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Physical Review A

Bell’s-inequality experiment employing four harmonic oscillators

Bernard Yurke, David Stoler

Published May 1, 1995• 7 citations
Publisher

Abstract

We show that Bell's-inequality experiments can be carried out using four harmonic oscillators, pairs of which interact with each other at suitable instants of time to allow for the exchange of energy. Between interactions, the oscillators are shifted in phase relative to each other in a controllable manner. Energy becomes distributed among the four harmonic oscillators in a manner that is incompatible with a local realism picture in which information is exchanged only when the oscillators interact with each other. Entanglement is dynamically generated from an initial product state.

DOI
10.1103/physreva.51.3437

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We show that Bell's-inequality experiments can be carried out using four harmonic oscillators, pairs of which interact with each other at suitable instants of time to allow for the exchange of energy. Between interactions, the oscillators are shifted in phase relative to each other in a controllable manner. Energy becomes distributed among the four harmonic oscillators in a manner that is incompatible with a local realism picture in which information is exchanged only when the oscillators interact with each other. Entanglement is dynamically generated from an initial product state.
We present a scheme for generating entanglement between two qubits associated with a pair of particles, which move and interact, in a one-dimensional harmonic-oscillator potential. The spatial wave functions of the particles are initially localized on either side of the oscillator. As they ``fall'' toward each other, we find that the interaction between the two particles produces an effective exchange interaction between the qubits. This allows us to produce a ``power-of-swap'' operation on the initial two-qubit quantum state without requiring any time-dependent control. We find that under certain conditions, we can generate maximally entangled states from a product state with a fidelity that is only weakly dependent on initialization errors.
In the usual Bell's-inequality experiments two particles carrying spin or polarization are prepared in an entangled state generated from the decay of an unstable quantum-mechanical system. These particles are then delivered to spin or polarization analyzers. The statistics of the measurements reported by the analyzers are incompatible with our notions of local realism. Here we show that Bell's-inequality violations occur even when the initial state is a direct product state. In fact, the two particles can come from two independent widely separated sources.
Bell's inequality is established based on local realism. The violation of Bell's inequality by quantum mechanics implies either locality or realism or both are untenable. Leggett's inequality is derived based on nonlocal realism. The violation of Leggett's inequality implies that quantum mechanics is neither local realistic nor nonlocal realistic. The incompatibility of nonlocal realism and quantum mechanics has been currently confirmed by photon experiments. In our work, we propose to test Leggett's inequality using the Aharonov-Casher effect. In our scheme, four entangled particles emitted from two sources manifest a two-qubit-typed correlation that may result in the violation of the Leggett inequality, while satisfying the no-signaling condition for spacelike separation. Our scheme is tolerant to some local inaccuracies due to the topological nature of the Aharonov-Casher phase. The experimental implementation of our scheme can be possibly realized by a calcium atomic polarization interferometer experiment.
Assuming that there is no way of sending signals propagating faster than light and that free will exists, the loophole-free observed violation of Bell's inequalities demonstrates that at least one of three fundamental hypotheses involved in the derivation and observation of the inequalities is false: Locality, Realism, or Ergodicity. An experiment is proposed to obtain some evidence about which one is the false one. It is based on recording the time evolution of the rate of non-random series of outcomes that are generated in a specially designed Bell's setup. The results of such experiment would be important not only to the foundations of Quantum Mechanics, but they would also have immediate practical impact on the efficient use of quantum-based random number generators and the security of Quantum Key Distribution using entangled states.
We introduce a family of entanglement witnesses for continuous variable systems, which rely on the sole assumption that their dynamics is that of coupled harmonic oscillators at the time of the test. Entanglement is inferred from the Tsirelson nonclassicality test on one of the normal modes, without any knowledge about the state of the other mode. In each round, the protocol requires measuring only the sign of one coordinate (e.g., position) at one among several times. This dynamic-based entanglement witness is more akin to a Bell inequality than to an uncertainty relation: in particular, it does not admit false positives from classical theory. Our criterion detects non-Gaussian states, some of which are missed by other criteria.