Determination of any pure spatial qudits from a minimum number of measurements by phase-stepping interferometry

We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension $d$, which is based on the classic phase-shift interferometry technique. In the proposed scheme a total of only $4d$ measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum-state tomography which require on the order of $d^2$. By using this technique, we have experimentally reconstructed a large number of states ranging from $d=2$ up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse-momentum position of single photons, once they are sent through an aperture with $d$ slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows one to reduce the number of measurement settings to four since the $d$ slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d-3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing one to implement the reconstruction method in a general experiment.

Figure 1

Experimental setup for reconstructing pure spatial qudits. Preparation: An expanded and collimated HeNe laser impinges onto a phase-only LCoS modulator. In conjunction with the $4f$ processor formed by the lenses ${\text{L}}_2$ and ${\text{L}}_3$  and the spatial filter $\text{SF}1$  the quantum state is encoded in the planes ${\pi }^{\prime }$ and ${\pi }^{\prime \prime }$. Tomography: A lens ${\text{L}}_{\mathrm{img}}$ on the image arm of the Mach-Zehnder interferometer images the plane ${\pi }^{\prime }$ onto the output plane ${\pi }^{\prime \prime \prime }$. The lens in the Fourier arm, ${\text{L}}_{\mathrm{ft}}$, performs the Fourier transform of the only slit that is not blocked by the spatial filter $\text{SF}2$.

Figure 2

Interferogram for a state of dimension $d=5$. The vertical lighted bands correspond to the image of the five slits. The horizontal lighted band corresponds to the Fourier transform of the filtered slit in the FA of the interferometer, which acts as the reference. The rectangles indicate the regions on which the measurements are performed, and $0,1,...,4$ are the corresponding slit numbers.

Figure 3

Bloch sphere showing the reconstruction fidelities of 1024 states uniformly distributed on the surface. The mean value of the fidelity is $\overline{F}=0.997$, and the standard deviation is ${\sigma }_F=0.003$.

Figure 4

Histogram of the reconstruction fidelities for 250 random states in $d=14$. The mean value of the fidelity is $\overline{F}=0.98$, and the standard deviation is ${\sigma }_F=0.01$.