Quantum depolarizing channel

2013-09-10T13:06:06Z

68.44.88.133: /* Outline of the proof of the additivity of Holevo information */

In [[mathematics]], specifically [[abstract algebra]], if (''G'', +) is an [[abelian group]] then <math>\scriptstyle \nu\colon G \to \mathbb{R}</math> is said to be a '''norm on the abelian group''' (''G'', +) if:
<ol>
<li><math>\scriptstyle \nu(g) > 0</math> if <math>\scriptstyle g\ne 0</math>,</li>
<li><math>\scriptstyle \nu(g+h) \le \nu(g) + \nu(h)</math>,</li>
<li><math>\scriptstyle \nu(mg) = |m| \nu(g)</math> if <math>\scriptstyle m \in \mathbb{Z}</math>.</li>
</ol>

The norm ''ν'' is '''discrete''' if there is some [[real number]] ''ρ'' > 0 such that ''ν''(''g'') > ''ρ'' whenever ''g'' ≠ 0.

== Free abelian groups ==
An abelian group is a [[free abelian group]] [[if and only if]] it has a discrete norm.<ref>{{citation
| last = Steprāns | first = Juris
| doi = 10.2307/2044776
| issue = 2
| journal = Proceedings of the American Mathematical Society
| mr = 770551
| pages = 347–349
| title = A characterization of free abelian groups
| volume = 93
| year = 1985}}</ref>

==References==
{{reflist}}

[[Category:Abelian group theory]]

{{Abstract-algebra-stub}}

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Quantum depolarizing channel