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arXiv:1102.5297 · Physical Review A

Testing quantum contextuality of continuous-variable states

Gerard McKeown, Matteo G. A. Paris, Mauro Paternostro

Published Jun 10, 2011• 17 citations
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Abstract

We investigate the violation of noncontextuality by a class of continuous-variable states, including variations of entangled coherent states and a two-mode continuous superposition of coherent states. We generalize the Kochen-Specker (KS) inequality discussed by Cabello [A. Cabello, Phys. Rev. Lett. 101, 210401 (2008)] by using effective bidimensional observables implemented through physical operations acting on continuous-variable states, in a way similar to an approach to the falsification of Bell-Clauser-Horne-Shimony-Holt inequalities put forward recently. We test for state-independent violation of KS inequalities under variable degrees of state entanglement and mixedness. We then demonstrate theoretically the violation of a KS inequality for any two-mode state by using pseudospin observables and a generalized quasiprobability function.

DOI
10.1103/physreva.83.062105

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We investigate the violation of noncontextuality by a class of continuous-variable states, including variations of entangled coherent states and a two-mode continuous superposition of coherent states. We generalize the Kochen-Specker (KS) inequality discussed by Cabello [A. Cabello, Phys. Rev. Lett. 101, 210401 (2008)] by using effective bidimensional observables implemented through physical operations acting on continuous-variable states, in a way similar to an approach to the falsification of Bell-Clauser-Horne-Shimony-Holt inequalities put forward recently. We test for state-independent violation of KS inequalities under variable degrees of state entanglement and mixedness. We then demonstrate theoretically the violation of a KS inequality for any two-mode state by using pseudospin observables and a generalized quasiprobability function.
Finding the boundary between quantum and classical domains is very much an open question that is fundamental to a great deal of research spanning across many physics communities. The division is not clear cut and a deeper understanding is necessary for the future generation and engineering of quantum devices. In this thesis we introduce and apply a variety of tests that will reveal non classicality in real physical systems. The systems that we consider in many instances can be deemed classical and so our study is well motivated by the endeavor to explain the quantum to classical transition. We investigate three main classes of nonclassicality tests: Bell non locality, Kochen-Specker quantum contextuality and violation of Leggett-Garg macrorealism. These tests are applied to specific continuous variable systems that live in infinite dimensional Hilbert spaces and represent various states of the electromagnetic field. Firstly we test non locality in multimode versions of the thermal state. We show numerous violations of relevant Bell inequalities under quite unfavorable conditions and include state generation schemes. We then demonstrate through the violation of a Kochen-Specker inequality the state-independent quantum contextuality of a range of continuous variable states. These states include variations of entangled coherent states, a two-mode squeezed state and states represented by a generalized quasi-probability function. In what follows in the thesis we study non-locality of the nonclassical state of a system consisting of a qubit and an oscillator exposed to the effects of non-Markovian evolutions. Here we show interesting features related to the various facets with which non-Markovianity manifests itself. Finally this work tests macroscopic realism of a coherent and thermal state. We successively show violation of a Leggett-Garg inequality for a range of coherent state amplitude and degree of thermal mixedness.
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.
There are two powerful arguments against the possibility of extending quantum mechanics, the violation of Bell inequalities and the Kochen-Specker theorem, but the connection between the two remains confused. Following the distinctive strategy proposed by Cabello [Phys. Rev. Lett. 127, 070401 (2021)], Bell inequalities can be violated by state-independent contextuality sets. However, the experimental realization of such ideas is challenging as it requires high-dimensional entanglement. Orbital angular momentum provides an unlimited state space and the number of effective dimensions can be readily tailored as required. We performed an experimental test of non-locality based on Bell inequalities from contextuality, using orbital angular momentum entanglement in a bipartite photonic system. Our experiment not only shows a new way to produce non-locality but also connects contextuality and non-locality, two fundamental quantum resources that are critical for quantum computation and secure communication tasks.
Quantum contextuality is one of the fundamental notions in quantum mechanics. Proofs of the Kochen–Specker theorem and noncontextuality inequalities are two means for revealing the contextuality phenomenon in quantum mechanics. It has been found that some proofs of the Kochen-Specker theorem, such as those based on rays, can be converted to a state-independent noncontextuality inequality, but it remains open whether this is true in general, i.e., whether any proof of the Kochen-Specker theorem can always be converted to a noncontextuality inequality. In this paper, we address this issue. We prove that all kinds of proofs of the Kochen-Specker theorem, based on rays or any other observables, can always be converted to state-independent noncontextuality inequalities. Besides, our constructive proof also provides a general approach for deriving a state-independent noncontextuality inequality from a proof of the KS theorem.
The Kochen-Specker theorem shows the incompatibility of noncontextual hidden variable theories with quantum mechanics. Quantum contextuality is a more general concept than quantum non-locality which is quite well tested in experiments using Bell inequalities. Within neutron interferometry we performed an experimental test of the Kochen-Specker theorem with an inequality, which identifies quantum contextuality, by using spin-path entanglement of single neutrons. Here entanglement is achieved not between different particles, but between degrees of freedom of a single neutron, i.e., between spin and path degree of freedom. Appropriate combinations of the spin analysis and the position of the phase shifter allow an experimental verification of the violation of an inequality derived from the Kochen-Specker theorem. The observed violation 2.291±0.008≰1 clearly shows that quantum mechanical predictions cannot be reproduced by noncontextual hidden variable theories.
The physical impact and the testability of the Kochen-Specker (KS) theorem is debated because of the fact that perfect compatibility in a single quantum system cannot be achieved in practical experiments with finite precision. Here, we follow the proposal of A. Cabello and M. T. Cunha [Phys. Rev. Lett. 106, 190401 (2011)], and present a compatibility-loophole-free experimental violation of an inequality of noncontextual theories by two spatially separated entangled qutrits. A maximally entangled qutrit-qutrit state with a fidelity as high as 0.975±0.001 is prepared and distributed to separated spaces, and these two photons are then measured locally, providing the compatibility requirement. The results show that the inequality for noncontextual theory is violated by 31 standard deviations. Our experiments pave the way to close the debate about the testability of the KS theorem. In addition, the method to generate high-fidelity and high-dimension entangled states will provide significant advantages in high-dimension quantum encoding and quantum communication.