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arXiv:0905.0115 · Europhysics Letters (EPL)

Entanglement of electron pairs extracted from a many-body system

A. Ramšak, Jernej Mravlje, T. Rejec, Anja Kopač Lautar

Published May 1, 2009• 7 citations
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Abstract

Entanglement of spins is analyzed for two electrons extracted from a mixed many-electron state by projecting onto the two-electron subspace. The concurrence formulae are expressed in a compact form for states with a well-defined square of the total spin projection. As an example, the thermal entanglement for a qubit pair with an anisotropic Heisenberg and the Dzyaloshinskii-Moriya interactions in an inhomogeneous magnetic field is given analytically. Remarkably, the concurrence of a pair of electrons with antiparallel spins and in a delocalised orbital state is given by the scalar product of the state with its spin-flipped state and not with the time-reversed state.

DOI
10.1209/0295-5075/86/40003

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Entanglement of spins is analyzed for two electrons extracted from a mixed many-electron state by projecting onto the two-electron subspace. The concurrence formulae are expressed in a compact form for states with a well-defined square of the total spin projection. As an example, the thermal entanglement for a qubit pair with an anisotropic Heisenberg and the Dzyaloshinskii-Moriya interactions in an inhomogeneous magnetic field is given analytically. Remarkably, the concurrence of a pair of electrons with antiparallel spins and in a delocalised orbital state is given by the scalar product of the state with its spin-flipped state and not with the time-reversed state.
In this paper, we consider two examples of an entanglement in two-qubit systems and an example of entanglement in quantum field theory (QFT). In the beginning, we study the entanglement of two spin states by a magnetic field. A nonzero entanglement appears for interacting spins. When the coupling between the spins is constant, we study the entanglement by several types of time-dependent magnetic fields. In the case of a constant difference between [Formula: see text] components of magnetic fields acting on each spin, we find several time-dependent coupling functions [Formula: see text] that also allow us to analyze analytically and numerically the entanglement measure. Considering two photons moving in an electron medium, we demonstrate that they can be entangled in a controlled way by applying an external magnetic field. The magnetic field affecting electrons of the medium affects photons and, thus, causes an entanglement of the photon beams. The third example is related to the effect of production of electron–positron pairs from the vacuum by a strong external electric field. Here, we have used a general nonperturbative expression for the density operator of the system under consideration. Applying a reduction procedure to this density operator, we construct mixed states of electron and positron subsystems. Calculating the von Neumann entropy of such states, we obtain the loss of information due to the reduction and, at the same time, the entanglement measure of electron and positron subsystems. This entanglement can be considered as an example of an entanglement in QFT.
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Several convenient formulas for the entanglement of two indistinguishable delocalized spin-1∕21∕21∕2 particles are introduced. These generalize the standard formula for concurrence, valid only in the limit of localized or distinguishable particles. Several illustrative examples are given.