Detecting entanglement between modes of light

We consider a subgroup of unitary transformations on a mode of light induced by a Mach-Zehnder Interferometer and an algebra of observables describing a photon-number detector preceded by the interferometer. We explore the uncertainty principles between such observables and their usefulness in performing a Bell-like experiment to show the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality under the physical assumption that the detector distinguishes between only zero photons and a nonzero number of photons. We show the local settings of the interferometer which lead to a maximal violation of the CHSH inequality.

Figure 1

Quantum beam splitter. Schematic diagram of a quantum beam splitter with two input ports (${\widehat{a}}_0$, ${\widehat{a}}_1$) and two output ports (${\widehat{a}}_2$, ${\widehat{a}}_3$), with corresponding reflectivities and transmittivities at the input ($r$, $t$) and output ($r^{\prime }$, $t^{\prime }$) ports, respectively.

Figure 2

Mach-Zehnder interferometer. MZI setup with a phase shift $\varphi$ in one of the arms of the interferometer. Two 50:50 beam splitters (${\mathrm{BS}}_1$ and ${\mathrm{BS}}_2$, with ${\mathrm{BS}}_2$ rotated ${180}^{\circ }$ with respect to ${\mathrm{BS}}_1$) with equal magnitudes of reflectivity and transmittivity are used along with two mirrors (${\mathrm{M}}_1$ and ${\mathrm{M}}_2$) for such an interferometer.

Figure 3

POVM element $M_k$: Numerically generated plot of the absolute values of matrix elements of operators $M_k$, restricted to their top left block with a size of $50\times 50$. Each graph shows data for $k\in \{0,10,\cdots ,40\}$. $T\alpha =0.1$ is kept constant, and the graphs show the data for (a) $|R^{\prime }|=0.866$, $\left|T\right|=0.5$; (b) $|R^{\prime }|=0.954$, $\left|T\right|=5\times {10}^{-3}$; (c) $|R^{\prime }|=0.987$, $\left|T\right|=5\times {10}^{-5}$; and (d) $|R^{\prime }|=0.999987$, $\left|T\right|=5\times {10}^{-7}$. The color bar (on the right) indicates the absolute values of the respective matrix entries. The axes of each graph are the column and row indices of the matrix.

Figure 4

Overlap between $M_i$ and $M_j$: Numerically generated plot (in logarithmic scale) for the overlap $\text{Tr}\left(M_i{M}_j\right)$ between POVMs, where $i,j\in \{0,\cdots ,39\}$. $T\alpha$ is kept constant, and the graphs show the data for (a) $|R^{\prime }|=0.866$, $\left|T\right|=0.5$; (b) $|R^{\prime }|=0.954$, $\left|T\right|=5\times {10}^{-3}$; (c) $|R^{\prime }|=0.987$, $\left|T\right|=5\times {10}^{-5}$; and (d) $|R^{\prime }|=0.999987$, $\left|T\right|=5\times {10}^{-7}$. The color bar (on the right) indicates the natural logarithm of the absolute value of the overlap. The axes denote the labels ($i$, $j$) of POVM effects $M_i$ and $M_j$, i.e., the effects corresponding to detection of the $i\mathrm{th}$ and $j\mathrm{th}$ photons at the detector.

Figure 5

Displacement operator: matrix elements. Numerically generated plot for the absolute value of the matrix element ($|C_{n,k}|=|\langle n\left|\widehat{D}\left(\beta \right)\right|k\rangle |$, with $n,k\in \{0,\cdots ,49\}$) of the displacement operator $\widehat{D}\left(\beta \right)$ for $\beta =3.8$. $|n\rangle$ and $|k\rangle$ are elements of the eigenbasis of the photon-number operator. The $x$ and $y$ axes represent $k$ and $n$, respectively, and hence the column and row indices of the matrix element. The color bar (on the right) indicates the absolute values of the plotted matrix entries.

Figure 6

Reparametrization of the summation area in (B5). The new variable $k$ takes non-negative values. For a given $k$, the variable $i$ takes values in the range $\{0,\cdots ,n\}$, except for $k<n$, for which the range of $i$ is $\{n-k,\cdots ,n\}$.