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arXiv:1908.03038 · IEEE Transactions on Information Theory

Gaussian Maximizers for Quantum Gaussian Observables and Ensembles

Alexander Semenovich Holevo

Published Apr 14, 2020• 19 citations
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Abstract

In this paper we prove two results related to the Gaussian optimizers conjecture for multimode bosonic system with gauge symmetry. First, we argue that the classical capacity of an arbitrary Gaussian observable is attained on a Gaussian ensemble of coherent states. This generalizes results previously known for heterodyne measurement in one mode. By using this fact and continuous variable version of ensemble-observable duality, we prove an old conjecture that accessible information of arbitrary Gaussian ensemble is attained on the multimode generalization of the heterodyne measurement.

DOI
10.1109/tit.2020.2987789

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In this paper we prove two results related to the Gaussian optimizers conjecture for multimode bosonic system with gauge symmetry. First, we argue that the classical capacity of an arbitrary Gaussian observable is attained on a Gaussian ensemble of coherent states. This generalizes results previously known for heterodyne measurement in one mode. By using this fact and continuous variable version of ensemble-observable duality, we prove an old conjecture that accessible information of arbitrary Gaussian ensemble is attained on the multimode generalization of the heterodyne measurement.
Accessible information, which is a basic quantity in quantum information theory, is computed for a general quantum Gaussian ensemble under certain “threshold conditions.” It is shown that the maximizing measurement is Gaussian, constituting a far-reaching generalization of optical heterodyning. This substantially extends the previous result concerning the gauge-invariant case even for a single bosonic mode. A simple sufficient condition is provided that implies the threshold condition for a general Gaussian ensemble. The tightness of the sufficient condition is demonstrated in the single-mode case.
Abstract The present paper is devoted to investigation of the classical capacity of infinite-dimensional continuous variable quantum measurement channels. A number of usable conditions are introduced that enable us to apply previously obtained general results to specific models, in particular, to the multi-mode bosonic Gaussian measurement channels. An explicit formula for the classical capacity of the Gaussian measurement channel is obtained in this paper without assuming the global gauge symmetry, solely under certain ‘threshold condition’. The result is illustrated by the capacity computation for a one-mode squeezed-noise heterodyne measurement channel.
The quantum Gaussian optimizers conjecture says that q-p norm of a Bosonic Gaussian channel is attained on "Gaussian" operators. Recently R.L. Frank and E.H. Lieb confirmed the hypothesis in the case q=p for gauge-covariant channels with arbitrary number of modes s [3]. In the present note we remark that in the case q=p our results in [6] and [9] in fact allow to prove the hypothesis for all Gaussian channels. Thus the condition of gauge covariance, rather crucial in the case q=1,s>1, plays no role in the case q=p.
We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein--Podolsky--Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.
We derive explicit formulas for the capacity of multimode quantum Gaussian channels which serve as a fundamental model for optical version of multiple-input multiple-output channels. We show that it is always optimal to increase the number of modes under fixed power constraint. We derive an analytical formula for the ensemble-averaged Holevo capacity in the case of random passive transformations. The analogous results are also obtained for capacities achievable under homodyne and heterodyne detection. We further discuss the generalization of the model to include weak active transformations.
Given a quantum state ensemble E\mathcal{E}E and quantum observable M,\mathcal{M},M, one can define the Shannon information I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) and introduce the two fundamental information quantities. The accessible information A(E)A(\mathcal{E})A(E) of the ensemble E\mathcal{E}E is defined as supremum of I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) over all observables M\mathcal{M}M . The information capacity C(M)C(\mathcal{M})C(M) of the observable M\mathcal{M}M is essentially supremum of I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) over ensembles E\mathcal{E}E (where E\mathcal{E}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(E,M),I(\mathcal{E},\mathcal{M}),I(E,M), where E\mathcal{E}E is Gaussian ensemble, is maximized by a Gaussian observable, while I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) for Gaussian observable M\mathcal{M}M is maximized by a Gaussian state ensemble. Thus, we have here still another confirmation of the famous ‘‘hypothesis of quantum Gaussian optimizers’’.