Shannon Dimensionality of Quantum Channels and Its Application to Photon Entanglement

We introduce the concept of Shannon dimensionality $D$ as a new way to quantify bipartite entanglement as measured in an experiment. This is applied to orbital-angular-momentum entanglement of two photons, using two state analyzers composed of a rotatable angular-sector phase plate that is lens coupled to a single-mode fiber. We can deduce the value of $D$ directly from the observed two-photon coincidence fringe. In our experiment, $D$ varies between 2 and 6, depending on the experimental conditions. We predict how the Shannon dimensionality evolves when the number of angular sectors imprinted in the phase plate is increased and anticipate that $D\simeq 50$ is experimentally within reach.

Figure 1

Experimental setup. Orbital-angular-momentum entangled photons are emitted at 826 nm by a BBO crystal, cut for type-I collinear phase matching. A thin GaP wafer serves to eliminate the pump beam. The two-photon field can be clipped with an aperture. The twin photons are spatially separated by a beam splitter and imaged on the angular phase plates ($f_2=4f_1=40\mathrm{cm}$). Just behind the phase plates, the frequency-degenerate photons are selected by interference filters (not shown), centered around 826 nm with a 10 nm width. The phase plates (shown are quarter-sector plates) are oriented at angles $\alpha$ and $\beta$, and photon counts are rendered by a coincidence circuit.

Figure 2

Coincidence count rate vs the relative orientation of the two state analyzers. Points denote experimental data; curves are theoretical predictions. (a) Half-sector plate. The parabolic fringe (circles) is a signature of a dimensionality larger than 2: We find $D=3.0$. Truncating the number of modes, by closing the aperture, gradually reduces the parabola into a sine of dimensionality 2.0 (triangles). (b) Quarter-sector plate. The piecewise parabolic fringe yields an experimental dimensionality of 5.8 (circles), where theory predicts $D=6$.

Figure 3

Maximum dimensionality that can be accessed with sector phase plates having $2N$ angular sectors alternatingly phase shifted by $\pi$. The insets show the optimized plates for $N=1$, $N=2$, and $N=3$.