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arXiv:1704.08720 · Journal of Mathematical Physics

Norms of quantum Gaussian multi-mode channels

Rupert L. Frank, Élliott H. Lieb

Published Jun 1, 2017• 11 citations
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Abstract

We compute the Sp→Sp norm of a general Gaussian gauge-covariant multi-mode channel for any 1 ≤ p < ∞, where Sp is a Schatten space. As a consequence, we verify the Gaussian optimizer conjecture and the multiplicativity conjecture in these cases.
DOI
10.1063/1.4989809

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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 give a survey of the two remarkable analytical problems of quantum information theory. The main part is a detailed report of the recent (partial) solution of the quantum Gaussian optimizers problem which establishes an optimal property of Glauber's coherent states -- a particular instance of pure quantum Gaussian states. We elaborate on the notion of quantum Gaussian channel as a noncommutative generalization of Gaussian kernel to show that the coherent states, and under certain conditions only they, minimize a broad class of the concave functionals of the output of a Gaussian channel. Thus, the output states corresponding to the Gaussian input are "the least chaotic", majorizing all the other outputs. The solution, however, is essentially restricted to the gauge-invariant case where a distinguished complex structure plays a special role. We also comment on the related famous additivity conjecture, which was solved in principle in the negative some five years ago. This refers to the additivity or multiplicativity (with respect to tensor products of channels) of information quantities related to the classical capacity of quantum channel, such as (1→p)(1\rightarrow p)(1→p)-norms or the minimal von Neumann or Rényi output entropies. A remarkable corollary of the present solution of the quantum Gaussian optimizers problem is that these additivity properties, while not valid in general, do hold in the important and interesting class of the gauge-covariant Gaussian channels.
We address the question of the multiplicativity of the maximal ppp-norm output purities of bosonic Gaussian channels under Gaussian inputs. We focus on general Gaussian channels resulting from the reduction of unitary dynamics in larger Hilbert spaces. It is shown that the maximal output purity of tensor products of single-mode channels under Gaussian inputs is multiplicative for any $p∊(1,\ensuremath{\infty})$ for products of arbitrary identical channels as well as for a large class of products of different channels. In the case of p=2p=2p=2, multiplicativity is shown to be true for arbitrary products of generic channels acting on any number of modes.
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