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The '''Reeh–Schlieder theorem''' is a result of relativistic local [[quantum field theory]], stating that the [[vacuum]] is a [[cyclic vector]] for the field algebra of any open set in [[Minkowski space]]. It was published by [[Helmut Reeh]] and [[Siegfried Schlieder]] (1918-2003) in 1961.
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One may remark the states created by applying elements of the local algebra
 
:<math>\mathcal{A}(\mathcal{O})</math>
 
to the [[vacuum state]] are, therefore,
not strictly localized in its region <math>\mathcal{O}</math>, but can in effect approximate any state. In a quantitative sense, the localization remains true. The long range effects of the operators of the
local algebra will diminish rapidly with distance, as seen by the cluster properties of the [[Wightman functions]]. And with increasing distance, creating a unit vector localized outside requires operators of ever increasing
[[operator norm]].  
 
This theorem is also cited in connection with [[quantum entanglement]]. But it is subject to some doubt whether
the '''Reeh–Schlieder theorem''' can usefully be seen as the [[quantum field theory]] analog to [[quantum entanglement]], since the  
[[exponential growth|exponentially-increasing]] energy needed for long range actions will prohibit any macroscopic effects. However, B.Reznik showed that vacuum entanglement can be distilled into EPR pairs used in quantum information tasks.
 
== External links ==
*Siegfried Schlieder, ''Some remarks about the localization of states in a quantum field theory'',  Comm. Math. Phys.  1, no. 4 (1965), 265–280 [http://projecteuclid.org/getRecord?id=euclid.cmp/1103758945 online] at [[Project Euclid]]
*[http://www.arxiv.org/abs/hep-th/0001154 hep-th/0001154 Christian Jaekel, "The Reeh–Schlieder property for ground states"]
*[http://arxiv.org/abs/quant-ph/0008006 Benni Reznik, "Distillation of vacuum entanglement to EPR pairs"]
 
{{DEFAULTSORT:Reeh-Schlieder theorem}}
[[Category:Quantum field theory]]
[[Category:Theorems in quantum physics]]

Revision as of 04:28, 15 February 2014



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