dorsal/arxiv
View SchemaThe Holevo capacity of infinite dimensional channels and the additivity problem
| Authors | M. E. Shirokov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408009 |
| URL | https://arxiv.org/abs/quant-ph/0408009 |
| DOI | 10.1007/s00220-005-1457-8 |
| Journal | Comm. Math. Phys., Vol. 262, (2006), 137--159. |
Abstract
The notion of the Holevo capacity for arbitrarily constrained infinite dimensional quantum channels is introduced. It is shown that despite nonexistence of an optimal ensemble in this case it is possible to define the notion of the output optimal average state for such a channel. The characterization of the output optimal average state and a "minimax" expression for the Holevo capacity are obtained. This makes it possible to prove equivalence of several additivity properties for infinite dimensional quantum channels. The notion of the $\chi$-function for an infinite dimensional channel is considered, its strong concavity and lower semicontinuity are shown. The problem of continuity of the Holevo capacity is also discussed. It is shown that the Holevo capacity is continuous function of a channel in the finite dimensional case while in general it is only lower semicontinuous. This conclusion is confirmed by the example. The main result of this note is the statement that additivity of the Holevo capacity for all finite dimensional channels implies additivity of the Holevo capacity for all infinite dimensional channels with arbitrary constraints. The subadditivity of the $\chi$-function for two infinite dimensional channels with one of them noiseless or entanglement breaking is also proved.
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"abstract": "The notion of the Holevo capacity for arbitrarily constrained infinite\ndimensional quantum channels is introduced. It is shown that despite\nnonexistence of an optimal ensemble in this case it is possible to define the\nnotion of the output optimal average state for such a channel. The\ncharacterization of the output optimal average state and a \"minimax\" expression\nfor the Holevo capacity are obtained. This makes it possible to prove\nequivalence of several additivity properties for infinite dimensional quantum\nchannels.\n The notion of the $\\chi$-function for an infinite dimensional channel is\nconsidered, its strong concavity and lower semicontinuity are shown.\n The problem of continuity of the Holevo capacity is also discussed. It is\nshown that the Holevo capacity is continuous function of a channel in the\nfinite dimensional case while in general it is only lower semicontinuous. This\nconclusion is confirmed by the example.\n The main result of this note is the statement that additivity of the Holevo\ncapacity for all finite dimensional channels implies additivity of the Holevo\ncapacity for all infinite dimensional channels with arbitrary constraints. The\nsubadditivity of the $\\chi$-function for two infinite dimensional channels with\none of them noiseless or entanglement breaking is also proved.",
"arxiv_id": "quant-ph/0408009",
"authors": [
"M. E. Shirokov"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1007/s00220-005-1457-8",
"journal_ref": "Comm. Math. Phys., Vol. 262, (2006), 137--159.",
"title": "The Holevo capacity of infinite dimensional channels and the additivity problem",
"url": "https://arxiv.org/abs/quant-ph/0408009"
},
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