Generalized Einstein-Podolsky-Rosen states

Based on Bohr’s ideas two families of states $ωθ$ and $ωϕ$ sharing the same perfect correlation of the Einstein-Podolsky-Rosen state and maximally violating Bell’s inequalities are constructed. Unlike their finite-dimensional counterpartner, these states are not unitarily equivalent. Hence they cannot be transformed into each other by physical operations of local algebras. This is a new entanglement phenomenon emerging in infinite-dimensional systems. Due to the uncertainty principle the existence of unbounded observables depends on the states $ωθ$ and $ωϕ$⁠. Therefore the manipulation of unbounded observables in quantum information processes cannot provide information for all states. Especially, the canonical unbounded observables $q̂i$ and $p̂i$⁠, $i=1,2$ of individual particles do not exist in the GNS representations associated with $ωθ$ and $ωϕ$⁠, and hence properties of individual particles cannot be obtained in such states.