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Zeitschrift für Naturforschung A

Energy Level Crossing and Entanglement

Willi-Hans Steeb

Published Aug 1, 2009• 2 citations
Publisher

Abstract

Abstract We consider a Hamilton operator in a finite dimensional Hilbert space with energy level crossing. We discuss the question how energy level crossing and entanglement of states in this Hilbert space are intertwined. Since energy level crossing is related to symmetries of the Hamilton operator we also derive these symmetries and give the reduction to the invariant Hilbert subspaces.

DOI
10.1515/zna-2009-7-806

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Abstract We consider a Hamilton operator in a finite dimensional Hilbert space with energy level crossing. We discuss the question how energy level crossing and entanglement of states in this Hilbert space are intertwined. Since energy level crossing is related to symmetries of the Hamilton operator we also derive these symmetries and give the reduction to the invariant Hilbert subspaces.
Abstract We consider a hierarchy of Hamilton operators Ĥ N in finite dimensional Hilbert spaces C2N \mathbb{C}^{2^N } C2N. We show that the eigenstates of Ĥ N are fully entangled for N even. We also calculate the unitary operator U N(t) = exp(—Ĥ N t/ħ) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.
Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum jjj with a spin 1∕21∕21∕2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.
We study the interplay between crossing symmetry and entanglement in 2→22 \to 22→2 scattering within local quantum field theories that possess an SU(N)SU(N)SU(N) global symmetry. In particular, we recast scattering amplitudes of fixed helicity as quantum operations on the Hilbert space of internal quantum numbers, where the external states play the role of qudits. The entire space of SU(N)SU(N)SU(N)-invariant scattering operators between qudits is spanned by a minimal set of three quantum gates. Recoupling relations among quantum gates are shown to follow directly from the crossing properties of the underlying amplitudes and reveal that entanglement generated from separable states in one channel is necessarily intertwined with another. Consequently, we argue any interacting quantum field theory that realizes an SU(N)SU(N)SU(N) global symmetry must generate entanglement in at least one scattering channel.