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arXiv:cond-mat/0307198

Dephasing of entangled electron-hole pairs in a degenerate electron gas

J. L. van Velsen, Kindermann, M., C. W. J. Beenakker

Published Jul 9, 2003• 1 citation
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Abstract

A tunnel barrier in a degenerate electron gas was recently discovered as a source of entangled electron-hole pairs. Here, we investigate the loss of entanglement by dephasing. We calculate both the maximal violation E_max of the Bell inequality and the degree of entanglement (concurrence) C. If the initially maximally entangled electron-hole pair is in a Bell state, then the Bell inequality is violated for arbitrary strong dephasing. The same relation E_max=2\sqrt{1+C^{2}} then holds as in the absence of dephasing. More generally, for a maximally entangled superposition of Bell states, the Bell inequality is satisfied for a finite dephasing strength and the entanglement vanishes for somewhat stronger (but still finite) dephasing strength. There is then no one-to-one relation between E_max and C.

DOI
10.48550/arxiv.cond-mat/0307198

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We demonstrate theoretically that the shot noise produced by a tunnel barrier in a two-channel conductor violates a Bell inequality. The nonlocality is shown to originate from entangled electron-hole pairs created by tunneling events-without requiring electron-electron interactions. The degree of entanglement (concurrence) equals 2(T1T2)(1/2)(T1+T2)(-1), with T1,T2<<1 the transmission eigenvalues. A pair of edge channels in the quantum Hall effect is proposed as an experimental realization.
Electron-hole pairs produced by tunneling in a degenerate electron gas lose their spin entanglement by spin-orbit coupling, which transforms the fully entangled Bell state into a partially entangled mixed density matrix of the electron and hole spins. We calculate the dependence of the entanglement (quantified by the concurrence) on the spin-orbit coupling time ${\ensuremath{\tau}}_{\mathrm{so}}$ and on the diffusion time (or dwell time) ${\ensuremath{\tau}}_{\text{dwell}}$ of electrons and holes in the conductors (with conductances ⪢e2∕h⪢{e}^{2}∕h⪢e2∕h) at the two sides of the tunnel barrier (with conductance ⪡e2∕h⪡{e}^{2}∕h⪡e2∕h). The entanglement disappears when the ratio ${\ensuremath{\tau}}_{\text{dwell}}∕{\ensuremath{\tau}}_{\mathrm{so}}$ exceeds a critical value of order unity. The results depend on the type of conductor (disordered wire or chaotic quantum dot), but they are independent of other microscopic parameters (number of channels, level spacing). Our analytical treatment relies on an ``isotropy approximation'' (no preferential basis in spin space), which allows us to express the concurrence entirely in terms of spin correlators. We test this approximation for the case of chaotic dynamics with a computer simulation (using the spin-kicked rotator) and find good agreement.
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It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.
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