We identify classical light fields as physical examples of nonquantum entanglement. A natural measure of degree of polarization emerges from this identification, and we discuss its systematic application to any optical field, whether beamlike or not.

The growing recognition that entanglement is not exclusively a quantum property, and does not even originate with Schrödinger's famous remark about it [Proc. Camb. Phil. Soc. 31, 555 (1935)], prompts examination of its role in marking the quantum-classical boundary. We have done this by subjecting correlations of classical optical fields to new Bell-analysis experiments, and report here values of the Bell parameter greater than B = 2.54. This is many standard deviations outside the limit B = 2 established by the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [Phys. Rev. Lett. 23, 880 (1969)], in agreement with our theoretical classical prediction, and not far from the Tsirelson limit B = 2.828.... These results cast a new light on the standard quantum-classical boundary description, and suggest a reinterpretation of it.Introduction: For many decades the term "entanglement" has been attached to the world of quantum mechanics [1]. However, it is true that non-quantum optical entanglement can exist (realized very early by Spreeuw [2]) and its applications have concrete consequences. These are based on entanglements between two, or more than two, degrees of freedom, which are easily avalable classically [2][3][4][5][6]. Multi-entanglements of the same kind are also being explored quantum mechanically [7]. Applications in the classical domain have included, for example, resolution of a long-standing issue concerning Mueller matrices [8], an alternative interpretation of the degree of polarization [9], introduction of the Bell measure as a new index of coherence in optics [10], and innovations in polarization metrology [11]. Here we present theoretical and experimental results extending these results by showing that probabilistic classical optical fields can exhibit violations of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [12] of quantum strength. This is evidence of a new kind that asks for reconsideration of the common understanding that Bell violation signals quantum physics. We emphasize that our discussion focuses on non-quantum entanglement of nondeterministic classical optical fields, and does not engage issues such as non-locality that are important for some applications in quantum information.The observations and applications of non-quantum wave entanglement noted above [2-6, 8-11] exploited nonseparable correlations among two or more modes or degrees of freedom (DOF) of optical wave fields. Nonseparable correlations among modes are an example of entanglement [13], but are not enough for our present purpose. In addition, we want to conform to three criteria that Shimony has identified for Bell tests [1], facts of quantum Nature that must be satisfied when examining possible tests of the quantum-classical border. Fortuitously, the ergodic stochastic optical fields of the classical theory of partial coherence and partial polarization (see Wolf [15]) satisfy these criteria fully (see Suppl. Materials [16]), and we have used such fields as our test bed. Background Theory: We will deal here only wit...

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Quantum optics and classical optics are linked in ways that are becoming apparent as a result of numerous recent detailed examinations of the relationships that elementary notions of optics have with each other. These elementary notions include interference, polarization, coherence, complementarity and entanglement. All of them are present in both quantum and classical optics. They have historic origins, and at least partly for this reason not all of them have quantitative definitions that are universally accepted. This makes further investigation into their engagement in optics very desirable. We pay particular attention to effects that arise from the mere co-existence of separately identifiable and readily available vector spaces. Exploitation of these vector-space relationships are shown to have unfamiliar theoretical implications and new options for observation. It is our goal to bring emerging quantum-classical links into wider view and to indicate directions in which forthcoming and future work will promote discussion and lead to unified understanding.

We study the pairwise concurrences, a measure of entanglement, of the ground states for the frustrated Heisenberg ring to explore the relation between entanglement and quantum phase transition associated with the momentum jump. The groundstate concurrences between any two sites are obtained analytically and numerically. It shows that the summation of all possible pairwise concurrences is an appropriate candidate to depict the phase transition. We also investigate the role that the momentum takes in the jump of concurrence at the critical points. We find that an abrupt momentum change rusults in the maximal concurrence difference of two degenerate ground states.