Generation of Orbital Angular Momentum Bell States and Their Verification via Accessible Nonlinear Witnesses

The controlled generation of entangled states and their subsequent detection are integral aspects of quantum information science. In this Letter, we implement a simple and precise technique that produces any of the four Bell states in the orbital angular momentum degree of freedom. We then use these states to perform the first experimental demonstration of an accessible nonlinear entanglement witness. Such a witness determines entanglement by using the same measurements as required for a linear witness but can detect, in this case, twice as many states as a single linear witness can. We anticipate that our method of state preparation and nonlinear witnesses will have further uses in areas of quantum science, such as superdense coding and quantum key distribution.

Figure 1

Experimental setup. In the quantum state preparation stage, entangled photon pairs are generated by parametric down-conversion and are separated into two paths by a mirror. If a Dove prism is introduced into the upper path, anticorrelated states are produced. If no prism is placed in the upper path, correlated states are produced. The angle of the prism in the lower path controls the relative phase of the two modes.

Figure 2

Real parts of the density matrices of anticorrelated states with phase $\varphi =0$ (a) and $\varphi =\pi$ (b) between the modes and correlated states with phase $\phi =0$ (c) and $\phi =\pi$ (d) between the modes. These are extremely close to the maximally entangled Bell states, with differences occurring due to the relative coefficients of the OAM values $\ell =0$ and $\ell =\pm 2$ that we choose to detect. The anticorrelated states both have fidelity 97% with their respective maximally entangled Bell states $|{\Psi }^{\pm }⟩$, and the correlated states both have fidelity 85% with their respective maximally entangled states $|{\Phi }^{\pm }⟩$. The average of the absolute values of the imaginary parts is $0.01\pm 0.01$.

Figure 3

Experimentally recorded expectation values of the nonlinear entanglement witness and the two linear witnesses for correlated (a) and anticorrelated (b) entangled states for $\ell =2$. A negative expectation value indicates that the state is entangled. Note that the nonlinear witness correctly detects entanglement under almost all cases, whereas each of the linear witnesses often fails to detect entanglement, even though it is present. For the correlated states, we measured $w_{\infty }^{{\Phi }^{+}}$, $w_L^{{\Phi }^{+}}$, and $w_L^{{\Phi }^{-}}$, and for the anticorrelated states, we measured $w_{\infty }^{{\Psi }^{+}}$, $w_L^{{\Psi }^{+}}$, and $w_L^{{\Psi }^{-}}$. The circles give the experimental data points, and the lines are theoretical predictions obtained by using average parameters obtained from the data. The vertical error bars were obtained by applying $\sqrt{N}$ fluctuations to the measured coincidence counts and then averaging over 100 iterations to obtain the standard deviation. The horizontal error bars are estimated to be $\pi /24$.