dorsal/arxiv
View SchemaContinuous ensembles and the $\chi$-capacity of infinite-dimensional channels
| Authors | A. S. Holevo, M. E. Shirokov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408176 |
| URL | https://arxiv.org/abs/quant-ph/0408176 |
| Journal | Theory of Probability and its Applications, Vol. 50, No.1, (2005), 86--114 |
Abstract
The paper is devoted to systematic study of the $\chi$-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite, in general, ``continuous'' state ensembles, defined as probability measures on the set of all quantum states. By using compactness criteria from probability theory and operator theory it is shown that the set of all generalized ensembles with the average (barycenter) in a compact set of states is itself a compact subset of the set of all probability measures. With this in hand we give a sufficient condition for the existence of an optimal generalized ensemble for a constrained quantum channel. This condition can be verified in the case of Bosonic Gaussian channels with constrained mean energy. The importance of the above condition is shown by considering example of a constrained channel with no optimal generalized ensemble. In the case of convex constraints a characterization of the optimal generalized ensemble is obtained extending the `` maximal distance'' property.
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"abstract": "The paper is devoted to systematic study of the $\\chi$-capacity (underlying\nthe classical capacity) of infinite dimensional quantum channels. An essential\nfeature of this case is the natural appearance of the input constraints and\ninfinite, in general, ``continuous\u0027\u0027 state ensembles, defined as probability\nmeasures on the set of all quantum states. By using compactness criteria from\nprobability theory and operator theory it is shown that the set of all\ngeneralized ensembles with the average (barycenter) in a compact set of states\nis itself a compact subset of the set of all probability measures. With this in\nhand we give a sufficient condition for the existence of an optimal generalized\nensemble for a constrained quantum channel. This condition can be verified in\nthe case of Bosonic Gaussian channels with constrained mean energy. The\nimportance of the above condition is shown by considering example of a\nconstrained channel with no optimal generalized ensemble. In the case of convex\nconstraints a characterization of the optimal generalized ensemble is obtained\nextending the `` maximal distance\u0027\u0027 property.",
"arxiv_id": "quant-ph/0408176",
"authors": [
"A. S. Holevo",
"M. E. Shirokov"
],
"categories": [
"quant-ph"
],
"journal_ref": "Theory of Probability and its Applications, Vol. 50, No.1, (2005),\n 86--114",
"title": "Continuous ensembles and the $\\chi$-capacity of infinite-dimensional channels",
"url": "https://arxiv.org/abs/quant-ph/0408176"
},
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