Information Capacity of State Ensembles and Observables

Given a quantum state ensemble \(\mathcal{E}\) and quantum observable \(\mathcal{M},\) one can define the Shannon information \(I(\mathcal{E},\mathcal{M})\) and introduce the two fundamental information quantities. The accessible information \(A(\mathcal{E})\) of the ensemble \(\mathcal{E}\) is defined as supremum of \(I(\mathcal{E},\mathcal{M})\) over all observables \(\mathcal{M}\) . The information capacity \(C(\mathcal{M})\) of the observable \(\mathcal{M}\) is essentially supremum of \(I(\mathcal{E},\mathcal{M})\) over ensembles \(\mathcal{E}\) (where \(\mathcal{E}\) may be subject to certain additional constraints). Computation of these quantities presents important and rather involved optimization problems which are closely connected via the so called ensemble-observable duality. This paper is devoted to consideration of these two quantities for quantum Gaussian systems. In this case ensemble-observable duality admits quite an explicit description. The present paper surveys results recently obtained in these directions. It turns out that in both cases the maximizer is quantum Gaussian: the quantity \(I(\mathcal{E},\mathcal{M}),\) where \(\mathcal{E}\) is Gaussian ensemble, is maximized by a Gaussian observable, while \(I(\mathcal{E},\mathcal{M})\) for Gaussian observable \(\mathcal{M}\) is maximized by a Gaussian state ensemble. Thus, we have here still another confirmation of the famous ‘‘hypothesis of quantum Gaussian optimizers’’.