operator \(\tau\) and a Markov probability kernel \(K\) satisfying Eq. (2). For each \(\alpha\), \(B\mapsto K(B\mid\alpha)\) is a probability measure on \((X,\Sigma)\). For each \(B\), \(\alpha\mapsto K(B\mid\alpha)\) is measurable.
Source
This is an editorial formulation distinguishing Gaussian POVM densities from a single covariant Gaussian observable. Holevo’s Secs. 2–3 describe the latter class, including the displaced-seed form in Eq. (9) and the type-1 construction [Hol21]. The universal post-processing claim is not attributed to Holevo.
Progress
The answer is negative. For one mode, set \(|\psi_0\rangle=|0\rangle\) and \(|\psi_1\rangle=\exp[(a^2-a^{\dagger2})/2]|0\rangle\). Randomly choose either displaced-seed measurement and retain the choice in the outcome. The density in Eq. (3) is Gaussian and integrates to \(I\).
\begin{equation} E(j,d^2\alpha)=\frac12D(\alpha)|\psi_j\rangle\langle\psi_j|D(\alpha)^\dagger\,\frac{d^2\alpha}{\pi},\qquad j\in\{0,1\}. \tag{3} \end{equation}Normalization follows from the covariant seed construction [Hol21].
Here is a direct proof that Eq. (2) cannot hold for Eq. (3). For this counterexample, the outcome spaces are standard Borel. Weighting a putative joint POVM by a faithful state gives a probability measure that admits conditional disintegration. A positive mixture can have a rank-one density only when almost all contributing operators have that same one-dimensional support. Thus a post-processing of a fixed seed could produce these rank-one densities only from displacements of one pure seed. Displacements preserve covariance. The vacuum and squeezed-vacuum densities have different covariance matrices, giving a contradiction. This counterexample and its proof are supplied here, rather than claimed as a theorem of the cited paper.
Comment
The explicit counterexample completely resolves this formulation. It is an editorial argument, not a separately peer-reviewed result. A POVM with Gaussian fine-grained densities need not have Gaussian densities after coarse-graining. This distinction does not contradict the published classification of Gaussian observables by their operator characteristic functions.