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arXiv:0908.0623 · Physical Review A

Engineering of triply entangled states in a single-neutron system

Yuji Hasegawa, Rudolf Loidl, G. Badurek, Katharina Durstberger-Rennhofer, Stephan Sponar, Helmut Rauch

Published Mar 25, 2010• 37 citations
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Abstract

Entanglement between degrees of freedom, namely between the spin, path, and (total) energy degrees of freedom, in a single-neutron system is exploited. We implemented a triply entangled Greenberger-Horne-Zeilinger (GHZ)--like state and coherently manipulated relative phases of two-level quantum subsystems. An equality derived by Mermin was applied to analyze the generated GHZ-like state: We determined the four expectation values and finally obtained $M=2.558\ifmmode\pm\else\textpm\fi{}0.004\ensuremath{\leqslant}\ensuremath{\llap{\not\;}}2$. This demonstrates a violation of Mermin-like inequality for triply entangled GHZ-like state in a single-particle system, which, in turn, clearly contradicts the noncontextual assumption and confirms quantum contextuality.

DOI
10.1103/physreva.81.032121

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Entanglement between degrees of freedom, namely between the spin, path, and (total) energy degrees of freedom, in a single-neutron system is exploited. We implemented a triply entangled Greenberger-Horne-Zeilinger (GHZ)--like state and coherently manipulated relative phases of two-level quantum subsystems. An equality derived by Mermin was applied to analyze the generated GHZ-like state: We determined the four expectation values and finally obtained M=2.558\ifmmode\pm\else\textpm\fi{}0.004\ensuremath{\leqslant}\ensuremath{\llap{\not\;}}2. This demonstrates a violation of Mermin-like inequality for triply entangled GHZ-like state in a single-particle system, which, in turn, clearly contradicts the noncontextual assumption and confirms quantum contextuality.
Entanglement between degrees of freedom, namely between the spin, path and (total) energy degrees of freedom, for single neutrons is exploited. We implemented a triply entangled Greenberger-Horne-Zeilinger(GHZ)-like state and coherently manipulated relative phases of two-level quantum subsystems. An inequality derived by Mermin was applied to analyze the generated GHZ-like state: we determined the four expectation values and finally obtained [Formula: see text]. This demonstrates the violation of a Mermin-like inequality for triply entangled GHZ-like state in a single-particle system, which, in turn, exhibits a clear inconsistency between noncontextual assumptions and quantum mechanics and confirms quantum contextuality.
Entanglement occupies a peculiar position in quantum mechanics (QM). It occurs in quantum systems that consist of space-like separated parts or—more generally—in systems whose observables belong to disjoint Hilbert spaces. The latter is the case with single-neutron systems. Here, we report on a neutron polarimetric experiment, where a triply entangled Greenberger–Horne–Zeilinger state is exploited. The entanglement of spin, momentum and total energy degree of freedom is generated utilizing a suitable combination of radio-frequency and static magnetic fields. An average deviation of expectation values from theory—ideal circumstances—of 0.016(1) confirms the predictions of QM with high accuracy, demonstrating the high-efficiency manipulation of the entangled single-neutron system.
Entanglement is a remarkable peculiarity in quantum mechanics. It occurs in quantum systems that consist of space-like separated parts, or in systems whose observables belong to disjoint Hilbert spaces. The latter is the case in single-neutron systems. Entangled states are renowned for exhibiting non-classical correlations between observables of individual sub-systems. In a perfect Si-crystal interferometer experiment entanglement between three degrees of freedom in a single-neutron system is created. The prepared entanglement of spin, path and energy is induced by interaction with an oscillating magnetic field. The generated Greenberger-Horne-Zeilinger (GHZ) state is analyzed with an inequality derived by Mermin, yielding a value M = 2.558(4) 2, which exhibits a clear violation of the classical assumption. In addition observation of a GHZ entanglement, consisting of spin, momentum and total energy, in a neutron polarimetric experiment is presented. Here the advantages of neutron polarimetry, such as high contrast or insensitivity to ambient disturbances, are utilized resulting in final value of M = 3.936(2) 2.
We performed an experimental test of the Kochen-Specker theorem based on an inequality derived from the Peres-Mermin proof, using spin-path (momentum) entanglement in a single neutron system. Following the strategy proposed by Cabello et al. [Phys. Rev. Lett. 100, 130404 (2008)10.1103/PhysRevLett.100.130404], a Bell-like state was generated, and three expectation values were determined. The observed violation 2.291 +/- 0.008 not less, dbl equals1 clearly shows that quantum mechanical predictions cannot be reproduced by noncontextual hidden-variable theories.
A neutron interferometer experiment which clearly demonstrates the violation of a Bell-like inequality is reported. Entanglement is achieved not between particles, but between the degrees of freedom, in this case, for a single particle. The total wave function of spin-1/2 neutron is described in a tensor product Hilbert space. Appropriate combinations of the direction of spin analysis and the position of the phase shifter allow experimental verification of the violation of a Bell-like inequality. Manipulation of the wave function in one Hilbert space influences the result of the measurement in the other Hilbert space: manipulation without touch! The violation of the Bell-like inequality is discussed in terms of quantum contextuality and an entanglement-induced correlation.
We report a single-neutron optical experiment to demonstrate the violation of a Bell-like inequality. Entanglement is achieved not between particles, but between the degrees of freedom, in this case, for a single-particle. The spin-{\small 1/2} property of neutrons are utilized. The total wave function of the neutron is described in a tensor product Hilbert space. A Bell-like inequality is derived not by a non-locality but by a contextuality. Joint measurements of the spinor and the path properties lead to the violation of a Bell-like inequality. Manipulation of the wavefunction in one Hilbert space influences the result of the measurement in the other Hilbert space. A discussion is given on the quantum contextuality and an entanglement-induced correlation in our experiment.