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arXiv:cond-mat/0310199 · Kluwer Academic Publishers eBooks

Entanglement Production in a Chaotic Quantum Dot

C. W. J. Beenakker, M. Kindermann, Charles M. Marcus, Amir Yacoby

Published Mar 9, 2006• 3 citations
arXivHTMLPDF

Abstract

It has recently been shown theoretically that elastic scattering in the Fermi sea produces quantum mechanically entangled states. The mechanism is similar to entanglement by a beam splitter in optics, but a key distinction is that the electronic mechanism works even if the source is in local thermal equilibrium. An experimental realization was proposed using tunneling between two edge channels in a strong magnetic field. Here we investigate a low-magnetic field alternative, using multiple scattering in a quantum dot. Two pairs of single-channel point contacts define a pair of qubits. If the scattering is chaotic, a universal statistical description of the entanglement production (quantified by the concurrence) is possible. The mean concurrence turns out to be almost independent on whether time-reversal symmetry is broken or not. We show how the concurrence can be extracted from a Bell inequality using low-frequency noise measurements, without requiring the tunneling assumption of earlier work.

DOI
10.1007/1-4020-2193-3_10

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It has recently been shown theoretically that elastic scattering in the Fermi sea produces quantum mechanically entangled states. The mechanism is similar to entanglement by a beam splitter in optics, but a key distinction is that the electronic mechanism works even if the source is in local thermal equilibrium. An experimental realization was proposed using tunneling between two edge channels in a strong magnetic field. Here we investigate a low-magnetic field alternative, using multiple scattering in a quantum dot. Two pairs of single-channel point contacts define a pair of qubits. If the scattering is chaotic, a universal statistical description of the entanglement production (quantified by the concurrence) is possible. The mean concurrence turns out to be almost independent on whether time-reversal symmetry is broken or not. We show how the concurrence can be extracted from a Bell inequality using low-frequency noise measurements, without requiring the tunneling assumption of earlier work.
We investigate the statistics of orbital entanglement production between electrons in a chaotic quantum dot with two-channel leads. Through a random-matrix simulation, we obtain the probability density of two entanglement measures, concurrence and entanglement formation, by varying the transparency of the contacts in the presence and absence of time-reversal invariance of the electron dynamics inside the cavity. The results suggest that orbital entanglement production is optimized by increasing the asymmetry between the transparency of the contacts, especially when time-reversal invariance is broken.
Building on a previous proposal for the entanglement of electron-hole pairs in the Fermi sea, we show how three qubits can be entangled without using electron-electron interactions. As in the two-qubit case, this electronic scheme works even if the sources are in (local) thermal equilibrium---in contrast to the photonic analog. The three qubits are represented by four edge-channel excitations in the quantum Hall effect (two hole excitations plus two electron excitations with identical channel index). The entangler consists of an adiabatic point contact flanked by a pair of tunneling point contacts. The irreducible three-qubit entanglement is characterized by the tangle, which is expressed in terms of the transmission matrices of the tunneling point contacts. The maximally entangled Greenberger-Horne-Zeilinger (GHZ) state is obtained for channel-independent tunnel probabilities. We show how low-frequency noise measurements can be used to determine an upper and lower bound to the tangle. The bounds become tighter the closer the electron-hole state is to the GHZ state.
The production of orbitally entangled electrons in quantum-chaotic dots is investigated from a statistical point of view. The degree of entanglement is quantified through the concurrence and the entanglement of formation. We calculate the complete statistical distributions of the entanglement measures by using random matrix theory. Simple analytical expressions are provided for the concurrence distributions. We identify clear signatures of time-reversal invariance in the production of entanglement at the level of the entanglement-measure distributions, such as the ability of producing maximally entangled (Bell) states, which passed unnoticed in previous works, where only the first two moments of the distributions were studied.
The voltage probe model is a model of incoherent scattering in quantum transport. Here we use this model to study the effect of spin-flip scattering on electrical conduction through a quantum dot with chaotic dynamics. The spin decay rate $\ensuremath{\gamma}$ is quantified by the correlation of spin-up and spin-down current fluctuations (spin-flip noise). The resulting decoherence reduces the ability of the quantum dot to produce spin-entangled electron-hole pairs. For $\ensuremath{\gamma}$ greater than a critical value ${\ensuremath{\gamma}}_{c}$, the entanglement production rate vanishes identically. The statistical distribution $P({\ensuremath{\gamma}}_{c})$ of the critical decay rate in an ensemble of chaotic quantum dots is calculated using the methods of random-matrix theory. For small ${\ensuremath{\gamma}}_{c}$ this distribution is $\ensuremath{\propto}{\ensuremath{\gamma}}_{c}^{\ensuremath{-}1+\ensuremath{\beta}∕2}$, depending on the presence $(\ensuremath{\beta}=1)$ or absence $(\ensuremath{\beta}=2)$ of time-reversal symmetry. To make contact with experimental observables, we derive a one-to-one relationship between the entanglement production rate and the spin-resolved shot noise, under the assumption that the density matrix is isotropic in the spin degrees of freedom. Unlike the Bell inequality, this relationship holds for both pure and mixed states. In the tunneling regime, the electron-hole pairs are entangled if and only if the correlator of parallel spin currents is at least twice larger than the correlator of antiparallel spin currents.
We study orbital entanglement production in a chaotic quantum dot with two-channel leads by varying the opacity of the contacts in the unitary and orthogonal Wigner-Dyson ensembles. We computed the occurrence probability of entangled states (squared norm) and its concurrence (entanglement level). We also define an entanglement production factor to properly evaluate the entanglement behavior in the system considering effective aspects. The results are numerically obtained through (i) integrations over random matrix ensembles (exact results) for the scenario of one contact ideally fixed and (ii) random matrix simulations for arbitrary contact opacities (sampling). Those outcomes are in mutual agreement and indicate that the optimum effective production of orbital entanglement is achieved when both contacts are ideal and the time-reversal symmetry is broken.
We study numerically the production of orbital and spin entangled states in chaotic quantum dots for noninteracting electrons. The introduction of spin-orbit coupling permits us to identify signatures of time-reversal symmetry correlations in the entanglement production previously unnoticed, resembling weak-(anti)localization quantum corrections to the conductance. We find the entanglement to be strongly dependent on spin-orbit coupling, showing universal features for broken time-reversal and spin-rotation symmetries.