Testing the Clauser-Horne-Shimony-Holt inequality using observables with arbitrary spectrum

The Clauser-Horne-Shimony-Holt inequality applies when measurements with binary outcomes are performed on physical systems under the assumption of local realism. Testing such inequalities in the quantum realm usually involves either measurements of two-valued quantum observables or predefining a context-dependent binning procedure. Here we establish the conditions to test the Clauser-Horne-Shimony-Holt inequality using any quantum observable. Our result applies to observables with an arbitrary spectrum, and no prior knowledge of their underlying Hilbert space's dimension is required. Finally, we demonstrate the proposed general measurement strategy, that can be seen as positive operator-valued measurements performed on the system, using the formalism of modular variables applied to the transverse degrees of freedom of single photons.

Figure 1

Expectation value of the Bell operator (19) according to the state (21) as a function of $a_{\overline{x}}/\ell$, showing the violation of the CHSH inequality, (a) for an infinitely squeezed modular momentum distribution (${\sigma }_{\overline{p}}\to 0$) located at $a_{\overline{p}}f=0$, and (b) for the values ${\sigma }_{\overline{p}}=0.1/\ell$, $a_{\overline{x}}=0.1/\ell$. Each line corresponds to a different width of the modular momentum distribution with increasing order from the top curve to the bottom curve: ${\sigma }_{\overline{x}}=0.001/\ell$ (blue, top), ${\sigma }_{\overline{x}}=0.01/\ell$ (red), ${\sigma }_{\overline{x}}=0.02/\ell$ (green), ${\sigma }_{\overline{x}}=0.03/\ell$ (purple), ${\sigma }_{\overline{x}}=0.04/\ell$ (orange), ${\sigma }_{\overline{x}}=0.05/\ell$ (pink), ${\sigma }_{\overline{x}}=0.06/\ell$ (cyan), ${\sigma }_{\overline{x}}=0.07/\ell$ (brown), ${\sigma }_{\overline{x}}=0.08/\ell$ (magenta, bottom). The black curve in (a) shows the function $2\sqrt{2}{cos}^2(2\pi a_{\overline{x}}/\ell )$ and the black dashed line indicates the local-realism threshold.

Figure 2

(a) Circuit diagram of an interferometer leading to the measurement of the expectation value of the operators ${\widehat{A}}_{\varphi }$, according to the state $|\Psi \rangle$ by coupling the continuous state $|\Psi \rangle$ to an ancilla qubit. $\widehat{H}$ represents the Hadamard gate, and the operators $\widehat{Z}\left(\Delta p\right)=e^{i\widehat{x}\Delta p/\hslash }$ and $\widehat{U}\left(\varphi \right)=e^{i{\widehat{p}}^2\varphi {\ell }^2/h^2}$ are implemented conditionally depending on the state of the ancilla. The probabilities $p_{\pm }^{\varphi }$ are obtained by measuring the ancilla in the basis $|\pm \rangle =\left(\right|0\rangle \pm |1\rangle )/\sqrt{2}$. (b) Optical implementation of the above circuit using a balanced Mach-Zehnder-type interferometer and linear optical elements acting on the transverse field of the photons. U1 and U2 indicate the free propagation of the photons and W a linear phase shift with $\Delta p=h/\ell$. The latter can be realized equivalently using a spatial light modulator (SLM). Here, $p_{\pm }^{\varphi }$ denote the photon counting probabilities for the detection of a photon in output $+$ or $-$.

Figure 3

Example plots of (a) the grating transmission function $T\left(x\right)$ with slit distance $L$ and $\kappa$ such that ${\sigma }_{\overline{x}}=0.01/L$, and (b) the single photon transverse wave function directly after it has passed through the diffraction grating defined by $T\left(x\right)$ and the Gaussian envelope with width $\sigma =5/L$. A plot of the $\overline{x}$-dependent part of the modular wave function ${\tilde{\Psi }}_P(\overline{x},\overline{p})$ [see Eq. (29)], namely $\tilde{T}\left(\overline{x}\right)$, is given by that of $T\left(x\right)$ restricted to the domain $[0,L[$.

Figure 4

Proposed experimental setup to create a spatially entangled two-photon state $|\Psi \rangle$ of the form (20). L, laser; NC, nonlinear crystals (type I); HWP, half-wave plate; WP1/WP2, combination of half- and quarter-wave plates; PBS, polarizing beam splitter; and G, diffraction grating.