Violation of Bell--CHSH--CH Inequalities by Superposition of Two Coherent States (π/2 Out of Phase)

Abstract

Einstein et al. [1] realized that in many states, when measuring either linear momentum or position of one of the two particles, one can infer precisely either momentum or position of the other. They advocated that if the local realism is taken for granted, then quantum theory is an incomplete description of the physical world. Following this line, Bell demonstrated that a contradiction arises between the EPR assumptions of realism and locality and quantum physics and termed as Bell's theorem. Quantum nonlocality con rms the interpretation and validity of quantum mechanics against the local-realistic theories by violations of the constraints on the correlation between local measurement outcomes. Mathematical expression of such a constraint is known as Bell inequality [2], of which many variants exist [3 5]. For example, well known Clauser, Horne, Shimony, and Holt (CHSH) inequality [3] and Clauser and Horne (CH) [4] are used for the veri cation of nonlocal correlations in a two-dimensional Hilbert space. The Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. According to the quantum mechanics they are entangled, while local realism would limit the correlation of subsequent measurements of the particles. Nonlocal correlations play crucial role for the device-independent versions of quantum information protocols, such as cryptography, random number generation, state estimation, and entangled measurement certi cation. Quantum continuous variables (CV) [6] of light have been successfully used to realize some of the standard informational tasks traditionally based on qubits. Bell-inequality tests were performed by using qubits [7] which satisfy the space-like separation between two local parties. But, the di culty of the detection loophole [8, 9] demanded to look at other approaches for Bell-inequality tests. Continuous variable (CV) states are of recent in-