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arXiv:1204.5850 · Physical Review B

Setup of three Mach-Zehnder interferometers for production and observation of Greenberger-Horne-Zeilinger entanglement of electrons

Andrey A. Vyshnevyy, Gordey B. Lesovik, Thibaut Jonckheere, Thierry Martin

Published Apr 9, 2013• 7 citations
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Abstract

We propose a new single-step scheme for the generation of a Greenberger-Horne-Zeilinger (GHZ) entangled state of three single-electron excitations (flying qubits). We also present a method to get a generalized GHZ state. Our idea relies upon the most recent progress in the field of on-demand electron sources and mesoscopic Mach-Zehnder interferometry. We also provide the recipe for the unambiguous detection of this GHZ state via correlation measurements at the output, which imply the violation of a Bell-type inequality which is generalized to the case of three particles. We explain how such measurements can be achieved in the context of Mach-Zehnder interferometry and draw an actual prototype device which could be achieved with point contacts and metallic gates placed on a GaAs sample, in the integer quantum Hall effect regime.

DOI
10.1103/physrevb.87.165417

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We propose a new single-step scheme for the generation of a Greenberger-Horne-Zeilinger (GHZ) entangled state of three single-electron excitations (flying qubits). We also present a method to get a generalized GHZ state. Our idea relies upon the most recent progress in the field of on-demand electron sources and mesoscopic Mach-Zehnder interferometry. We also provide the recipe for the unambiguous detection of this GHZ state via correlation measurements at the output, which imply the violation of a Bell-type inequality which is generalized to the case of three particles. We explain how such measurements can be achieved in the context of Mach-Zehnder interferometry and draw an actual prototype device which could be achieved with point contacts and metallic gates placed on a GaAs sample, in the integer quantum Hall effect regime.
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.
We study Greenberger-Horne-Zeilinger--type (GHZ-type) and WWW-type three-mode entangled coherent states. Both types of entangled coherent states violate Mermin's version of the Bell inequality with threshold photon detection (i.e., without photon counting). Such an experiment can be performed using linear optics elements and threshold detectors with significant Bell violations for GHZ-type entangled coherent states. However, to demonstrate Bell-type inequality violations for WWW-type entangled coherent states, additional nonlinear interactions are needed. We also propose an optical scheme to generate WWW-type entangled coherent states in free-traveling optical fields. The required resources for the generation are a single-photon source, a coherent state source, beam splitters, phase shifters, photodetectors, and Kerr nonlinearities. Our scheme does not necessarily require strong Kerr nonlinear interactions; i.e., weak nonlinearities can be used for the generation of the WWW-type entangled coherent states. Furthermore, it is also robust against inefficiencies of the single-photon source and the photon detectors.
We propose and analyze a mesoscopic device producing on-demand entangled pairs of electrons. The system consists of two capacitively coupled Mach-Zehnder interferometers implemented in a quantum Hall structure. A pair of electron wave packets is injected into the chiral edge states of two (of the four) incoming arms; scattering on the incoming interferometers splits the wave packets into four components, of which two interact. The resulting interaction phase associated with this component leads to the entanglement of the state; the latter is scattered at the outgoing beam splitter and analyzed in a Bell violation test measuring the presence of particles in the four outgoing leads. We study the two-particle case and determine the conditions to reach and observe full entanglement. We extend our two-particle analysis to include the underlying Fermi seas in the quantum Hall device; the change in shape of the wave function, the generation of electron-hole pairs in the interaction regime, and a time delay between the pulses all reduce the degree of visible entanglement and the violation of the Bell inequality, effects which we analyze quantitatively. We determine the device settings optimizing the entanglement and the Bell test and find that violation is still possible in the presence of the Fermi seas, with a maximal Bell parameter reaching B=2.18>2\mathcal{B}=2.18>2B=2.18>2 in our setup.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.
A scheme is proposed for generating maximally entangled Greenberger-Horne-Zeilinger (GHZ) atomic states for testing quantum nonlocality. In the scheme, three atoms are simultaneously sent through a nonresonant cavity in a vacuum state. They are initially in the same state, and thus there is no energy exchange in the process. The cavity-assisted collision results in a phase-shift which depends upon the collective atomic excitations. In principle, the scheme can be generalized to generate N-atom GHZ states. The scheme is insensitive to cavity decay and requires only one cavity, providing new prospects for testing fundamental aspects of quantum mechanics and for quantum information processing.
We demonstrate how a new exactly soluble cavity QED model can be used to test local realism associated with three-particle entangled states, as proposed by Mermin. These states embody the correlations described by Greenberger, Horne, and Zeilinger (GHZ) and provide a stronger refutation of local hidden variable theories than is possible with Bell inequalities. It is shown that two photons distributed in the model’s four cavity modes coupled with a two-level atom can effectively produce three independent spin-1/2 particle states. We discuss preparation of the desired perfectly correlated GHZ state and a method to perform measurement on various field modes and atomic level correlations equivalent to a spin component along any direction.