The additivity of the minimal output entropy and that of the χ-capacity are known to be equivalent for finite-dimensional irreducibly covariant quantum channels. In this paper, we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper [Giovannetti et al., Commun. Math. Phys. 334(3), 1553-1571 (2015)] to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particular, this implies a multimode generalization of the “threshold condition,” obtained for single mode in Schäfer et al. [Phys. Rev. Lett. 111, 030503 (2013)], and the proof of the fact that under this condition the classical “Gaussian capacity” resulting from optimization over only Gaussian inputs is equal to the full classical capacity. Complex structures correspond to different squeezings, each with its own normal modes, vacuum and coherent states, and the gauge. Thus our results apply, e.g., to multimode channels with a squeezed Gaussian noise under the standard input energy constraint, provided the squeezing is not too large as to violate the generalized threshold condition. We also investigate the restrictiveness of the gauge-covariance condition for single- and multimode bosonic Gaussian channels.
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January 2016
Research Article|
August 07 2015
On the constrained classical capacity of infinite-dimensional covariant quantum channels
A. S. Holevo
A. S. Holevo
Steklov Mathematical Institute
, 119991 Moscow, Russia
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J. Math. Phys. 57, 015203 (2016)
Article history
Received:
March 23 2015
Accepted:
July 24 2015
Citation
A. S. Holevo; On the constrained classical capacity of infinite-dimensional covariant quantum channels. J. Math. Phys. 1 January 2016; 57 (1): 015203. https://doi.org/10.1063/1.4928050
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