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arXiv:1807.02762 · Journal of Physics A Mathematical and Theoretical

Arbitrarily large violations of non-contextuality in single mode photon states with positive Wigner function

S M Roy

Published Apr 24, 2019• 0 citations
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Abstract

Abstract Banaszek, Wódkiewicz and others (Banaszek and Wdkiewicz 1998 Phys. Rev . A 58 4345; Chen et al 2002 Phys. Rev. Lett . 88 040406; Chen and Zhang 2002 Phys. Rev . A 65 044102) made the surprising discovery that Einstein–Bell locality inequalities can be violated by the two mode squeezed vacuum by a factor , in spite of the fact that the state has a positive Wigner function. I use here the more general Gleason–Kochen–Specker assumption of non-contextuality (Gleason 1957 J. Math. Mech . 6 885) to express classicality. I then derive non-contextuality Bell inequalities for correlations of N pseudo spins embedded in an infinite dimensional continuous variable Hilbert space, and show that their maximum possible quantum violation is by a factor 2 ( N −1)/2 . I find quantum states for which this maximum violation is reached. I also show that the familiar displaced squeezed vacuum for a single optical mode, which has a positive Wigner function, can violate the inequality by a factor for odd . The arbitrarily large non-classicality means that realizations of the pseudo-spin measurements even in a single mode photon state might afford similar opportunities in quantum information tasks as entangled N qubit systems with large N .

DOI
10.1088/1751-8121/ab1c2f

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Abstract Banaszek, Wódkiewicz and others (Banaszek and Wdkiewicz 1998 Phys. Rev . A 58 4345; Chen et al 2002 Phys. Rev. Lett . 88 040406; Chen and Zhang 2002 Phys. Rev . A 65 044102) made the surprising discovery that Einstein–Bell locality inequalities can be violated by the two mode squeezed vacuum by a factor , in spite of the fact that the state has a positive Wigner function. I use here the more general Gleason–Kochen–Specker assumption of non-contextuality (Gleason 1957 J. Math. Mech . 6 885) to express classicality. I then derive non-contextuality Bell inequalities for correlations of N pseudo spins embedded in an infinite dimensional continuous variable Hilbert space, and show that their maximum possible quantum violation is by a factor 2 ( N −1)/2 . I find quantum states for which this maximum violation is reached. I also show that the familiar displaced squeezed vacuum for a single optical mode, which has a positive Wigner function, can violate the inequality by a factor for odd . The arbitrarily large non-classicality means that realizations of the pseudo-spin measurements even in a single mode photon state might afford similar opportunities in quantum information tasks as entangled N qubit systems with large N .
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.
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.
Abstract It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen–Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension d ⩾ 217 with the ratio of quantum value to classical bias being O ( d / log d ) . We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen–Specker proofs always consist of substructures of state-dependent contextuality proofs called 01-gadgets. We show a one-to-one connection between 01-gadgets in C d and Hardy paradoxes for the maximally entangled state in C d ⊗ C d . We use this connection to construct large violation 01-gadgets between arbitrary vectors in C d , as well as novel Hardy paradoxes for the maximally entangled state in C d ⊗ C d , and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in R d is not a graph monotone, a result that may be of independent interest.
The rapidly developing interdisciplinary field of quantum information, which merges quantum and information science, studies non-classical aspects of quantum systems. These studies are motivated by the promise that the non-classicality can be used to solve tasks more efficiently than classical methods would allow. In many quantum informational studies, non-classical behaviour is attributed to the notion of entanglement. In this thesis we use photons to experimentally investigate fundamental questions such as: What happens to the entanglement in a system when it is affected by noise? In our study of noisy entanglement we pursue the challenging task of creating bound entanglement. Bound entangled states are created through an irreversible process that requires entanglement. Once in the bound regime, entanglement cannot be distilled out through local operations assisted by classical communication. We show that it is possible to experimentally produce four-photon bound entangled states and that a violation of a Bell inequality can be achieved. Moreover, we demonstrate an entanglement-unlocking protocol by relaxing the condition of local operations. We also explore the non-classical nature of quantum mechanics in several single-photon experiments. In these experiments, we show the violation of various inequalities that were derived under the assumption of non-contextuality. Using qutrits we construct and demonstrate the simplest possible test that offers a discrepancy between classical and quantum theory. Furthermore, we perform an experiment in the spirit of the Kochen-Specker theorem to illustrate the state-independence of this theorem. Here, we investigate whether or not measurement outcomes exhibit fully contextual correlations. That is, no part of the correlations can be attributed to the non-contextual theory. Our results show that only a small part of the experimental generated correlations are amenable to a non-contextual interpretation.
We construct an explicit Wigner function for the NNN-mode squeezed state. Based on a previous observation that the Wigner function describes correlations in the joint measurement of the phase-space displaced parity operator, we investigate the nonlocality of the multipartite entangled state by the violation of the \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski-Brukner NNN-qubit Bell inequality. We find that quantum predictions for such a squeezed state violate these inequalities by an amount that grows with the number NNN.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and they are key resources for quantum advantage in information-processing tasks.Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure.Contextuality, on the other hand, is on a more controversial foundation.In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality.First, I show that Bell inequalities can be derived solely from the assumptions of no signaling and no fine-tuning of the causal model.This removes two extra assumptions from a recent result from Wood and Spekkens and, remarkably, does not require any assumption related to independence of measurement settings-unlike all other derivations of Bell inequalities.I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning.Thus, the quantum violation of classical causality goes beyond the case of spacelike-separated systems and already manifests in scenarios involving single systems.