Reeh–Schlieder theorem

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.

One may remark the states created by applying elements of the local algebra

$\mathcal{A}(\mathcal{O})$

to the vacuum state are, therefore, not strictly localized in its region $\mathcal{O}$, 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 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 online at Project Euclid
• hep-th/0001154 Christian Jaekel, "The Reeh–Schlieder property for ground states"
• Benni Reznik, "Distillation of vacuum entanglement to EPR pairs"