Information Capacity of State Ensembles and Observables

Abstract

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’’.